text: //------------------------------------------------------------------------------
  // GraphBLAS.h: definitions for the GraphBLAS package
  //------------------------------------------------------------------------------
  
  // SuiteSparse:GraphBLAS, Timothy A. Davis, (c) 2017-2018, All Rights Reserved.
  // http://suitesparse.com   See GraphBLAS/Doc/License.txt for license.
  
  //------------------------------------------------------------------------------
  
  // SuiteSparse:GraphBLAS is an full implementation of the GraphBLAS standard,
  // which defines a set of sparse matrix operations on an extended algebra of
  // semirings, using an almost unlimited variety of operators and types.  When
  // applied to sparse adjacency matrices, these algebraic operations are
  // equivalent to computations on graphs.  GraphBLAS provides a powerful and
  // expressive framework creating graph algorithms based on the elegant
  // mathematics of sparse matrix operations on a semiring.
  
  // This GraphBLAS.h file contains GraphBLAS definitions for user applications
  // to #include.  Functions and variables with the prefix GB_ need to be defined
  // in this file and are thus technically visible to the user, but they must not
  // be accessed in user code.  They are here only so that the ANSI C11 _Generic
  // feature can be used in the user-accessible polymorphic functions.  For
  // example GrB_free is a macro that uses _Generic to select the right method,
  // depending on the type of its argument.
  
  // This implementation fully conforms to the GraphBLAS API Specification, but
  // also includes functions and features that are extensions to the spec.  These
  // are cataloged here and tagged with "SPEC."
  
  // All functions and definitions that are extensions to the spec are given
  // names of the form GxB_* for functions and built-in objects, or GXB_ for
  // macros, so it is clear which are in the spec and which are extensions.
  // Extensions with the name GxB_* or GXB_* are user-accessible in
  // SuiteSparse:GraphBLAS but cannot be guaranteed to appear in all GraphBLAS
  // implementations.  In the future, if any GxB_* functions are included as-is
  // in the GraphBLAS API spec with GrB_* names, the prior GxB_* variants that
  // appear here will be kept for backward compatibility.  If they must change
  // for inclusion in the spec, a reasonable attempt will be made to keep the
  // prior GxB_* variant alongside the GrB_* version, also for backward
  // compatibility.
  
  #ifndef GRAPHBLAS_H
  #define GRAPHBLAS_H
  
  //------------------------------------------------------------------------------
  // GraphBLAS version
  //------------------------------------------------------------------------------
  
  // SPEC: the following macros are extensions to the spec
  
  // There are two version numbers that user codes can check against with
  // compile-time #if tests:  the version of this GraphBLAS implementation,
  // and the version of the GraphBLAS specification it conforms to.  User code
  // can use tests like this:
  //
  //      #if GXB >= GXB_VERSION (2,0,3)
  //      ... use features in GraphBLAS specification 2.0.3 ...
  //      #else
  //      ... only use features in early specifications
  //      #endif
  //
  //      #if GXB_IMPLEMENTATION > GXB_VERSION (1,4,0)
  //      ... use features from version 1.4.0 of a GraphBLAS package
  //      #endif
  
  // X_GRAPHBLAS: names this particular implementation:
  #define GXB_SUITESPARSE_GRAPHBLAS
  
  // GXB_VERSION: a single integer for comparing spec and version levels
  #define GXB_VERSION(major,minor,sub) \
      (((major)*1000ULL + (minor))*1000ULL + (sub))
  
  // The version of this implementation:
  #define GXB_DATE "July 4, 2018"
  #define GXB_IMPLEMENTATION_MAJOR 2
  #define GXB_IMPLEMENTATION_MINOR 0
  #define GXB_IMPLEMENTATION_SUB   3
  
  #define GXB_IMPLEMENTATION \
          GXB_VERSION (GXB_IMPLEMENTATION_MAJOR, \
                       GXB_IMPLEMENTATION_MINOR, \
                       GXB_IMPLEMENTATION_SUB)
  
  // The 'about' string the describes this particular implementation of GraphBLAS:
  #define GXB_ABOUT \
  "SuiteSparse:GraphBLAS, Timothy A. Davis, (c) 2017-2018, "                   \
  "All Rights Reserved.\n"                                                     \
  "http://suitesparse.com  Dept of Computer Sci. & Eng, Texas A&M University\n"
  
  // The GraphBLAS license for this particular implementation of GraphBLAS:
  #define GXB_LICENSE \
  "SuiteSparse:GraphBLAS, Copyright 2017-2018, Timothy A. Davis\n"             \
  "\n"                                                                         \
  "Licensed under the Apache License, Version 2.0 (the \"License\");\n"        \
  "you may not use SuiteSparse:GraphBLAS except in compliance with the\n"      \
  "License.  You may obtain a copy of the License at\n"                        \
  "\n"                                                                         \
  "    http://www.apache.org/licenses/LICENSE-2.0  \n"                         \
  "\n"                                                                         \
  "Unless required by applicable law or agreed to in writing, software\n"      \
  "distributed under the License is distributed on an \"AS IS\" BASIS,\n"      \
  "WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n" \
  "See the License for the specific language governing permissions and\n"      \
  "limitations under the License.\n"
  
  //------------------------------------------------------------------------------
  // GraphBLAS C API version
  //------------------------------------------------------------------------------
  
  // This implementation conforms to the following GraphBLAS API version:
  #define GXB_SPEC_DATE "Nov 14, 2017"
  #define GXB_MAJOR 1
  #define GXB_MINOR 1
  #define GXB_SUB   0
  
  #define GXB GXB_VERSION(GXB_MAJOR, GXB_MINOR, GXB_SUB)
  
  // The 'spec' string describes the GraphBLAS spec:
  #define GXB_SPEC \
  "GraphBLAS C API, provisional release, by Aydin Buluc, Timothy\n"   \
  "Mattson, Scott McMillan, Jose' Moreira, Carl Yang.  Based on\n"    \
  "\"GraphBLAS Mathematics\" by Jeremy Kepner.\n"
  
  //------------------------------------------------------------------------------
  // include files required by GraphBLAS
  //------------------------------------------------------------------------------
  
  #include <stdio.h>
  #include <string.h>
  #include <stdlib.h>
  #include <stdbool.h>
  #include <stdint.h>
  #include <inttypes.h>
  #include <stddef.h>
  #include <limits.h>
  #include <math.h>
  
  #ifdef _OPENMP
  #include <omp.h>
  #endif
  
  //------------------------------------------------------------------------------
  // GraphBLAS error and informational codes
  //------------------------------------------------------------------------------
  
  // All GraphBLAS functions return a code that indicates if it was successful
  // or not.  If more information is required, the GrB_error function can be
  // called, which returns a string that provides more information on the last
  // return value from GraphBLAS.
  
  typedef enum
  {
  
      GrB_SUCCESS,                // all is well
  
      //--------------------------------------------------------------------------
      // informational codes, not an error:
      //--------------------------------------------------------------------------
  
      // The GraphBLAS spec lists this as an 'error' code, but it just means that
      // A(i,j) is not present in the matrix, having been requested by
      // GrB_*_extractElement.  The function cannot return the proper value
      // because the value of 'implicit zeros' depends on the semiring.  For the
      // conventational plus-times semiring, the implied 'zero' actually has the
      // value of zero.  For the max-plus semiring, it has the value -infinity.
      // A matrix does not keep track of its semiring, and the user can change
      // the semiring used to operate on the matrix.  How mathematically
      // well-defined that change of semiring is depends the user; GraphBLAS will
      // not change the explicit values in the matrix if the semiring changes.
      // As a result, GraphBLAS needs to return not a value, but an indication
      // that the value of A(i,j) is implicit.  The user application can use this
      // indicator (GrB_NO_VALUE) to use the semiring's addititive identity, or
      // it can take other action, as it chooses.  In either case, it is safe to
      // ask for values that are not there, which is why this return condition is
      // not really an 'error' code but an informational code.
  
      GrB_NO_VALUE,               // A(i,j) requested but not there
  
      //--------------------------------------------------------------------------
      // API errors:
      //--------------------------------------------------------------------------
  
      // In non-blocking mode, these errors are caught right away.
  
      GrB_UNINITIALIZED_OBJECT,   // object has not been initialized
      GrB_INVALID_OBJECT,         // object is corrupted
      GrB_NULL_POINTER,           // input pointer is NULL
      GrB_INVALID_VALUE,          // generic error code; some value is bad
      GrB_INVALID_INDEX,          // a row or column index is out of bounds;
                                  // used for indices passed as scalars, not
                                  // in a list.
      GrB_DOMAIN_MISMATCH,        // object domains are not compatible
      GrB_DIMENSION_MISMATCH,     // matrix dimensions do not match
      GrB_OUTPUT_NOT_EMPTY,       // output matrix already has values in it
  
      //--------------------------------------------------------------------------
      // execution errors:
      //--------------------------------------------------------------------------
  
      // In non-blocking mode, these errors can be deferred.
  
      GrB_OUT_OF_MEMORY,          // out of memory
      GrB_INSUFFICIENT_SPACE,     // output array not large enough
      GrB_INDEX_OUT_OF_BOUNDS,    // a row or column index is out of bounds;
                                  // used for indices in a list of indices.
      GrB_PANIC                   // SuiteSparse:GraphBLAS never panics
  
  }
  GrB_Info ;
  
  //==============================================================================
  //=== GraphBLAS context methods ================================================
  //==============================================================================
  
  // GrB_init must called before any other GraphBLAS operation.  GrB_finalize
  // must be called as the last GraphBLAS operation.
  
  // GrB_init defines the mode that GraphBLAS will use:  blocking or
  // non-blocking.  With blocking mode, all operations finish before returning to
  // the user application.  With non-blocking mode, operations can be left
  // pending, and are computed only when needed.
  
  // The GrB_wait ( ) function forces all pending operations to complete.
  // Blocking mode is as if GrB_wait is called whenever a GraphBLAS method or
  // operation returns to the user.
  
  // The non-blocking mode is unpredictable if user-defined functions have side
  // effects or if they rely on global variables, which are not under the control
  // of GraphBLAS.  Suppose the user application creates a user-defined operator
  // that accesses a global variable.  That operator is then used in a GraphBLAS
  // operation, which is left pending.  If the user application then changes the
  // global variable before pending operations complete, the pending operations
  // will be eventually computed with this different value.
  
  // The non-blocking mode can have side effects if user-defined functions have
  // side effects or if they rely on global variables, which are not under the
  // control of GraphBLAS.  Suppose the user creates a user-defined operator that
  // accesses a global variable.  That operator is then used in a GraphBLAS
  // operation, which is left pending.  If the user then changes the global
  // variable before pending operations complete, the pending operations will be
  // eventually computed with this different value.
  
  // Worse yet, a user-defined operator might be freed before it is needed to
  // finish a pending operation.  This causes undefined behavior.  To avoid this,
  // call GrB_wait before modifying any global variables relied upon by
  // user-defined operators, or before freeing any user-defined types, operators,
  // monoids, or semirings.
  
  typedef enum
  {
      GrB_NONBLOCKING,    // methods may return with pending computations
      GrB_BLOCKING        // no computations are ever left pending
  }
  GrB_Mode ;
  
  GrB_Info GrB_init           // start up GraphBLAS
  (
      const GrB_Mode mode     // blocking or non-blocking mode
  ) ;
  
  // In non-blocking mode, GraphBLAS operations need not complete until their
  // results are required.  GrB_wait ensures all pending operations are finished.
  
  GrB_Info GrB_wait ( ) ;     // finish all pending computations
  
  // GrB_finalize does not call GrB_wait; any pending computations are abandoned.
  
  GrB_Info GrB_finalize ( ) ;     // finish GraphBLAS
  
  //==============================================================================
  //=== GraphBLAS sequence termination ===========================================
  //==============================================================================
  
  // Each GraphBLAS method and operation returns a GrB_Info error code.
  // GrB_error returns additional information on the error in a thread-safe
  // null-terminated string.  The string returned by GrB_error is statically
  // allocated in thread local storage and must not be free'd.
  
  const char *GrB_error ( ) ;     // return a string describing the last error
  
  //==============================================================================
  //=== GraphBLAS types, operators, monoids, and semirings =======================
  //==============================================================================
  
  //------------------------------------------------------------------------------
  // GraphBLAS types
  //------------------------------------------------------------------------------
  
  // A GraphBLAS GrB_Type defines the type of scalar values that a matrix
  // contains, and the type of scalar operands for a unary or binary operator.
  // There are eleven built-in types, and a user application can define any types
  // of its own as well.  The built-in types correspond to built-in types in C
  // (#include <stdbool.h> #include <stdint.h>), and the classes in MATLAB, as
  // listed in the "extern GrB_Type ..." table below The user application can
  // also define new types based on any typedef in the C language whose values
  // are held in a contiguous region of memory.
  
  typedef struct GB_Type_opaque *GrB_Type ;
  
  // GraphBLAS predefined types and the counterparts in pure C and in MATLAB
  extern GrB_Type
      GrB_BOOL   ,        // in C: bool       in MATLAB: logical
      GrB_INT8   ,        // in C: int8_t     in MATLAB: int8
      GrB_UINT8  ,        // in C: uint8_t    in MATLAB: uint8
      GrB_INT16  ,        // in C: int16_t    in MATLAB: int16
      GrB_UINT16 ,        // in C: uint16_t   in MATLAB: uint16
      GrB_INT32  ,        // in C: int32_t    in MATLAB: int32
      GrB_UINT32 ,        // in C: uint32_t   in MATLAB: uint32
      GrB_INT64  ,        // in C: int64_t    in MATLAB: int64
      GrB_UINT64 ,        // in C: uint64_t   in MATLAB: uint64
      GrB_FP32   ,        // in C: float      in MATLAB: single
      GrB_FP64   ;        // in C: double     in MATLAB: double
  
  // GrB_Type_new is implemented both as a macro and a function.  Both are
  // user-callable.  The default is to use the macro, since this allows the name
  // of the type to be saved as a string, for subsequent error reporting by
  // GrB_error.  It is also provided as a function so that applications that
  // require a function instead of macro can access it.  User code can simply do
  // #undef GrB_Type_new before using the function.  This approach also places
  // the function GrB_Type_new in the linkable SuiteSparse:GraphBLAS library so
  // that it is visible for linking with applications in languages other than
  // ANSI C99.  The function version does not allow the name of the ctype to be
  // saved in the new GraphBLAS type, however.  It is given a generic name.
  
  // If SuiteSparse:GraphBLAS is compiled with -DNMACRO then the macro versions
  // of GrB_Type_new, GrB_UnaryOp_new, GrB_BinaryOp_new, and GxB_SelectOp_new
  // are not made available.  The function versions are always used instead.
  // #define NMACRO
  
  #undef GrB_Type_new
  
  GrB_Info GrB_Type_new           // create a new GraphBLAS type
  (
      GrB_Type *type,             // handle of user type to create
      size_t sizeof_ctype         // size = sizeof (ctype) of the C type
  ) ;
  
  // user code should not directly use GB_STR or GB_XSTR
  // GB_STR: convert the content of x into a string "x"
  #define GB_XSTR(x) GB_STR(x)
  #define GB_STR(x) #x
  
  // GrB_Type_new as a user-callable macro, which allows the name of the ctype
  // to be added to the new type. 
  #ifndef NMACRO
  #define GrB_Type_new(utype, sizeof_ctype) \
      GB_Type_new (utype, sizeof_ctype, GB_STR(sizeof_ctype))
  #endif
  
  GrB_Info GB_Type_new            // not user-callable; use GrB_Type_new instead
  (
      GrB_Type *type,             // handle of user type to create
      const size_t sizeof_ctype,  // size of the user type
      const char *name            // name of the type, as "sizeof (ctype)"
  ) ;
  
  // SPEC: GxB_Type_size is an extension to the spec
  
  GrB_Info GxB_Type_size          // determine the size of the type
  (
      size_t *size,               // the sizeof the type
      GrB_Type type               // type to determine the sizeof
  ) ;
  
  GrB_Info GrB_Type_free          // free a user-defined type
  (
      GrB_Type *type              // handle of user-defined type to free
  ) ;
  
  
  //------------------------------------------------------------------------------
  // GraphBLAS unary and binary operators
  //------------------------------------------------------------------------------
  
  // GraphBLAS defines built-in unary and binary operators, and the user may also
  // define new ones via function pointers.  When a user function z=f(x,y) or
  // z=f(x) is called by GraphBLAS, the pointers x, y, and z are guaranteed to be
  // non-NULL and to point to unique valid space of the expected type.
  
  // GraphBLAS provides 256 built-in binary operators z=f(x,y) and 45 built-in
  // unary operators z=f(x) that operate on the 11 built-in types.  Built-in
  // types are statically allocated and need not be freed when the application
  // finishes.
  
  //------------------------------------------------------------------------------
  // unary operators
  //------------------------------------------------------------------------------
  
  // GrB_UnaryOp: a function z=f(x).  The function f must have the signature:
  
  //      void f (void *z, const void *x) ;
  
  // The pointers are void * but they are always of pointers to objects of type
  // ztype and xtype, respectively.  The function must typecast its arguments as
  // needed from void* to ztype* and xtype*.
  
  typedef struct GB_UnaryOp_opaque *GrB_UnaryOp ;
  
  //------------------------------------------------------------------------------
  // built-in unary operators, z = f(x)
  //------------------------------------------------------------------------------
  
  // There are 67 unary operators: 6 kinds * 11 types and GrB_LNOT.
  
  // SPEC: ONE and ABS unary operators are extensions to the spec
  // as are the LNOT_TYPE operators.
  
  extern GrB_UnaryOp
      // For these three functions z=f(x), z and x have the same type.
      // The suffix in the name is the type of x and z.
      // z = x             z = -x             z = 1/x             z = ! (x != 0)
      // identity          additive           multiplicative      logical
      //                   inverse            inverse             negation
      GrB_IDENTITY_BOOL,   GrB_AINV_BOOL,     GrB_MINV_BOOL,      GxB_LNOT_BOOL,
      GrB_IDENTITY_INT8,   GrB_AINV_INT8,     GrB_MINV_INT8,      GxB_LNOT_INT8,
      GrB_IDENTITY_UINT8,  GrB_AINV_UINT8,    GrB_MINV_UINT8,     GxB_LNOT_UINT8,
      GrB_IDENTITY_INT16,  GrB_AINV_INT16,    GrB_MINV_INT16,     GxB_LNOT_INT16,
      GrB_IDENTITY_UINT16, GrB_AINV_UINT16,   GrB_MINV_UINT16,    GxB_LNOT_UINT16,
      GrB_IDENTITY_INT32,  GrB_AINV_INT32,    GrB_MINV_INT32,     GxB_LNOT_INT32,
      GrB_IDENTITY_UINT32, GrB_AINV_UINT32,   GrB_MINV_UINT32,    GxB_LNOT_UINT32,
      GrB_IDENTITY_INT64,  GrB_AINV_INT64,    GrB_MINV_INT64,     GxB_LNOT_INT64,
      GrB_IDENTITY_UINT64, GrB_AINV_UINT64,   GrB_MINV_UINT64,    GxB_LNOT_UINT64,
      GrB_IDENTITY_FP32,   GrB_AINV_FP32,     GrB_MINV_FP32,      GxB_LNOT_FP32,
      GrB_IDENTITY_FP64,   GrB_AINV_FP64,     GrB_MINV_FP64,      GxB_LNOT_FP64,
  
      // z = 1             z = abs(x)
      // one               absolute value
      //
      GxB_ONE_BOOL,        GxB_ABS_BOOL,
      GxB_ONE_INT8,        GxB_ABS_INT8,
      GxB_ONE_UINT8,       GxB_ABS_UINT8,
      GxB_ONE_INT16,       GxB_ABS_INT16,
      GxB_ONE_UINT16,      GxB_ABS_UINT16,
      GxB_ONE_INT32,       GxB_ABS_INT32,
      GxB_ONE_UINT32,      GxB_ABS_UINT32,
      GxB_ONE_INT64,       GxB_ABS_INT64,
      GxB_ONE_UINT64,      GxB_ABS_UINT64,
      GxB_ONE_FP32,        GxB_ABS_FP32,
      GxB_ONE_FP64,        GxB_ABS_FP64,
  
      // Boolean negation, z = !x, where both x and x are boolean.  There is no
      // suffix since z and x are only boolean.  This operator is identical to
      // GxB_LNOT_BOOL; it just has a different name.
      GrB_LNOT ;
  
  //------------------------------------------------------------------------------
  // methods for unary operators
  //------------------------------------------------------------------------------
  
  // GrB_UnaryOp_new is implemented both as a macro and a function.  Both are
  // user-callable.  The default is to use the macro, since this allows the name
  // of the unary function to be kept in the new operator as a string.  See the
  // discussion of GrB_Type_new above.
  
  #undef GrB_UnaryOp_new
  
  GrB_Info GrB_UnaryOp_new            // create a new user-defined unary operator
  (
      GrB_UnaryOp *unaryop,           // handle for the new unary operator
      void *function,                 // pointer to the unary function
      const GrB_Type ztype,           // type of output z
      const GrB_Type xtype            // type of input x
  ) ;
  
  #ifndef NMACRO
  #define GrB_UnaryOp_new(op,f,z,x) GB_UnaryOp_new (op,f,z,x, GB_STR(f))
  #endif
  
  GrB_Info GB_UnaryOp_new             // not user-callable; use GrB_UnaryOp_new
  (
      GrB_UnaryOp *unaryop,           // handle for the new unary operator
      void *function,                 // pointer to the unary function
      const GrB_Type ztype,           // type of output z
      const GrB_Type xtype,           // type of input x
      const char *name                // name of the underlying function
  ) ;
  
  // SPEC: GxB_UnaryOp_ztype is an extension to the spec
  
  GrB_Info GxB_UnaryOp_ztype          // return the type of z
  (
      GrB_Type *ztype,                // return type of output z
      const GrB_UnaryOp unaryop       // unary operator
  ) ;
  
  // SPEC: GxB_UnaryOp_xtype is an extension to the spec
  
  GrB_Info GxB_UnaryOp_xtype          // return the type of x
  (
      GrB_Type *xtype,                // return type of input x
      const GrB_UnaryOp unaryop       // unary operator
  ) ;
  
  GrB_Info GrB_UnaryOp_free           // free a user-created unary operator
  (
      GrB_UnaryOp *unaryop            // handle of unary operator to free
  ) ;
  
  //------------------------------------------------------------------------------
  // binary operators
  //------------------------------------------------------------------------------
  
  // GrB_BinaryOp: a function z=f(x,y).  The function f must have the signature:
  
  //      void f (void *z, const void *x, const void *y) ;
  
  // The pointers are void * but they are always of pointers to objects of type
  // ztype, xtype, and ytype, respectively.  See Demo/usercomplex.c for examples.
  
  typedef struct GB_BinaryOp_opaque *GrB_BinaryOp ;
  
  //------------------------------------------------------------------------------
  // built-in binary operators, z = f(x,y)
  //------------------------------------------------------------------------------
  
  // There are three sets of built-in binary operators.  For the first set of
  // 17 kinds of operators, x,y,z all have the same type, and they are available
  // for all 11 types, for a total of 17*11 = 187 operators.  All of them have
  // a "_TYPE" suffix that denotes the type of x,y,z:
  
  //      8 general: FIRST, SECOND, MIN, MAX, PLUS, MINUS, TIMES, DIV
  //      6 comparison: ISEQ, ISNE, ISGT, ISLT, ISGE, ISLE
  //      3 logical: LOR, LAND, LXOR
  
  // For the second set, there are 6 comparison operators where, x,y have the
  // same type but z is always boolean, for a total of 6*11 = 66 operators.
  // All of them have a "_TYPE" suffix that denotes the type of x,y (not z):
  
  //      6 comparison: EQ, NE, GT, LT, GE, LE
  
  // The final set of operators is for boolean x,y,z only, and they have no
  // suffix:
  
  //      3 logical: LOR, LAND, LXOR
  
  // Thus there are 187+66+3 = 256 built-in binary operators.  Some are redundant
  // but are included to keep the name space of operators uniform.
  
  // For eight binary operators z=f(x,y), x, y, and z are all the same type:
  // FIRST, SECOND, MIN, MAX, PLUS, MINUS, TIMES, and DIV, for all 11 types.
  
  extern GrB_BinaryOp
      // z = x            z = y               z = min(x,y)        z = max (x,y)
      GrB_FIRST_BOOL,     GrB_SECOND_BOOL,    GrB_MIN_BOOL,       GrB_MAX_BOOL,
      GrB_FIRST_INT8,     GrB_SECOND_INT8,    GrB_MIN_INT8,       GrB_MAX_INT8,
      GrB_FIRST_UINT8,    GrB_SECOND_UINT8,   GrB_MIN_UINT8,      GrB_MAX_UINT8,
      GrB_FIRST_INT16,    GrB_SECOND_INT16,   GrB_MIN_INT16,      GrB_MAX_INT16,
      GrB_FIRST_UINT16,   GrB_SECOND_UINT16,  GrB_MIN_UINT16,     GrB_MAX_UINT16,
      GrB_FIRST_INT32,    GrB_SECOND_INT32,   GrB_MIN_INT32,      GrB_MAX_INT32,
      GrB_FIRST_UINT32,   GrB_SECOND_UINT32,  GrB_MIN_UINT32,     GrB_MAX_UINT32,
      GrB_FIRST_INT64,    GrB_SECOND_INT64,   GrB_MIN_INT64,      GrB_MAX_INT64,
      GrB_FIRST_UINT64,   GrB_SECOND_UINT64,  GrB_MIN_UINT64,     GrB_MAX_UINT64,
      GrB_FIRST_FP32,     GrB_SECOND_FP32,    GrB_MIN_FP32,       GrB_MAX_FP32,
      GrB_FIRST_FP64,     GrB_SECOND_FP64,    GrB_MIN_FP64,       GrB_MAX_FP64,
  
      // z = x+y          z = x-y             z = x*y             z = x/y
      GrB_PLUS_BOOL,      GrB_MINUS_BOOL,     GrB_TIMES_BOOL,     GrB_DIV_BOOL,
      GrB_PLUS_INT8,      GrB_MINUS_INT8,     GrB_TIMES_INT8,     GrB_DIV_INT8,
      GrB_PLUS_UINT8,     GrB_MINUS_UINT8,    GrB_TIMES_UINT8,    GrB_DIV_UINT8,
      GrB_PLUS_INT16,     GrB_MINUS_INT16,    GrB_TIMES_INT16,    GrB_DIV_INT16,
      GrB_PLUS_UINT16,    GrB_MINUS_UINT16,   GrB_TIMES_UINT16,   GrB_DIV_UINT16,
      GrB_PLUS_INT32,     GrB_MINUS_INT32,    GrB_TIMES_INT32,    GrB_DIV_INT32,
      GrB_PLUS_UINT32,    GrB_MINUS_UINT32,   GrB_TIMES_UINT32,   GrB_DIV_UINT32,
      GrB_PLUS_INT64,     GrB_MINUS_INT64,    GrB_TIMES_INT64,    GrB_DIV_INT64,
      GrB_PLUS_UINT64,    GrB_MINUS_UINT64,   GrB_TIMES_UINT64,   GrB_DIV_UINT64,
      GrB_PLUS_FP32,      GrB_MINUS_FP32,     GrB_TIMES_FP32,     GrB_DIV_FP32,
      GrB_PLUS_FP64,      GrB_MINUS_FP64,     GrB_TIMES_FP64,     GrB_DIV_FP64 ;
  
  // Six comparison operators z=f(x,y) return the same type as their inputs.
  // Each of them compute z = (x OP y), where x, y, and z all have the same type.
  // The value z is either 1 for true or 0 for false, but it is a value with the
  // same type as x and y.  Z is not bool (unless x and y are also bool).  These
  // operators compute the same thing as the 6 sets of EQ, NE, GT, LT, GE, and LE
  // operators.  They just return their result z as the same type as x and y,
  // instead of returning a value z that is boolean.  Since their ztype is
  // non-boolean, they can be used as multiply operators in a semring with
  // non-boolean monoids (PLUS, for example).
  
  extern GrB_BinaryOp
      // z = (x == y)     z = (x != y)        z = (x > y)         z = (x < y)
      GxB_ISEQ_BOOL,      GxB_ISNE_BOOL,      GxB_ISGT_BOOL,      GxB_ISLT_BOOL,
      GxB_ISEQ_INT8,      GxB_ISNE_INT8,      GxB_ISGT_INT8,      GxB_ISLT_INT8,
      GxB_ISEQ_UINT8,     GxB_ISNE_UINT8,     GxB_ISGT_UINT8,     GxB_ISLT_UINT8,
      GxB_ISEQ_INT16,     GxB_ISNE_INT16,     GxB_ISGT_INT16,     GxB_ISLT_INT16,
      GxB_ISEQ_UINT16,    GxB_ISNE_UINT16,    GxB_ISGT_UINT16,    GxB_ISLT_UINT16,
      GxB_ISEQ_INT32,     GxB_ISNE_INT32,     GxB_ISGT_INT32,     GxB_ISLT_INT32,
      GxB_ISEQ_UINT32,    GxB_ISNE_UINT32,    GxB_ISGT_UINT32,    GxB_ISLT_UINT32,
      GxB_ISEQ_INT64,     GxB_ISNE_INT64,     GxB_ISGT_INT64,     GxB_ISLT_INT64,
      GxB_ISEQ_UINT64,    GxB_ISNE_UINT64,    GxB_ISGT_UINT64,    GxB_ISLT_UINT64,
      GxB_ISEQ_FP32,      GxB_ISNE_FP32,      GxB_ISGT_FP32,      GxB_ISLT_FP32,
      GxB_ISEQ_FP64,      GxB_ISNE_FP64,      GxB_ISGT_FP64,      GxB_ISLT_FP64,
  
      // z = (x >= y)     z = (x <= y)
      GxB_ISGE_BOOL,      GxB_ISLE_BOOL,
      GxB_ISGE_INT8,      GxB_ISLE_INT8,
      GxB_ISGE_UINT8,     GxB_ISLE_UINT8,
      GxB_ISGE_INT16,     GxB_ISLE_INT16,
      GxB_ISGE_UINT16,    GxB_ISLE_UINT16,
      GxB_ISGE_INT32,     GxB_ISLE_INT32,
      GxB_ISGE_UINT32,    GxB_ISLE_UINT32,
      GxB_ISGE_INT64,     GxB_ISLE_INT64,
      GxB_ISGE_UINT64,    GxB_ISLE_UINT64,
      GxB_ISGE_FP32,      GxB_ISLE_FP32,
      GxB_ISGE_FP64,      GxB_ISLE_FP64 ;
  
  // Six comparison operators z=f(x,y) return their result as boolean, but where
  // x and y have the same type (any one of the 11 built-in types).  The suffix
  // in their names refers to the type of x and y since z is always boolean.  If
  // used as multiply operators in a semiring, they can only be combined with
  // boolean monoids.  The _BOOL versions of these operators give the same
  // results as their IS*_BOOL counterparts.
  
  extern GrB_BinaryOp
      // z = (x == y)     z = (x != y)        z = (x > y)         z = (x < y)
      GrB_EQ_BOOL,        GrB_NE_BOOL,        GrB_GT_BOOL,        GrB_LT_BOOL,
      GrB_EQ_INT8,        GrB_NE_INT8,        GrB_GT_INT8,        GrB_LT_INT8,
      GrB_EQ_UINT8,       GrB_NE_UINT8,       GrB_GT_UINT8,       GrB_LT_UINT8,
      GrB_EQ_INT16,       GrB_NE_INT16,       GrB_GT_INT16,       GrB_LT_INT16,
      GrB_EQ_UINT16,      GrB_NE_UINT16,      GrB_GT_UINT16,      GrB_LT_UINT16,
      GrB_EQ_INT32,       GrB_NE_INT32,       GrB_GT_INT32,       GrB_LT_INT32,
      GrB_EQ_UINT32,      GrB_NE_UINT32,      GrB_GT_UINT32,      GrB_LT_UINT32,
      GrB_EQ_INT64,       GrB_NE_INT64,       GrB_GT_INT64,       GrB_LT_INT64,
      GrB_EQ_UINT64,      GrB_NE_UINT64,      GrB_GT_UINT64,      GrB_LT_UINT64,
      GrB_EQ_FP32,        GrB_NE_FP32,        GrB_GT_FP32,        GrB_LT_FP32,
      GrB_EQ_FP64,        GrB_NE_FP64,        GrB_GT_FP64,        GrB_LT_FP64,
  
      // z = (x >= y)     z = (x <= y)
      GrB_GE_BOOL,        GrB_LE_BOOL,
      GrB_GE_INT8,        GrB_LE_INT8,
      GrB_GE_UINT8,       GrB_LE_UINT8,
      GrB_GE_INT16,       GrB_LE_INT16,
      GrB_GE_UINT16,      GrB_LE_UINT16,
      GrB_GE_INT32,       GrB_LE_INT32,
      GrB_GE_UINT32,      GrB_LE_UINT32,
      GrB_GE_INT64,       GrB_LE_INT64,
      GrB_GE_UINT64,      GrB_LE_UINT64,
      GrB_GE_FP32,        GrB_LE_FP32,
      GrB_GE_FP64,        GrB_LE_FP64 ;
  
  // Three binary operators operate on each of the types, converting them
  // internally to boolean and returning a value 1 or 0 in the same type, for
  // true or false.  Each of them compute z = ((x != 0) OP (y != 0)), where x, y,
  // and z all the same type.  These operators are useful as multiply operators
  // when combined with non-boolean monoids of the same type.
  
  extern GrB_BinaryOp
      // z = (x || y)     z = (x && y)        z = (x != y)
      GxB_LOR_BOOL,       GxB_LAND_BOOL,      GxB_LXOR_BOOL,
      GxB_LOR_INT8,       GxB_LAND_INT8,      GxB_LXOR_INT8,
      GxB_LOR_UINT8,      GxB_LAND_UINT8,     GxB_LXOR_UINT8,
      GxB_LOR_INT16,      GxB_LAND_INT16,     GxB_LXOR_INT16,
      GxB_LOR_UINT16,     GxB_LAND_UINT16,    GxB_LXOR_UINT16,
      GxB_LOR_INT32,      GxB_LAND_INT32,     GxB_LXOR_INT32,
      GxB_LOR_UINT32,     GxB_LAND_UINT32,    GxB_LXOR_UINT32,
      GxB_LOR_INT64,      GxB_LAND_INT64,     GxB_LXOR_INT64,
      GxB_LOR_UINT64,     GxB_LAND_UINT64,    GxB_LXOR_UINT64,
      GxB_LOR_FP32,       GxB_LAND_FP32,      GxB_LXOR_FP32,
      GxB_LOR_FP64,       GxB_LAND_FP64,      GxB_LXOR_FP64 ;
  
  // Finally, three binary operate only on boolean types: LOR, LAND, LXOR.  The
  // naming convention differs (_BOOL is not appended to the name).  They are
  // the same as GxB_LOR_BOOL, GxB_LAND_BOOL, and GxB_LXOR_BOOL; they just
  // have a simpler name.
  
  extern GrB_BinaryOp
      // z = (x || y)     z = (x && y)        z = (x != y)
      GrB_LOR,            GrB_LAND,           GrB_LXOR ;
  
  // Some of the boolean operators compute the same thing but have unique names.
  // For example, x*y and x&&y give the same results for boolean x and y.
  // Operations such as x < y when x and y are boolean are treated as if true=1
  // and false=0.  Below is the truth table for all 17 binary operators with
  // boolean inputs.  This table is defined by how C typecasts boolean values for
  // non-boolean operations.  For example, if x, y, and z are boolean, x = true,
  // and y = true, then z = x + y = true + true = true.  DIV (x/y) is defined
  // below.
  
  //                                                     is  is  is  is  is  is
  //  x y    1st 2nd min max +   -   *   /   or  and xor eq  ne  gt  lt  ge  le
  //  0 0    0   0   0   0   0   0   0   0   0   0   0   1   0   0   0   1   1
  //  0 1    0   1   0   1   1   1   0   0   1   0   1   0   1   0   1   0   1
  //  1 0    1   0   0   1   1   1   0   1   1   0   1   0   1   1   0   1   0
  //  1 1    1   1   1   1   1   0   1   1   1   1   0   1   0   0   0   1   1
  
  // SPEC: the definition of divide-by-zero is an extension to the spec
  
  // GraphBLAS includes a GrB_DIV_BOOL operator in its specification, but does
  // not define what boolean "division" means.  SuiteSparse:GraphBLAS makes the
  // following interpretation.
  
  // GraphBLAS does not generate exceptions for divide-by-zero, so the results
  // are defined just as they are in MATLAB.  Floating-point divide-by-zero
  // follows the IEEE 754 standard: 1/0 is +Inf, -1/0 is -Inf, and 0/0 is NaN.
  // For integer division by zero, if x is positive, x/0 is the largest integer,
  // -x/0 is the integer minimum (zero for unsigned integers), and 0/0 is zero.
  // For example, for int8, 1/0 is 127, and -1/0 is -128.  For uint8, 1/0 is 255
  // and 0/0 is zero.
  
  // Boolean division is not in MATLAB.  For SuiteSparse:GraphBLAS, boolean
  // division is treated as if it were an unsigned integer type with true=1 and
  // false=0, and with the max and min value being 1 and 0.  As a result,
  // GrB_IDENTITY_BOOL, GrB_AINV_BOOL, and GrB_MINV_BOOL all give the same result
  // (z = x).
  
  // With this convention for boolean "division", there are 10 unique binary
  // operators that are purely boolean; 13 *_BOOL operators are redundant but are
  // included in GraphBLAS so that the name space of operators is complete:
  
  //      z = x           FIRST, DIV
  //      z = y           SECOND
  //      z = (x && y)    AND, MIN, TIMES
  //      z = (x || y)    OR, MAX, PLUS
  //      z = (x != y)    XOR, MINUS, NE, ISNE
  //      z = (x == y)    EQ, ISEQ
  //      z = (x >  y)    GT, ISGT
  //      z = (x <  y)    LT, ISLT
  //      z = (x >= y)    GE, ISGE
  //      z = (x >= y)    LE, ISLE
  
  // Three more that have no_BOOL suffix are also redundant with the operators
  // of the form GxB_*_BOOL.
  // (GrB_LOR, GrB_LAND, and GrB_LXOR).
  
  // There are thus 256 built-in binary operators with unique names, 16 of which
  // are redundant, giving 240 built-in binary operators that compute unique
  // results.
  
  //------------------------------------------------------------------------------
  // methods for binary operators
  //------------------------------------------------------------------------------
  
  // GrB_BinaryOp_new is implemented both as a macro and a function.  Both are
  // user-callable.  The default is to use the macro, since this allows the name
  // of the unary function to be kept in the new operator as a string.  See the
  // discussion of GrB_Type_new above.
  
  #undef GrB_BinaryOp_new
  
  GrB_Info GrB_BinaryOp_new
  (
      GrB_BinaryOp *binaryop,         // handle for the new binary operator
      void *function,                 // pointer to the binary function
      const GrB_Type ztype,           // type of output z
      const GrB_Type xtype,           // type of input x
      const GrB_Type ytype            // type of input y
  ) ;
  
  #ifndef NMACRO
  #define GrB_BinaryOp_new(op,f,z,x,y) GB_BinaryOp_new (op,f,z,x,y, GB_STR(f))
  #endif
  
  GrB_Info GB_BinaryOp_new            // not user-callable; use GrB_BinaryOp_new
  (
      GrB_BinaryOp *binaryop,         // handle for the new binary operator
      void *function,                 // pointer to the binary function
      const GrB_Type ztype,           // type of output z
      const GrB_Type xtype,           // type of input x
      const GrB_Type ytype,           // type of input y
      const char *name                // name of the underlying function
  ) ;
  
  // SPEC: GxB_BinaryOp_ztype is an extension to the spec
  
  GrB_Info GxB_BinaryOp_ztype         // return the type of z
  (
      GrB_Type *ztype,                // return type of output z
      const GrB_BinaryOp binaryop     // binary operator to query
  ) ;
  
  // SPEC: GxB_BinaryOp_xtype is an extension to the spec
  
  GrB_Info GxB_BinaryOp_xtype         // return the type of x
  (
      GrB_Type *xtype,                // return type of input x
      const GrB_BinaryOp binaryop     // binary operator to query
  ) ;
  
  // SPEC: GxB_BinaryOp_ytype is an extension to the spec
  
  GrB_Info GxB_BinaryOp_ytype         // return the type of y
  (
      GrB_Type *ytype,                // return type of input y
      const GrB_BinaryOp binaryop     // binary operator to query
  ) ;
  
  GrB_Info GrB_BinaryOp_free          // free a user-created binary operator
  (
      GrB_BinaryOp *binaryop          // handle of binary operator to free
  ) ;
  
  //------------------------------------------------------------------------------
  // Select operators
  //------------------------------------------------------------------------------
  
  // SPEC: GxB_SelectOp and all related functions are an extenstion to the spec.
  
  // GxB_SelectOp is an operator used by GxB_select to select entries from an
  // input matrix A that are kept in the output C.  If an entry A(i,j) in the
  // matrix A, of size nrows-by-ncols, has the value aij, then it calls the
  // select function as result = f (i, j, nrows, ncols, aij, k).  If the function
  // returns true, the entry is kept in the output C.  If f returns false, the
  // entry is not kept in C.  The type of x for the GxB_SelectOp operator may be
  // any of the 11 built-in types, or any user-defined type.  It may also be
  // GrB_NULL, to indicate that the function is type-generic and does not depend
  // at all on the value aij.  In this case, x is passed to f as a NULL pointer.
  // The k parameter is a const void * pointer to a user-defined object that is
  // passed to GxB_select.  It is not used by GxB_select except to pass it to the
  // function f.
  
  // GxB_SelectOp:  a function z=f(i,j,m,n,x,k) for the GxB_Select operation.
  // The function f must have the signature:
  
  //      bool f (const GrB_Index i, const GrB_Index j,
  //              const GrB_Index nrows, const GrB_Index ncols,
  //              const void *x, const void *k) ;
  
  typedef struct GB_SelectOp_opaque *GxB_SelectOp ;
  
  //------------------------------------------------------------------------------
  // built-in select operators
  //------------------------------------------------------------------------------
  
  // GxB_select (C, Mask, accum, op, A, k, desc) always returns a matrix C of the
  // same size as A (or A' if GrB_TRAN is in the descriptor).
  
  extern GxB_SelectOp
      GxB_TRIL,       // C=tril(A,k):   returns true if ((j-i) <= k)
      GxB_TRIU,       // C=triu(A,k):   returns true if ((j-i) >= k)
      GxB_DIAG,       // C=diag(A,k):   returns true if ((j-i) == k)
      GxB_OFFDIAG,    // C=A-diag(A,k): returns true if ((j-i) != k)
      GxB_NONZERO ;   // C=A(A~=0):     returns true if aij is nonzero,
                      //                for any built-in or user-defined type
  
  // For GxB_TRIL, GxB_TRIU, GxB_DIAG, and GxB_OFFDIAG, the parameter k is a
  // const void * pointer to a single scalar value of type int64_t.  These
  // select operators do not depend on the values of A, but just their position.
  
  // For GxB_NONZERO, the result depends only on the value of A(i,j), and the k
  // parameter may be GrB_NULL.  It works on all built-in types and all
  // user-defined types.  When applied to user-defined types the operator it
  // returns true if all bits in the user-defined value are zero, which can be
  // tested regardless of how the user-defined type is defined.
  
  //------------------------------------------------------------------------------
  // select operators
  //------------------------------------------------------------------------------
  
  // GxB_SelectOp_new is implemented both as a macro and a function.  Both are
  // user-callable.  The default is to use the macro, since this allows the name
  // of the select function to be kept in the new operator as a string.  See the
  // discussion of GrB_Type_new above.
  
  #undef GxB_SelectOp_new
  
  GrB_Info GxB_SelectOp_new       // create a new user-defined select operator
  (
      GxB_SelectOp *selectop,     // handle for the new select operator
      void *function,             // pointer to the select function
      const GrB_Type xtype        // type of input x, or NULL if type-generic
  ) ;
  
  #ifndef NMACRO
  #define GxB_SelectOp_new(op,f,x) GB_SelectOp_new (op,f,x, GB_STR(f))
  #endif
  
  GrB_Info GB_SelectOp_new        // not user-callable; use GxB_SelectOp_new
  (
      GxB_SelectOp *selectop,     // handle for the new select operator
      void *function,             // pointer to the select function
      const GrB_Type xtype,       // type of input x
      const char *name            // name of the underlying function
  ) ;
  
  GrB_Info GxB_SelectOp_xtype     // return the type of x
  (
      GrB_Type *xtype,            // return type of input x
      const GxB_SelectOp selectop // select operator
  ) ;
  
  GrB_Info GxB_SelectOp_free      // free a user-created select operator
  (
      GxB_SelectOp *selectop      // handle of select operator to free
  ) ;
  
  //------------------------------------------------------------------------------
  // GraphBLAS Monoid
  //------------------------------------------------------------------------------
  
  // A monoid is an associative operator z=op(x,y) where all three types of z, x,
  // and y are identical.  The monoid also has an identity element, such that
  // op(x,identity) = op(identity,x) = x.
  
  typedef struct GB_Monoid_opaque *GrB_Monoid ;
  
  // Create a new Monoid with a specific type of identity, which must match
  // the binary_op type.  The binary_op's three types must all be the same.
  
  GrB_Info GrB_Monoid_new_BOOL        // create a new boolean monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const bool identity             // identity value of the monoid
  ) ;
  
  GrB_Info GrB_Monoid_new_INT8        // create a new int8 monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const int8_t identity           // identity value of the monoid
  ) ;
  
  GrB_Info GrB_Monoid_new_UINT8       // create a new uint8 monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const uint8_t identity          // identity value of the monoid
  ) ;
  
  GrB_Info GrB_Monoid_new_INT16       // create a new int16 monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const int16_t identity          // identity value of the monoid
  ) ;
  
  GrB_Info GrB_Monoid_new_UINT16      // create a new uint16 monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const uint16_t identity         // identity value of the monoid
  ) ;
  
  GrB_Info GrB_Monoid_new_INT32       // create a new int32 monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const int32_t identity          // identity value of the monoid
  ) ;
  
  GrB_Info GrB_Monoid_new_UINT32      // create a new uint32 monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const uint32_t identity         // identity value of the monoid
  ) ;
  
  GrB_Info GrB_Monoid_new_INT64       // create a new int64 monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const int64_t identity          // identity value of the monoid
  ) ;
  
  GrB_Info GrB_Monoid_new_UINT64      // create a new uint64 monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const uint64_t identity         // identity value of the monoid
  ) ;
  
  GrB_Info GrB_Monoid_new_FP32        // create a new float monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const float identity            // identity value of the monoid
  ) ;
  
  GrB_Info GrB_Monoid_new_FP64        // create a new double monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const double identity           // identity value of the monoid
  ) ;
  
  GrB_Info GrB_Monoid_new_UDT         // create a monoid with a user-defined type
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const void *identity            // identity value of the monoid
  ) ;
  
  // Type-generic method for creating a new monoid:
  
  /*
  
  GrB_Info GrB_Monoid_new             // create a monoid
  (
      GrB_Monoid *monoid,             // handle of monoid to create
      const GrB_BinaryOp op,          // binary operator of the monoid
      const <type> identity           // identity value of the monoid
  ) ;
  
  */
  
  #define GrB_Monoid_new(monoid,op,identity)          \
      _Generic                                        \
      (                                               \
          (identity),                                 \
          const bool     : GrB_Monoid_new_BOOL   ,    \
                bool     : GrB_Monoid_new_BOOL   ,    \
          const int8_t   : GrB_Monoid_new_INT8   ,    \
                int8_t   : GrB_Monoid_new_INT8   ,    \
          const uint8_t  : GrB_Monoid_new_UINT8  ,    \
                uint8_t  : GrB_Monoid_new_UINT8  ,    \
          const int16_t  : GrB_Monoid_new_INT16  ,    \
                int16_t  : GrB_Monoid_new_INT16  ,    \
          const uint16_t : GrB_Monoid_new_UINT16 ,    \
                uint16_t : GrB_Monoid_new_UINT16 ,    \
          const int32_t  : GrB_Monoid_new_INT32  ,    \
                int32_t  : GrB_Monoid_new_INT32  ,    \
          const uint32_t : GrB_Monoid_new_UINT32 ,    \
                uint32_t : GrB_Monoid_new_UINT32 ,    \
          const int64_t  : GrB_Monoid_new_INT64  ,    \
                int64_t  : GrB_Monoid_new_INT64  ,    \
          const uint64_t : GrB_Monoid_new_UINT64 ,    \
                uint64_t : GrB_Monoid_new_UINT64 ,    \
          const float    : GrB_Monoid_new_FP32   ,    \
                float    : GrB_Monoid_new_FP32   ,    \
          const double   : GrB_Monoid_new_FP64   ,    \
                double   : GrB_Monoid_new_FP64   ,    \
          const void *   : GrB_Monoid_new_UDT    ,    \
                void *   : GrB_Monoid_new_UDT         \
      )                                               \
      (monoid, op, identity) ;
  
  // SPEC: GxB_Monoid_operator is an extension to the spec
  
  GrB_Info GxB_Monoid_operator        // return the monoid operator
  (
      GrB_BinaryOp *op,               // returns the binary op of the monoid
      const GrB_Monoid monoid         // monoid to query
  ) ;
  
  // SPEC: GxB_Monoid_identity is an extension to the spec
  
  GrB_Info GxB_Monoid_identity        // return the monoid identity
  (
      void *identity,                 // returns the identity of the monoid
      const GrB_Monoid monoid         // monoid to query
  ) ;
  
  GrB_Info GrB_Monoid_free            // free a user-created monoid
  (
      GrB_Monoid *monoid              // handle of monoid to free
  ) ;
  
  //------------------------------------------------------------------------------
  // GraphBLAS Semiring
  //------------------------------------------------------------------------------
  
  // A semiring defines all the operators required to define the multiplication
  // of two sparse matrices in GraphBLAS, C=A*B.  The "add" operator is a
  // commutative and associative monoid, and the binary "multiply" operator
  // defines a function z=fmult(x,y) where the type of z matches the exactly with
  // the monoid type.
  
  typedef struct GB_Semiring_opaque *GrB_Semiring ;
  
  GrB_Info GrB_Semiring_new           // create a semiring
  (
      GrB_Semiring *semiring,         // handle of semiring to create
      const GrB_Monoid add,           // add monoid of the semiring
      const GrB_BinaryOp multiply     // multiply operator of the semiring
  ) ;
  
  // SPEC: GxB_Semiring_add is an extension to the spec
  
  GrB_Info GxB_Semiring_add           // return the add monoid of a semiring
  (
      GrB_Monoid *add,                // returns add monoid of the semiring
      const GrB_Semiring semiring     // semiring to query
  ) ;
  
  // SPEC: GxB_Semiring_multiply is an extension to the spec
  
  GrB_Info GxB_Semiring_multiply      // return multiply operator of a semiring
  (
      GrB_BinaryOp *multiply,         // returns multiply operator of the semiring
      const GrB_Semiring semiring     // semiring to query
  ) ;
  
  GrB_Info GrB_Semiring_free          // free a user-created semiring
  (
      GrB_Semiring *semiring          // handle of semiring to free
  ) ;
  
  //==============================================================================
  //=== GraphBLAS Matrix and Vector objects ======================================
  //==============================================================================
  
  // Sparse matrices and vectors are the primary objects in GraphBLAS.  All other
  // objects exist to support them, and all the operations do their work on them.
  
  // A sparse matrix is nrows-by-ncols and stored in a compressed sparse column
  // form.  The row indices are kept sorted.  Also present is a list of pending
  // tuples, held in (i,j,x) form in an unsorted format.  These are pending
  // updates to the matrix, having been put there by the setElement method and/or
  // assign operations.  The row and column indices of a matrix are of type
  // GrB_Index, and they range from 0 to the dimesion minus 1.  That is, they are
  // zero-based.
  
  // Like all GraphBLAS objects, the GrB_Vector and GrB_Matrix are opaque to
  // the user; their structure may change in future releases.
  
  // GrB_Index: row or column index, or matrix dimension.  This typedef is used
  // for row and column indices, or matrix and vector dimensions.
  
  typedef uint64_t GrB_Index ;
  
  typedef struct GB_Matrix_opaque *GrB_Matrix ;
  
  typedef struct GB_Vector_opaque *GrB_Vector ;
  
  //==============================================================================
  //=== GraphBLAS Vector methods =================================================
  //==============================================================================
  
  // These methods create, free, copy, and clear a vector.  The size, nvals,
  // and type methods return basic information about a vector.
  
  GrB_Info GrB_Vector_new     // create a new vector with no entries
  (
      GrB_Vector *v,          // handle of vector to create
      const GrB_Type type,    // type of vector to create
      const GrB_Index n       // vector dimension is n-by-1
  ) ;
  
  GrB_Info GrB_Vector_dup     // make an exact copy of a vector
  (
      GrB_Vector *w,          // handle of output vector to create
      const GrB_Vector u      // input vector to copy
  ) ;
  
  GrB_Info GrB_Vector_clear   // clear a vector of all entries;
  (                           // type and dimension remain unchanged.
      GrB_Vector v            // vector to clear
  ) ;
  
  GrB_Info GrB_Vector_size    // get the dimension of a vector
  (
      GrB_Index *n,           // vector dimension is n-by-1
      const GrB_Vector v      // vector to query
  ) ;
  
  GrB_Info GrB_Vector_nvals   // get the number of entries in a vector
  (
      GrB_Index *nvals,       // vector has nvals entries
      const GrB_Vector v      // vector to query
  ) ;
  
  // SPEC: GxB_Vector_type is an extension to the spec
  
  GrB_Info GxB_Vector_type    // get the type of a vector
  (
      GrB_Type *type,         // returns the type of the vector
      const GrB_Vector v      // vector to query
  ) ;
  
  
  //------------------------------------------------------------------------------
  // GrB_Vector_build
  //------------------------------------------------------------------------------
  
  // GrB_Vector_build:  w = sparse (I,1,X) in MATLAB notation, but using any
  // associative operator to assemble duplicate entries.
  
  // Build a vector w from a set of (i,x) tuples.  The type and dimension of the
  // vector is already defined in w (via GrB_Vector_new), which must initially
  // have no entries.  I [0..nvals-1] is the list of row indices, and X
  // [0..nvals-1] is the list of numerical values.  The kth tuple is (I[k],X[k]),
  // and tuples can appear in any order.  Values are typecasted from X into the
  // type of the dup operator, as needed (user-defined types cannot be cast).
  // Duplicates are assembled together with the dup operator.  If two tuples
  // (i,x1) and (i,x2) have the same row index, then w(i) = dup (x1,x2).  All
  // three types of dup must be the same.  The types of C, X, and dup must be
  // compatible.
  
  // SPEC: extension: well-defined behavior of a non-associative dup operator.
  
  // The GraphBLAS spec requires dup to be associative and does not define the
  // order in which duplicates are assembled.  Currently this implementation
  // assembles duplicates in the order they appear in I and X.  For example, if
  // (i,x1), (i,x2), and (i,x3) appear in that order in I and X, then w(i) =
  // dup(dup(x1,x2),x3).  This means that using the non-associative FIRST
  // operator as dup means that w(i) is set equal to the first entry in the list,
  // x1, and SECOND gives the last one, x3.  SuiteSparse:GraphBLAS guarantees
  // this ordering.  However, per the spec, this order of assembly is not
  // guaranteed in all implementations.  Thus dup must be associative and results
  // are not guaranteed in all implementations if it is not.
  
  GrB_Info GrB_Vector_build_BOOL      // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const bool *X,                  // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Vector_build_INT8      // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const int8_t *X,                // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Vector_build_UINT8     // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const uint8_t *X,               // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Vector_build_INT16     // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const int16_t *X,               // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Vector_build_UINT16    // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const uint16_t *X,              // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Vector_build_INT32     // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const int32_t *X,               // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Vector_build_UINT32    // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const uint32_t *X,              // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Vector_build_INT64     // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const int64_t *X,               // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Vector_build_UINT64    // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const uint64_t *X,              // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Vector_build_FP32      // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const float *X,                 // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Vector_build_FP64      // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const double *X,                // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Vector_build_UDT       // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const void *X,                  // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  // Type-generic version:  X can be a pointer to any supported C type or void *
  // for a user-defined type.
  
  /*
  
  GrB_Info GrB_Vector_build           // build a vector from (I,X) tuples
  (
      GrB_Vector w,                   // vector to build
      const GrB_Index *I,             // array of row indices of tuples
      const <type> *X,                // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  */
  
  #define GrB_Vector_build(w,I,X,nvals,dup)               \
      _Generic                                            \
      (                                                   \
          (X),                                            \
          const bool      *: GrB_Vector_build_BOOL   ,    \
                bool      *: GrB_Vector_build_BOOL   ,    \
          const int8_t    *: GrB_Vector_build_INT8   ,    \
                int8_t    *: GrB_Vector_build_INT8   ,    \
          const uint8_t   *: GrB_Vector_build_UINT8  ,    \
                uint8_t   *: GrB_Vector_build_UINT8  ,    \
          const int16_t   *: GrB_Vector_build_INT16  ,    \
                int16_t   *: GrB_Vector_build_INT16  ,    \
          const uint16_t  *: GrB_Vector_build_UINT16 ,    \
                uint16_t  *: GrB_Vector_build_UINT16 ,    \
          const int32_t   *: GrB_Vector_build_INT32  ,    \
                int32_t   *: GrB_Vector_build_INT32  ,    \
          const uint32_t  *: GrB_Vector_build_UINT32 ,    \
                uint32_t  *: GrB_Vector_build_UINT32 ,    \
          const int64_t   *: GrB_Vector_build_INT64  ,    \
                int64_t   *: GrB_Vector_build_INT64  ,    \
          const uint64_t  *: GrB_Vector_build_UINT64 ,    \
                uint64_t  *: GrB_Vector_build_UINT64 ,    \
          const float     *: GrB_Vector_build_FP32   ,    \
                float     *: GrB_Vector_build_FP32   ,    \
          const double    *: GrB_Vector_build_FP64   ,    \
                double    *: GrB_Vector_build_FP64   ,    \
          const void      *: GrB_Vector_build_UDT    ,    \
                void      *: GrB_Vector_build_UDT         \
      )                                                   \
      (w, I, ((const void *) (X)), nvals, dup)
  
  //------------------------------------------------------------------------------
  // GrB_Vector_setElement
  //------------------------------------------------------------------------------
  
  // Set a single scalar in a vector, w(i) = x, typecasting from the type of x to
  // the type of w as needed.
  
  GrB_Info GrB_Vector_setElement_BOOL     // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const bool x,                       // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  GrB_Info GrB_Vector_setElement_INT8     // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const int8_t x,                     // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  GrB_Info GrB_Vector_setElement_UINT8    // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const uint8_t x,                    // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  GrB_Info GrB_Vector_setElement_INT16    // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const int16_t x,                    // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  GrB_Info GrB_Vector_setElement_UINT16   // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const uint16_t x,                   // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  GrB_Info GrB_Vector_setElement_INT32    // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const int32_t x,                    // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  GrB_Info GrB_Vector_setElement_UINT32   // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const uint32_t x,                   // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  GrB_Info GrB_Vector_setElement_INT64    // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const int64_t x,                    // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  GrB_Info GrB_Vector_setElement_UINT64   // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const uint64_t x,                   // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  GrB_Info GrB_Vector_setElement_FP32     // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const float x,                      // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  GrB_Info GrB_Vector_setElement_FP64     // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const double x,                     // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  GrB_Info GrB_Vector_setElement_UDT      // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const void *x,                      // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  // Type-generic version:  x can be any supported C type or void * for a
  // user-defined type.
  
  /*
  
  GrB_Info GrB_Vector_setElement          // w(i) = x
  (
      GrB_Vector w,                       // vector to modify
      const <type> x,                     // scalar to assign to w(i)
      const GrB_Index i                   // row index
  ) ;
  
  */
  
  #define GrB_Vector_setElement(w,x,i)                \
      _Generic                                        \
      (                                               \
          (x),                                        \
          const bool      : GrB_Vector_setElement_BOOL   ,  \
                bool      : GrB_Vector_setElement_BOOL   ,  \
          const int8_t    : GrB_Vector_setElement_INT8   ,  \
                int8_t    : GrB_Vector_setElement_INT8   ,  \
          const uint8_t   : GrB_Vector_setElement_UINT8  ,  \
                uint8_t   : GrB_Vector_setElement_UINT8  ,  \
          const int16_t   : GrB_Vector_setElement_INT16  ,  \
                int16_t   : GrB_Vector_setElement_INT16  ,  \
          const uint16_t  : GrB_Vector_setElement_UINT16 ,  \
                uint16_t  : GrB_Vector_setElement_UINT16 ,  \
          const int32_t   : GrB_Vector_setElement_INT32  ,  \
                int32_t   : GrB_Vector_setElement_INT32  ,  \
          const uint32_t  : GrB_Vector_setElement_UINT32 ,  \
                uint32_t  : GrB_Vector_setElement_UINT32 ,  \
          const int64_t   : GrB_Vector_setElement_INT64  ,  \
                int64_t   : GrB_Vector_setElement_INT64  ,  \
          const uint64_t  : GrB_Vector_setElement_UINT64 ,  \
                uint64_t  : GrB_Vector_setElement_UINT64 ,  \
          const float     : GrB_Vector_setElement_FP32   ,  \
                float     : GrB_Vector_setElement_FP32   ,  \
          const double    : GrB_Vector_setElement_FP64   ,  \
                double    : GrB_Vector_setElement_FP64   ,  \
          const void *    : GrB_Vector_setElement_UDT    ,  \
                void *    : GrB_Vector_setElement_UDT       \
      )                                               \
      (w, x, i)
  
  //------------------------------------------------------------------------------
  // GrB_Vector_extractElement
  //------------------------------------------------------------------------------
  
  // Extract a single entry from a vector, x = v(i), typecasting from the type of
  // v to the type of x as needed.
  
  // Returns GrB_SUCCESS if v(i) is present, and sets x to its value.
  // Returns GrB_NO_VALUE if v(i) is not present, and x is unmodified.
  
  GrB_Info GrB_Vector_extractElement_BOOL     // x = v(i)
  (
      bool *x,                        // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  GrB_Info GrB_Vector_extractElement_INT8     // x = v(i)
  (
      int8_t *x,                      // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  GrB_Info GrB_Vector_extractElement_UINT8    // x = v(i)
  (
      uint8_t *x,                     // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  GrB_Info GrB_Vector_extractElement_INT16    // x = v(i)
  (
      int16_t *x,                     // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  GrB_Info GrB_Vector_extractElement_UINT16   // x = v(i)
  (
      uint16_t *x,                    // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  GrB_Info GrB_Vector_extractElement_INT32    // x = v(i)
  (
      int32_t *x,                     // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  GrB_Info GrB_Vector_extractElement_UINT32   // x = v(i)
  (
      uint32_t *x,                    // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  GrB_Info GrB_Vector_extractElement_INT64    // x = v(i)
  (
      int64_t *x,                     // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  GrB_Info GrB_Vector_extractElement_UINT64   // x = v(i)
  (
      uint64_t *x,                    // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  GrB_Info GrB_Vector_extractElement_FP32     // x = v(i)
  (
      float *x,                       // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  GrB_Info GrB_Vector_extractElement_FP64     // x = v(i)
  (
      double *x,                      // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  GrB_Info GrB_Vector_extractElement_UDT      // x = v(i)
  (
      void *x,                        // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  // Type-generic version:  x can be a pointer to any supported C type or void *
  // for a user-defined type.
  
  /*
  
  GrB_Info GrB_Vector_extractElement  // x = v(i)
  (
      <type> *x,                      // scalar extracted
      const GrB_Vector v,             // vector to extract an entry from
      const GrB_Index i               // row index
  ) ;
  
  */
  
  #define GrB_Vector_extractElement(x,v,i)                \
      _Generic                                            \
      (                                                   \
          (x),                                            \
          bool     *: GrB_Vector_extractElement_BOOL   ,  \
          int8_t   *: GrB_Vector_extractElement_INT8   ,  \
          uint8_t  *: GrB_Vector_extractElement_UINT8  ,  \
          int16_t  *: GrB_Vector_extractElement_INT16  ,  \
          uint16_t *: GrB_Vector_extractElement_UINT16 ,  \
          int32_t  *: GrB_Vector_extractElement_INT32  ,  \
          uint32_t *: GrB_Vector_extractElement_UINT32 ,  \
          int64_t  *: GrB_Vector_extractElement_INT64  ,  \
          uint64_t *: GrB_Vector_extractElement_UINT64 ,  \
          float    *: GrB_Vector_extractElement_FP32   ,  \
          double   *: GrB_Vector_extractElement_FP64   ,  \
          void     *: GrB_Vector_extractElement_UDT       \
      )                                                   \
      (x, v, i)
  
  //------------------------------------------------------------------------------
  // GrB_Vector_extractTuples
  //------------------------------------------------------------------------------
  
  // Extracts all tuples from a vector, like [I,~,X] = find (v) in MATLAB.  If
  // any parameter I and/or X is NULL, then that component is not extracted.  The
  // size of the I and X arrays (those that are not NULL) is given by nvals,
  // which must be at least as large as GrB_nvals (&nvals, v).  The values in the
  // vector are typecasted to the type of X, as needed.
  
  // If any parameter I and/or X is NULL, that component is not extracted.  So to
  // extract just the row indices, pass I as non-NULL, and X as NULL.  This is
  // like [I,~,~] = find (v) in MATLAB.
  
  GrB_Info GrB_Vector_extractTuples_BOOL      // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      bool *X,                    // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  GrB_Info GrB_Vector_extractTuples_INT8      // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      int8_t *X,                  // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  GrB_Info GrB_Vector_extractTuples_UINT8     // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      uint8_t *X,                 // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  GrB_Info GrB_Vector_extractTuples_INT16     // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      int16_t *X,                 // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  GrB_Info GrB_Vector_extractTuples_UINT16    // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      uint16_t *X,                // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  GrB_Info GrB_Vector_extractTuples_INT32     // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      int32_t *X,                 // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  GrB_Info GrB_Vector_extractTuples_UINT32    // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      uint32_t *X,                // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  GrB_Info GrB_Vector_extractTuples_INT64     // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      int64_t *X,                 // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  GrB_Info GrB_Vector_extractTuples_UINT64    // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      uint64_t *X,                // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  GrB_Info GrB_Vector_extractTuples_FP32      // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      float *X,                   // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  GrB_Info GrB_Vector_extractTuples_FP64      // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      double *X,                  // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  GrB_Info GrB_Vector_extractTuples_UDT       // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      void *X,                    // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  // Type-generic version:  X can be a pointer to any supported C type or void *
  // for a user-defined type.
  
  /*
  
  GrB_Info GrB_Vector_extractTuples           // [I,~,X] = find (v)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      <type> *X,                  // array for returning values of tuples
      GrB_Index *nvals,           // I, X size on input; # tuples on output
      const GrB_Vector v          // vector to extract tuples from
  ) ;
  
  */
  
  #define GrB_Vector_extractTuples(I,X,nvals,v)           \
      _Generic                                            \
      (                                                   \
          (X),                                            \
          bool     *: GrB_Vector_extractTuples_BOOL   ,   \
          int8_t   *: GrB_Vector_extractTuples_INT8   ,   \
          uint8_t  *: GrB_Vector_extractTuples_UINT8  ,   \
          int16_t  *: GrB_Vector_extractTuples_INT16  ,   \
          uint16_t *: GrB_Vector_extractTuples_UINT16 ,   \
          int32_t  *: GrB_Vector_extractTuples_INT32  ,   \
          uint32_t *: GrB_Vector_extractTuples_UINT32 ,   \
          int64_t  *: GrB_Vector_extractTuples_INT64  ,   \
          uint64_t *: GrB_Vector_extractTuples_UINT64 ,   \
          float    *: GrB_Vector_extractTuples_FP32   ,   \
          double   *: GrB_Vector_extractTuples_FP64   ,   \
          void     *: GrB_Vector_extractTuples_UDT        \
      )                                                   \
      (I, X, nvals, v)
  
  //------------------------------------------------------------------------------
  // GrB_Vector_free
  //------------------------------------------------------------------------------
  
  GrB_Info GrB_Vector_free    // free a vector
  (
      GrB_Vector *v           // handle of vector to free
  ) ;
  
  //==============================================================================
  //=== GraphBLAS Matrix methods =================================================
  //==============================================================================
  
  // These methods create, free, copy, and clear a matrix.  The nrows, ncols,
  // nvals, and type methods return basic information about a matrix.
  
  GrB_Info GrB_Matrix_new     // create a new matrix with no entries
  (
      GrB_Matrix *A,          // handle of matrix to create
      const GrB_Type type,    // type of matrix to create
      const GrB_Index nrows,  // matrix dimension is nrows-by-ncols
      const GrB_Index ncols
  ) ;
  
  GrB_Info GrB_Matrix_dup     // make an exact copy of a matrix
  (
      GrB_Matrix *C,          // handle of output matrix to create
      const GrB_Matrix A      // input matrix to copy
  ) ;
  
  GrB_Info GrB_Matrix_clear   // clear a matrix of all entries;
  (                           // type and dimensions remain unchanged
      GrB_Matrix A            // matrix to clear
  ) ;
  
  GrB_Info GrB_Matrix_nrows   // get the number of rows of a matrix
  (
      GrB_Index *nrows,       // matrix has nrows rows
      const GrB_Matrix A      // matrix to query
  ) ;
  
  GrB_Info GrB_Matrix_ncols   // get the number of columns of a matrix
  (
      GrB_Index *ncols,       // matrix has ncols columns
      const GrB_Matrix A      // matrix to query
  ) ;
  
  GrB_Info GrB_Matrix_nvals   // get the number of entries in a matrix
  (
      GrB_Index *nvals,       // matrix has nvals entries
      const GrB_Matrix A      // matrix to query
  ) ;
  
  // SPEC: GxB_Matrix_type is an extension to the spec
  
  GrB_Info GxB_Matrix_type    // get the type of a matrix
  (
      GrB_Type *type,         // returns the type of the matrix
      const GrB_Matrix A      // matrix to query
  ) ;
  
  //------------------------------------------------------------------------------
  // GrB_Matrix_build
  //------------------------------------------------------------------------------
  
  // GrB_Matrix_build:  C = sparse (I,J,X) in MATLAB notation, but using any
  // associative operator to assemble duplicate entries.
  
  // Builds a matrix C from a set of (i,j,x) tuples.  The type and dimension of
  // the matrix is already defined in C (via GrB_Matrix_new), which must
  // initially have no entries.  I [0..nvals-1] is the list of row indices, J
  // [0..nvals-1] is the list of column indices, and X [0..nvals-1] is the list
  // of numerical values.  The kth triplet is (I[k],J[k],X[k]), and tuples can
  // appear in any order.  Values are typecasted from X into the type of C, as
  // needed (user-defined types cannot be cast).  Duplicates are assembled
  // together with the dup operator.  If two tuples (i,j,x1) and (i,j,x2) have
  // the same row index, then C(i,j) = dup(x1,x2).  All three types of dup must
  // be the same, and dup, C, and X must be compatible.
  
  // SPEC: extension: well-defined behavior of a non-associative dup operator.
  
  // The dup operator must be associative in general, and the GraphBLAS spec
  // states the order of assembly is not defined.  However, SuiteSparse:GraphBLAS
  // does guarantee an ordering; see the description of GrB_Vector_build for more
  // details.
  
  GrB_Info GrB_Matrix_build_BOOL      // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const bool *X,                  // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Matrix_build_INT8      // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const int8_t *X,                // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Matrix_build_UINT8     // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const uint8_t *X,               // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Matrix_build_INT16     // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const int16_t *X,               // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Matrix_build_UINT16    // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const uint16_t *X,              // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Matrix_build_INT32     // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const int32_t *X,               // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Matrix_build_UINT32    // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const uint32_t *X,              // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Matrix_build_INT64     // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const int64_t *X,               // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Matrix_build_UINT64    // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const uint64_t *X,              // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Matrix_build_FP32      // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const float *X,                 // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Matrix_build_FP64      // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const double *X,                // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  GrB_Info GrB_Matrix_build_UDT       // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const void *X,                  // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  // Type-generic version:  X can be a pointer to any supported C type or void *
  // for a user-defined type.
  
  /*
  
  GrB_Info GrB_Matrix_build           // build a matrix from (I,J,X) tuples
  (
      GrB_Matrix C,                   // matrix to build
      const GrB_Index *I,             // array of row indices of tuples
      const GrB_Index *J,             // array of column indices of tuples
      const <type> *X,                // array of values of tuples
      const GrB_Index nvals,          // number of tuples
      const GrB_BinaryOp dup          // binary function to assemble duplicates
  ) ;
  
  */
  
  #define GrB_Matrix_build(C,I,J,X,nvals,dup)             \
      _Generic                                            \
      (                                                   \
          (X),                                            \
          const bool      *: GrB_Matrix_build_BOOL   ,    \
                bool      *: GrB_Matrix_build_BOOL   ,    \
          const int8_t    *: GrB_Matrix_build_INT8   ,    \
                int8_t    *: GrB_Matrix_build_INT8   ,    \
          const uint8_t   *: GrB_Matrix_build_UINT8  ,    \
                uint8_t   *: GrB_Matrix_build_UINT8  ,    \
          const int16_t   *: GrB_Matrix_build_INT16  ,    \
                int16_t   *: GrB_Matrix_build_INT16  ,    \
          const uint16_t  *: GrB_Matrix_build_UINT16 ,    \
                uint16_t  *: GrB_Matrix_build_UINT16 ,    \
          const int32_t   *: GrB_Matrix_build_INT32  ,    \
                int32_t   *: GrB_Matrix_build_INT32  ,    \
          const uint32_t  *: GrB_Matrix_build_UINT32 ,    \
                uint32_t  *: GrB_Matrix_build_UINT32 ,    \
          const int64_t   *: GrB_Matrix_build_INT64  ,    \
                int64_t   *: GrB_Matrix_build_INT64  ,    \
          const uint64_t  *: GrB_Matrix_build_UINT64 ,    \
                uint64_t  *: GrB_Matrix_build_UINT64 ,    \
          const float     *: GrB_Matrix_build_FP32   ,    \
                float     *: GrB_Matrix_build_FP32   ,    \
          const double    *: GrB_Matrix_build_FP64   ,    \
                double    *: GrB_Matrix_build_FP64   ,    \
          const void      *: GrB_Matrix_build_UDT    ,    \
                void      *: GrB_Matrix_build_UDT         \
      )                                                   \
      (C, I, J, ((const void *) (X)), nvals, dup)
  
  //------------------------------------------------------------------------------
  // GrB_Matrix_setElement
  //------------------------------------------------------------------------------
  
  // Set a single entry in a matrix, C(i,j) = x in MATLAB notation, typecasting
  // from the type of x to the type of C, as needed.
  
  GrB_Info GrB_Matrix_setElement_BOOL     // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const bool x,                       // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_setElement_INT8     // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const int8_t x,                     // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_setElement_UINT8    // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const uint8_t x,                    // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_setElement_INT16    // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const int16_t x,                    // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_setElement_UINT16   // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const uint16_t x,                   // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_setElement_INT32    // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const int32_t x,                    // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_setElement_UINT32   // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const uint32_t x,                   // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_setElement_INT64    // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const int64_t x,                    // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_setElement_UINT64   // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const uint64_t x,                   // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_setElement_FP32     // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const float x,                      // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_setElement_FP64     // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const double x,                     // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_setElement_UDT      // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const void *x,                      // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  // Type-generic version:  x can be any supported C type or void * for a
  // user-defined type.
  
  /*
  
  GrB_Info GrB_Matrix_setElement          // C (i,j) = x
  (
      GrB_Matrix C,                       // matrix to modify
      const <type> x,                     // scalar to assign to C(i,j)
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  */
  
  #define GrB_Matrix_setElement(C,x,i,j)                    \
      _Generic                                              \
      (                                                     \
          (x),                                              \
          const bool      : GrB_Matrix_setElement_BOOL   ,  \
                bool      : GrB_Matrix_setElement_BOOL   ,  \
          const int8_t    : GrB_Matrix_setElement_INT8   ,  \
                int8_t    : GrB_Matrix_setElement_INT8   ,  \
          const uint8_t   : GrB_Matrix_setElement_UINT8  ,  \
                uint8_t   : GrB_Matrix_setElement_UINT8  ,  \
          const int16_t   : GrB_Matrix_setElement_INT16  ,  \
                int16_t   : GrB_Matrix_setElement_INT16  ,  \
          const uint16_t  : GrB_Matrix_setElement_UINT16 ,  \
                uint16_t  : GrB_Matrix_setElement_UINT16 ,  \
          const int32_t   : GrB_Matrix_setElement_INT32  ,  \
                int32_t   : GrB_Matrix_setElement_INT32  ,  \
          const uint32_t  : GrB_Matrix_setElement_UINT32 ,  \
                uint32_t  : GrB_Matrix_setElement_UINT32 ,  \
          const int64_t   : GrB_Matrix_setElement_INT64  ,  \
                int64_t   : GrB_Matrix_setElement_INT64  ,  \
          const uint64_t  : GrB_Matrix_setElement_UINT64 ,  \
                uint64_t  : GrB_Matrix_setElement_UINT64 ,  \
          const float     : GrB_Matrix_setElement_FP32   ,  \
                float     : GrB_Matrix_setElement_FP32   ,  \
          const double    : GrB_Matrix_setElement_FP64   ,  \
                double    : GrB_Matrix_setElement_FP64   ,  \
          const void *    : GrB_Matrix_setElement_UDT    ,  \
                void *    : GrB_Matrix_setElement_UDT       \
      )                                                     \
      (C, x, i, j)
  
  //------------------------------------------------------------------------------
  // GrB_Matrix_extractElement
  //------------------------------------------------------------------------------
  
  // Extract a single entry from a matrix, x = A(i,j), typecasting from the type
  // of A to the type of x, as needed.
  
  // Returns GrB_SUCCESS if A(i,j) is present, and sets x to its value.
  // Returns GrB_NO_VALUE if A(i,j) is not present, and x is unmodified.
  
  GrB_Info GrB_Matrix_extractElement_BOOL     // x = A(i,j)
  (
      bool *x,                            // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_extractElement_INT8     // x = A(i,j)
  (
      int8_t *x,                          // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_extractElement_UINT8    // x = A(i,j)
  (
      uint8_t *x,                         // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_extractElement_INT16    // x = A(i,j)
  (
      int16_t *x,                         // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_extractElement_UINT16   // x = A(i,j)
  (
      uint16_t *x,                        // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_extractElement_INT32    // x = A(i,j)
  (
      int32_t *x,                         // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_extractElement_UINT32   // x = A(i,j)
  (
      uint32_t *x,                        // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_extractElement_INT64    // x = A(i,j)
  (
      int64_t *x,                         // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_extractElement_UINT64   // x = A(i,j)
  (
      uint64_t *x,                        // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_extractElement_FP32     // x = A(i,j)
  (
      float *x,                           // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_extractElement_FP64     // x = A(i,j)
  (
      double *x,                          // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  GrB_Info GrB_Matrix_extractElement_UDT      // x = A(i,j)
  (
      void *x,                            // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  // Type-generic version:  x can be a pointer to any supported C type or void *
  // for a user-defined type.
  
  /*
  
  GrB_Info GrB_Matrix_extractElement      // x = A(i,j)
  (
      <type> *x,                          // extracted scalar
      const GrB_Matrix A,                 // matrix to extract a scalar from
      const GrB_Index i,                  // row index
      const GrB_Index j                   // column index
  ) ;
  
  */
  
  #define GrB_Matrix_extractElement(x,A,i,j)              \
      _Generic                                            \
      (                                                   \
          (x),                                            \
          bool     *: GrB_Matrix_extractElement_BOOL   ,  \
          int8_t   *: GrB_Matrix_extractElement_INT8   ,  \
          uint8_t  *: GrB_Matrix_extractElement_UINT8  ,  \
          int16_t  *: GrB_Matrix_extractElement_INT16  ,  \
          uint16_t *: GrB_Matrix_extractElement_UINT16 ,  \
          int32_t  *: GrB_Matrix_extractElement_INT32  ,  \
          uint32_t *: GrB_Matrix_extractElement_UINT32 ,  \
          int64_t  *: GrB_Matrix_extractElement_INT64  ,  \
          uint64_t *: GrB_Matrix_extractElement_UINT64 ,  \
          float    *: GrB_Matrix_extractElement_FP32   ,  \
          double   *: GrB_Matrix_extractElement_FP64   ,  \
          void     *: GrB_Matrix_extractElement_UDT       \
      )                                                   \
      (x, A, i, j)
  
  //------------------------------------------------------------------------------
  // GrB_Matrix_extractTuples
  //------------------------------------------------------------------------------
  
  // Extracts all tuples from a matrix, like [I,J,X] = find (A) in MATLAB.  If
  // any parameter I, J and/or X is NULL, then that component is not extracted.
  // The size of the I, J, and X arrays (those that are not NULL) is given by
  // nvals, which must be at least as large as GrB_nvals (&nvals, A).  The values
  // in the matrix are typecasted to the type of X, as needed.
  
  // If any parameter I, J, and/or X is NULL, that component is not extracted.
  // So to extract just the row and col indices, pass I and J as non-NULL,
  // and X as NULL.  This is like [I,J,~] = find (A).
  
  GrB_Info GrB_Matrix_extractTuples_BOOL      // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      bool *X,                    // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  GrB_Info GrB_Matrix_extractTuples_INT8      // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      int8_t *X,                  // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  GrB_Info GrB_Matrix_extractTuples_UINT8     // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      uint8_t *X,                 // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  GrB_Info GrB_Matrix_extractTuples_INT16     // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      int16_t *X,                 // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  GrB_Info GrB_Matrix_extractTuples_UINT16    // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      uint16_t *X,                // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  GrB_Info GrB_Matrix_extractTuples_INT32     // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      int32_t *X,                 // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  GrB_Info GrB_Matrix_extractTuples_UINT32    // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      uint32_t *X,                // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  GrB_Info GrB_Matrix_extractTuples_INT64     // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      int64_t *X,                 // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  GrB_Info GrB_Matrix_extractTuples_UINT64    // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      uint64_t *X,                // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  GrB_Info GrB_Matrix_extractTuples_FP32      // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      float *X,                   // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  GrB_Info GrB_Matrix_extractTuples_FP64      // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      double *X,                  // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  GrB_Info GrB_Matrix_extractTuples_UDT       // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      void *X,                    // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  // Type-generic version:  X can be a pointer to any supported C type or void *
  // for a user-defined type.
  
  /*
  
  GrB_Info GrB_Matrix_extractTuples           // [I,J,X] = find (A)
  (
      GrB_Index *I,               // array for returning row indices of tuples
      GrB_Index *J,               // array for returning col indices of tuples
      <type> *X,                  // array for returning values of tuples
      GrB_Index *nvals,           // I,J,X size on input; # tuples on output
      const GrB_Matrix A          // matrix to extract tuples from
  ) ;
  
  */
  
  #define GrB_Matrix_extractTuples(I,J,X,nvals,A)         \
      _Generic                                            \
      (                                                   \
          (X),                                            \
          bool     *: GrB_Matrix_extractTuples_BOOL   ,   \
          int8_t   *: GrB_Matrix_extractTuples_INT8   ,   \
          uint8_t  *: GrB_Matrix_extractTuples_UINT8  ,   \
          int16_t  *: GrB_Matrix_extractTuples_INT16  ,   \
          uint16_t *: GrB_Matrix_extractTuples_UINT16 ,   \
          int32_t  *: GrB_Matrix_extractTuples_INT32  ,   \
          uint32_t *: GrB_Matrix_extractTuples_UINT32 ,   \
          int64_t  *: GrB_Matrix_extractTuples_INT64  ,   \
          uint64_t *: GrB_Matrix_extractTuples_UINT64 ,   \
          float    *: GrB_Matrix_extractTuples_FP32   ,   \
          double   *: GrB_Matrix_extractTuples_FP64   ,   \
          void     *: GrB_Matrix_extractTuples_UDT        \
      )                                                   \
      (I, J, X, nvals, A)
  
  //------------------------------------------------------------------------------
  // GrB_Matrix_free
  //------------------------------------------------------------------------------
  
  GrB_Info GrB_Matrix_free    // free a matrix
  (
      GrB_Matrix *A           // handle of matrix to free
  ) ;
  
  //==============================================================================
  //=== GraphBLAS Descriptor =====================================================
  //==============================================================================
  
  // The Descriptor is used to modify the behavior of GraphBLAS operations.
  //
  // OUTP: can be DEFAULT or REPLACE.  If REPLACE, then C is cleared after taking
  //      part in the accum operation but before the mask.  In other words,
  //      C<Mask> = accum (C,T) is split into Z = accum(C,T) ; C=0 ; C<Mask> = Z.
  //
  // MASK: can be DEFAULT or SCMP.  If DEFAULT, the mask is used normally,
  //      where Mask(i,j)=1 means C(i,j) can be modified by C<Mask>=Z, and
  //      Mask(i,j)=0 means it cannot be modified even if Z(i,j) is has been
  //      computed and differs from C(i,j).  If SCMP, this is the same as
  //      taking the logical complement of the Mask.
  //
  // INP0: can be DEFAULT or TRAN.  If DEFAULT, the first input is used as-is.
  //      If TRAN, it is transposed.  Only matrices are transposed this way.
  //
  // INP1: the same as INP0 but for the second input
  
  typedef enum
  {
      GrB_OUTP,       // descriptor for output of a method
      GrB_MASK,       // descriptor for the mask input of a method
      GrB_INP0,       // descriptor for the first input of a method
      GrB_INP1        // descriptor for the second input of a method
  }
  GrB_Desc_Field ;
  
  // SPEC: GxB_DEFAULT is an extension to the spec
  
  typedef enum
  {
      GxB_DEFAULT,    // default behavior of the method
      GrB_REPLACE,    // clear the output before assigning new values to it
      GrB_SCMP,       // use the structural complement of the input
      GrB_TRAN        // use the transpose of the input
  }
  GrB_Desc_Value ;
  
  typedef struct GB_Descriptor_opaque *GrB_Descriptor ;
  
  GrB_Info GrB_Descriptor_new     // create a new descriptor
  (
      GrB_Descriptor *descriptor  // handle of descriptor to create
  ) ;
  
  GrB_Info GrB_Descriptor_set     // set a parameter in a descriptor
  (
      GrB_Descriptor desc,        // descriptor to modify
      const GrB_Desc_Field field, // parameter to change
      const GrB_Desc_Value val    // value to change it to
  ) ;
  
  // SPEC: GxB_Descriptor_get is an extension to the spec
  
  GrB_Info GxB_Descriptor_get     // get a parameter from a descriptor
  (
      GrB_Desc_Value *val,        // value of the parameter
      const GrB_Descriptor desc,  // descriptor to query; NULL means defaults
      const GrB_Desc_Field field  // parameter to query
  ) ;
  
  GrB_Info GrB_Descriptor_free    // free a descriptor
  (
      GrB_Descriptor *descriptor  // handle of descriptor to free
  ) ;
  
  //==============================================================================
  //=== GrB_free =================================================================
  //==============================================================================
  
  // GrB_free: free a GraphBLAS object.  Each GraphBLAS object has a specific
  // GrB_*_new and GrB_*_free method.  There is no generic GrB_new, but the
  // generic GrB_free method can free any GraphBLAS object.  It is safe to free
  // an object twice, and it is also safe to (attempt to) free a built-in object.
  // In that case, GrB_free silently does nothing and returns GrB_SUCCESS.  By
  // the GraphBLAS spec, GrB_*_free functions can return GrB_SUCCESS or
  // GrB_PANIC; in this implementation they never panic.
  
  #define GrB_free(object)                         \
      _Generic                                     \
      (                                            \
          (object),                                \
          GrB_Type       *: GrB_Type_free       ,  \
          GrB_UnaryOp    *: GrB_UnaryOp_free    ,  \
          GrB_BinaryOp   *: GrB_BinaryOp_free   ,  \
          GxB_SelectOp   *: GxB_SelectOp_free   ,  \
          GrB_Monoid     *: GrB_Monoid_free     ,  \
          GrB_Semiring   *: GrB_Semiring_free   ,  \
          GrB_Vector     *: GrB_Vector_free     ,  \
          GrB_Matrix     *: GrB_Matrix_free     ,  \
          GrB_Descriptor *: GrB_Descriptor_free    \
      )                                            \
      (object)
  
  //==============================================================================
  //=== GraphBLAS operations =====================================================
  //==============================================================================
  
  // Each GraphBLAS operation can be modified by an optional Mask, an optional
  // accum operator, and a descriptor.
  
  // The primary computation of an operation computes a matrix or vector T.  If
  // accum is NULL, Z=T.  Otherwise, Z=accum(C,T) is computed, where accum is a
  // binary operator applied in an element-wise add manner.  Next, C is
  // optionally cleared if the OUTP:REPLACE descriptor is enabled.  Finally,
  // C<Mask>=Z is computed.  If there is no Mask, C=Z, or if an empty Mask
  // (Mask==NULL) is complemented via the descriptor, C is not modified at all.
  // Otherwise C(Mask)=Z(Mask) is computed using MATLAB-style logical index, if
  // the Mask is not complemented.  Otherwise C(~Mask)=Z(~Mask) is computed.
  // This description is terse; see the User Guide for more details.
  
  // GrB_NULL is used for the accum argument when no accum operation is desired,
  // for the Mask argument when no Mask is desired, and for the descriptor
  // argument when the default descriptor is desired.
  
  #define GrB_NULL NULL
  
  // An object that has been freed is a GrB_INVALID_HANDLE, a NULL pointer.
  
  #define GrB_INVALID_HANDLE NULL
  
  //------------------------------------------------------------------------------
  // matrix and vector multiplication over a semiring
  //------------------------------------------------------------------------------
  
  // Each of these methods compute a matrix multiplication over a semiring.  The
  // inputs are typecasted into the inputs of the semiring's multiply operator.
  // The result T=A*B has the type of the multiplier output, which is also the 3
  // types of the 'add' operator.  The 'add' operator is a commutatitive and
  // associative monoid.
  
  GrB_Info GrB_mxm                    // C<Mask> = accum (C, A*B)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GrB_Semiring semiring,    // defines '+' and '*' for A*B
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Matrix B,             // second input: matrix B
      const GrB_Descriptor desc       // descriptor for C, Mask, A, and B
  ) ;
  
  GrB_Info GrB_vxm                    // w'<Mask> = accum (w, u'*A)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_Semiring semiring,    // defines '+' and '*' for u'*A
      const GrB_Vector u,             // first input:  vector u
      const GrB_Matrix A,             // second input: matrix A
      const GrB_Descriptor desc       // descriptor for w, mask, and A
  ) ;
  
  GrB_Info GrB_mxv                    // w<Mask> = accum (w, A*u)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_Semiring semiring,    // defines '+' and '*' for A*B
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Vector u,             // second input: vector u
      const GrB_Descriptor desc       // descriptor for w, mask, and A
  ) ;
  
  //------------------------------------------------------------------------------
  // element-wise matrix and vector operations: using set intersection
  //------------------------------------------------------------------------------
  
  // GrB_eWiseMult computes C<Mask> = accum (C, A.*B), where ".*" is MATLAB
  // notation, and where pairs of elements in two matrices (or vectors) are
  // pairwise "multiplied" with C(i,j) = mult (A(i,j),B(i,j)).  The
  // "multiplication" operator can be any binary operator.  This is not matrix
  // multiplication in the conventional linear algebra sense; see GrB_mxm and
  // related methods for that operation.  The pattern of the result T=A.*B is the
  // set intersection (not union) of A and B.  Entries outside of the
  // intersection are not computed.  This is primary difference with
  // GrB_eWiseAdd.
  
  // The input matrices A and/or B may be transposed first, via the descriptor.
  
  // For a semiring, the mult operator is the semiring's multiply operator; note
  // that this differs from the eWiseAdd methods which use the semiring's add
  // operator instead. For a monoid, the mult operator is the monoid operator.
  
  GrB_Info GrB_eWiseMult_Vector_Semiring       // w<Mask> = accum (w, u.*v)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_Semiring semiring,    // defines '.*' for t=u.*v
      const GrB_Vector u,             // first input:  vector u
      const GrB_Vector v,             // second input: vector v
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_eWiseMult_Vector_Monoid         // w<Mask> = accum (w, u.*v)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_Monoid monoid,        // defines '.*' for t=u.*v
      const GrB_Vector u,             // first input:  vector u
      const GrB_Vector v,             // second input: vector v
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_eWiseMult_Vector_BinaryOp       // w<Mask> = accum (w, u.*v)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_BinaryOp mult,        // defines '.*' for t=u.*v
      const GrB_Vector u,             // first input:  vector u
      const GrB_Vector v,             // second input: vector v
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_eWiseMult_Matrix_Semiring       // C<Mask> = accum (C, A.*B)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GrB_Semiring semiring,    // defines '.*' for T=A.*B
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Matrix B,             // second input: matrix B
      const GrB_Descriptor desc       // descriptor for C, Mask, A, and B
  ) ;
  
  GrB_Info GrB_eWiseMult_Matrix_Monoid         // C<Mask> = accum (C, A.*B)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GrB_Monoid monoid,        // defines '.*' for T=A.*B
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Matrix B,             // second input: matrix B
      const GrB_Descriptor desc       // descriptor for C, Mask, A, and B
  ) ;
  
  GrB_Info GrB_eWiseMult_Matrix_BinaryOp       // C<Mask> = accum (C, A.*B)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GrB_BinaryOp mult,        // defines '.*' for T=A.*B
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Matrix B,             // second input: matrix B
      const GrB_Descriptor desc       // descriptor for C, Mask, A, and B
  ) ;
  
  // All 6 of the above type-specific functions are captured in a single
  // type-generic function, GrB_eWiseMult:
  
  #define GrB_eWiseMult(C,Mask,accum,op,A,B,desc)                         \
      _Generic                                                            \
      (                                                                   \
          (C),                                                            \
          GrB_Matrix :                                                    \
              _Generic                                                    \
              (                                                           \
                  (op),                                                   \
                  const GrB_Semiring : GrB_eWiseMult_Matrix_Semiring ,    \
                        GrB_Semiring : GrB_eWiseMult_Matrix_Semiring ,    \
                  const GrB_Monoid   : GrB_eWiseMult_Matrix_Monoid   ,    \
                        GrB_Monoid   : GrB_eWiseMult_Matrix_Monoid   ,    \
                  const GrB_BinaryOp : GrB_eWiseMult_Matrix_BinaryOp ,    \
                        GrB_BinaryOp : GrB_eWiseMult_Matrix_BinaryOp      \
              ),                                                          \
          GrB_Vector :                                                    \
              _Generic                                                    \
              (                                                           \
                  (op),                                                   \
                  const GrB_Semiring : GrB_eWiseMult_Vector_Semiring ,    \
                        GrB_Semiring : GrB_eWiseMult_Vector_Semiring ,    \
                  const GrB_Monoid   : GrB_eWiseMult_Vector_Monoid   ,    \
                        GrB_Monoid   : GrB_eWiseMult_Vector_Monoid   ,    \
                  const GrB_BinaryOp : GrB_eWiseMult_Vector_BinaryOp ,    \
                        GrB_BinaryOp : GrB_eWiseMult_Vector_BinaryOp      \
              )                                                           \
      )                                                                   \
      (C, Mask, accum, op, A, B, desc)
  
  //------------------------------------------------------------------------------
  // element-wise matrix and vector operations: using set union
  //------------------------------------------------------------------------------
  
  // GrB_eWiseAdd computes C<Mask> = accum (C, A+B), where pairs of elements in
  // two matrices (or two vectors) are pairwise "added".  The "add" operator can
  // be any binary operator.  With the plus operator, this is the same matrix
  // addition in conventional linear algebra.  The pattern of the result T=A+B is
  // the set union (not intersection) of A and B.  Entries outside of the union
  // are not computed.  That is, if both A(i,j) and B(i,j) are present in the
  // pattern of A and B, then T(i,j) = A(i,j) "+" B(i,j).  If only A(i,j) is
  // present then T(i,j) = A (i,j) and the "+" operator is not used.  Likewise,
  // if only B(i,j) is in the pattern of B but A(i,j) is not in the pattern of A,
  // then T(i,j) = B(i,j).  This is primary difference between GrB_eWiseAdd and
  // GrB_eWiseMult; the same set of binary operators can be used in both methods,
  // and the action they take on entries in the intersection of the pattern of A
  // and B is identical.
  
  // The input matrices A and/or B may be transposed first, via the descriptor.
  
  // For a semiring, the mult operator is the semiring's add operator; note that
  // this differs from the eWiseMult methods which use the semiring's multiply
  // operator instead. For a monoid, the mult operator is the monoid operator.
  
  GrB_Info GrB_eWiseAdd_Vector_Semiring       // w<Mask> = accum (w, u+v)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_Semiring semiring,    // defines '+' for t=u+v
      const GrB_Vector u,             // first input:  vector u
      const GrB_Vector v,             // second input: vector v
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_eWiseAdd_Vector_Monoid         // w<Mask> = accum (w, u+v)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_Monoid monoid,        // defines '+' for t=u+v
      const GrB_Vector u,             // first input:  vector u
      const GrB_Vector v,             // second input: vector v
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_eWiseAdd_Vector_BinaryOp       // w<Mask> = accum (w, u+v)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_BinaryOp add,         // defines '+' for t=u+v
      const GrB_Vector u,             // first input:  vector u
      const GrB_Vector v,             // second input: vector v
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_eWiseAdd_Matrix_Semiring       // C<Mask> = accum (C, A+B)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GrB_Semiring semiring,    // defines '+' for T=A+B
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Matrix B,             // second input: matrix B
      const GrB_Descriptor desc       // descriptor for C, Mask, A, and B
  ) ;
  
  GrB_Info GrB_eWiseAdd_Matrix_Monoid         // C<Mask> = accum (C, A+B)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GrB_Monoid monoid,        // defines '+' for T=A+B
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Matrix B,             // second input: matrix B
      const GrB_Descriptor desc       // descriptor for C, Mask, A, and B
  ) ;
  
  GrB_Info GrB_eWiseAdd_Matrix_BinaryOp       // C<Mask> = accum (C, A+B)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GrB_BinaryOp add,         // defines '+' for T=A+B
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Matrix B,             // second input: matrix B
      const GrB_Descriptor desc       // descriptor for C, Mask, A, and B
  ) ;
  
  #define GrB_eWiseAdd(C,Mask,accum,op,A,B,desc)                          \
      _Generic                                                            \
      (                                                                   \
          (C),                                                            \
          GrB_Matrix :                                                    \
              _Generic                                                    \
              (                                                           \
                  (op),                                                   \
                  const GrB_Semiring : GrB_eWiseAdd_Matrix_Semiring ,     \
                        GrB_Semiring : GrB_eWiseAdd_Matrix_Semiring ,     \
                  const GrB_Monoid   : GrB_eWiseAdd_Matrix_Monoid   ,     \
                        GrB_Monoid   : GrB_eWiseAdd_Matrix_Monoid   ,     \
                  const GrB_BinaryOp : GrB_eWiseAdd_Matrix_BinaryOp ,     \
                        GrB_BinaryOp : GrB_eWiseAdd_Matrix_BinaryOp       \
              ),                                                          \
          GrB_Vector :                                                    \
              _Generic                                                    \
              (                                                           \
                  (op),                                                   \
                  const GrB_Semiring : GrB_eWiseAdd_Vector_Semiring ,     \
                        GrB_Semiring : GrB_eWiseAdd_Vector_Semiring ,     \
                  const GrB_Monoid   : GrB_eWiseAdd_Vector_Monoid   ,     \
                        GrB_Monoid   : GrB_eWiseAdd_Vector_Monoid   ,     \
                  const GrB_BinaryOp : GrB_eWiseAdd_Vector_BinaryOp ,     \
                        GrB_BinaryOp : GrB_eWiseAdd_Vector_BinaryOp       \
              )                                                           \
      )                                                                   \
      (C, Mask, accum, op, A, B, desc)
  
  //------------------------------------------------------------------------------
  // matrix and vector extract
  //------------------------------------------------------------------------------
  
  // Extract entries from a matrix or vector; T = A(I,J) in MATLAB notation.
  // This (like most GraphBLAS methods) is then followed by C<Mask>=accum(C,T).
  
  // The input matrix A may be transposed first, via the descriptor.
  
  // To extract all rows of a matrix or vector, as in A (:,J) in MATLAB, use
  // I=GrB_ALL as the input argument.  For all columns of a matrix, use
  // J=GrB_ALL.  GrB_ALL is a predefined pointer that is not NULL so that
  // out-of-memory conditions can be (I=NULL) distinguished from a request for
  // all rows (I=GrB_ALL).  The pointer GrB_ALL should never dereferenced, and it
  // must not be freed or modified.
  
  // Each of these can be used with their generic name, GrB_extract.
  
  extern const uint64_t *GrB_ALL ;
  
  GrB_Info GrB_Vector_extract         // w<mask> = accum (w, u(I))
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_Vector u,             // first input:  vector u
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Matrix_extract         // C<Mask> = accum (C, A(I,J))
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C, Mask, and A
  ) ;
  
  GrB_Info GrB_Col_extract            // w<mask> = accum (w, A(I,j))
  (
      GrB_Vector w,                   // input/output matrix for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index j,              // column index
      const GrB_Descriptor desc       // descriptor for w, mask, and A
  ) ;
  
  //------------------------------------------------------------------------------
  // GrB_extract: generic matrix/vector extraction
  //------------------------------------------------------------------------------
  
  // GrB_extract is a generic interface to the following functions:
  
  // GrB_Vector_extract (w,mask,acc,u,I,ni,d)      // w<m>    = acc (w, u(I))
  // GrB_Col_extract    (w,mask,acc,A,I,ni,j,d)    // w<m>    = acc (w, A(I,j))
  // GrB_Matrix_extract (C,Mask,acc,A,I,ni,J,nj,d) // C<Mask> = acc (C, A(I,J))
  
  #define GrB_extract(arg1,Mask,accum,arg4,...) \
      _Generic                                                \
      (                                                       \
          (arg1),                                             \
          GrB_Vector :                                        \
              _Generic                                        \
              (                                               \
                  (arg4),                                     \
                  const GrB_Vector : GrB_Vector_extract ,     \
                        GrB_Vector : GrB_Vector_extract ,     \
                  const GrB_Matrix : GrB_Col_extract    ,     \
                        GrB_Matrix : GrB_Col_extract          \
              ),                                              \
          GrB_Matrix : GrB_Matrix_extract                     \
      )                                                       \
      (arg1, Mask, accum, arg4, __VA_ARGS__)
  
  //------------------------------------------------------------------------------
  // matrix and vector subassign: C(I,J)<Mask> = accum (C(I,J), A)
  //------------------------------------------------------------------------------
  
  // Assign entries in a matrix or vector; C(I,J) = A in MATLAB notation.
  // Each of these can be used with their generic name, GxB_subassign.
  
  // SPEC: The GxB_*_subassign functions are extensions to the spec.
  
  // Each GxB_subassign function is very similar to its corresponding GrB_assign
  // function in the spec, but they differ in three ways:
  
  // (1) the mask in the GxB_subassign functions has the same dimensions as
  //      w(I) for vectors and C(I,J) for matrices.  In GrB_assign, the mask is
  //      the same size as w or C, respectively (except for GrB_Row_asssign and
  //      GrB_Col_assign, in which case the mask is the same size as a row or
  //      column of C, respectively).  The two masks are related.  If M is the
  //      mask for GrB_assign, then M(I,J) is the mask for GxB_subassign.  If
  //      there is no mask, or if I and J are both GrB_ALL, then the two masks
  //      are the same.
  
  //      For GrB_Row_assign and GrB_Col_assign, the mask vector is the same
  //      size as a row or column of C, respectively.  For the corresponding
  //      GxB_Row_subassign and GxB_Col_subassign operations, the mask is the
  //      same size as the subrow C(i,J) or subcolumn C(I,j), respectively.
  
  // (2) They differ in how C is affected in areas outside the C(I,J) submatrix.
  //      In GxB_subassign, C(I,J) is the only part of C that can be modified,
  //      and no part of C outside the submatrix is ever modified.  In
  //      GrB_assign, it is possible to modify C outside the submatrix, but only
  //      in one specific manner.  Suppose the mask M is present (or, suppose it
  //      is not present but GrB_SCMP is true).  After (optionally) complementing
  //      the mask, the value of M(i,j) can be 0 for some entry outside the
  //      C(I,J) submatrix.  If the GrB_REPLACE descriptor is true, the
  //      GrB_assign deletes this entry.  This case does not occur if GrB_REPLACE
  //      is false.  With GrB_assign, it is not possible to change entries
  //      outside the submatrix C(I,J), except to delete them in this
  //      circumstance.
  
  // (3) They differ in how duplicate indices are treated in I and J.  For both
  //      assign and subassign operations, results are not defined for
  //      GrB_Matrix_*assign, GrB_Vector_*assign, GrB_Row_*assign, and
  //      GrB_Col_*assign when duplicate indices appear in I and J.  The scalar
  //      expansion operations, GrB_*_assign_TYPE, are well-defined if duplicate
  //      indices appear (the results are the same as if duplicates are removed
  //      first from I and J).  However, the subassign scalar expansion
  //      operations, GxB_*_subassign_TYPE are not well-defined if duplicate
  //      indices appear in I and J.
  
  // GxB_subassign and GrB_assign are identical if GrB_REPLACE is set to its
  // default value of false, or if the masks happen to be the same.  The two
  // masks can be the same in two cases:  either there is no mask (and GrB_SCMP
  // is false), or I and J are both GrB_ALL.  In this case, the two algorithms
  // are identical and have the same performance.
  
  // GxB_subassign is much faster than GrB_assign, when the latter must examine
  // the entire matrix C to delete entries (when GrB__REPLACE is true), and it
  // must deal with a much larger Mask matrix.  However, both methods have
  // specific uses.  Consider using C(I,J)+=F for many submatrices F (for
  // example, when assembling a finite-element matrix).  If the Mask is meant as
  // a specification for which entries of C should appear in the final result,
  // then use GrB_assign.  If the Mask is meant to control which entries of the
  // submatrix C(I,J) are modified by the finite-element F, then use
  // GxB_subassign.  This is particularly useful is the Mask is a "template" that
  // follows along with the finite-element F, independent of where it is applied
  // C.  Using GrB_assign would be very difficult in this case since a new Mask,
  // the same size as C, would need to be constructed for each finite-element F.
  
  // In GraphBLAS notation, the two methods can be described as follows:
  
  // matrix and vector subassign: C(I,J)<Mask> = accum (C(I,J), A)
  // matrix and vector    assign: C<Mask>(I,J) = accum (C(I,J), A)
  
  // This notation does not include the details of the GrB_SCMP and GrB_REPLACE
  // descriptors, but it does illustrate the difference in the Mask.  In the
  // subassign, Mask is the same size as C(I,J) and A.  If I[0]=i and J[0]=j,
  // Then Mask(0,0) controls how C(i,j) is modified by the subassign, from the
  // value A(0,0).  In the assign, Mask is the same size as C, and Mask(i,j)
  // controls how C(i,j) is modified.
  
  // The GxB_subassign and GrB_assign functions have the same signatures; they
  // differ only in how they consider the Mask and the GrB_REPLACE descriptor,
  // and in how duplicate indices are treated for scalar expansion.
  
  GrB_Info GxB_Vector_subassign       // w(I)<mask> = accum (w(I),u)
  (
      GrB_Vector w,                   // input/output matrix for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w(I),t)
      const GrB_Vector u,             // first input:  vector u
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign       // C(I,J)<Mask> = accum (C(I,J),A)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),T)
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J), Mask, and A
  ) ;
  
  GrB_Info GxB_Col_subassign          // C(I,j)<mask> = accum (C(I,j),u)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Vector mask,          // optional mask for C(I,j), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(C(I,j),t)
      const GrB_Vector u,             // input vector
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index j,              // column index
      const GrB_Descriptor desc       // descriptor for C(I,j) and mask
  ) ;
  
  GrB_Info GxB_Row_subassign          // C(i,J)<mask'> = accum (C(i,J),u')
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Vector mask,          // optional mask for C(i,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(C(i,J),t)
      const GrB_Vector u,             // input vector
      const GrB_Index i,              // row index
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(i,J) and mask
  ) ;
  
  //------------------------------------------------------------------------------
  // GxB_Vector_subassign_[SCALAR]:  scalar expansion assignment to subvector
  //------------------------------------------------------------------------------
  
  // Assigns a single scalar to a subvector, w(I)<mask> = accum(w(I),x).  The
  // scalar x is implicitly expanded into a vector u of size ni-by-1, with each
  // entry in u equal to x, and then w(I)<mask> = accum(w(I),u) is done.
  
  // Each of these can be used with their generic name, GxB_subassign.
  
  GrB_Info GxB_Vector_subassign_BOOL  // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w(I),x)
      const bool x,                   // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Vector_subassign_INT8  // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const int8_t x,                 // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Vector_subassign_UINT8 // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const uint8_t x,                // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Vector_subassign_INT16 // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const int16_t x,                // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Vector_subassign_UINT16   // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const uint16_t x,               // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Vector_subassign_INT32    // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const int32_t x,                // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Vector_subassign_UINT32   // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const uint32_t x,               // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Vector_subassign_INT64    // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const int64_t x,                // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Vector_subassign_UINT64   // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const uint64_t x,               // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Vector_subassign_FP32     // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const float x,                  // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Vector_subassign_FP64     // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const double x,                 // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  GrB_Info GxB_Vector_subassign_UDT      // w(I)<mask> = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w(I), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const void *x,                  // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w(I) and mask
  ) ;
  
  //------------------------------------------------------------------------------
  // GxB_Matrix_subassign_[SCALAR]:  scalar expansion assignment to submatrix
  //------------------------------------------------------------------------------
  
  // Assigns a single scalar to a submatrix, C(I,J)<Mask> = accum(C(I,J),x).  The
  // scalar x is implicitly expanded into a matrix A of size ni-by-nj, with each
  // entry in A equal to x, and then C(I,J)<Mask> = accum(C(I,J),A) is done.
  
  // Each of these can be used with their generic name, GxB_subassign.
  
  GrB_Info GxB_Matrix_subassign_BOOL  // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const bool x,                   // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign_INT8  // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const int8_t x,                 // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign_UINT8 // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const uint8_t x,                // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign_INT16 // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const int16_t x,                // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign_UINT16   // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const uint16_t x,               // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign_INT32    // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const int32_t x,                // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign_UINT32   // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const uint32_t x,               // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign_INT64    // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const int64_t x,                // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign_UINT64   // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const uint64_t x,               // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign_FP32     // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const float x,                  // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign_FP64     // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const double x,                 // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  GrB_Info GxB_Matrix_subassign_UDT      // C(I,J)<Mask> = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C(I,J), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const void *x,                  // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(I,J) and Mask
  ) ;
  
  //------------------------------------------------------------------------------
  // GxB_subassign: generic submatrix/subvector assignment
  //------------------------------------------------------------------------------
  
  // GxB_subassign is a generic function that provides access to all specific
  // GxB_*_subassign* functions:
  
  // GxB_Vector_subassign   (w,m,acc,u,I,ni,d)     // w(I)<m>   =acc(w(I),u)
  // GxB_Matrix_subassign   (C,M,acc,A,I,ni,J,nj,d)// C(I,J)<M> =acc(C(I,J),A)
  // GxB_Col_subassign      (C,m,acc,u,I,ni,j,d)   // C(I,j)<m> =acc(C(I,j),u)
  // GxB_Row_subassign      (C,m,acc,u,i,J,nj,d)   // C(i,J)<m'>=acc(C(i,J),u')
  // GxB_Vector_subassign_T (w,m,acc,x,I,ni,d)     // w(I)<m>   =acc(w(I),x)
  // GxB_Matrix_subassign_T (C,M,acc,x,I,ni,J,nj,d)// C(I,J)<M> =acc(C(I,J),x)
  
  #define GxB_subassign(arg1,Mask,accum,arg4,arg5,...)               \
      _Generic                                                    \
      (                                                           \
          (arg1),                                                 \
          GrB_Vector :                                            \
              _Generic                                            \
              (                                                   \
                  (arg4),                                         \
                  const bool       : GxB_Vector_subassign_BOOL   ,   \
                        bool       : GxB_Vector_subassign_BOOL   ,   \
                  const int8_t     : GxB_Vector_subassign_INT8   ,   \
                        int8_t     : GxB_Vector_subassign_INT8   ,   \
                  const uint8_t    : GxB_Vector_subassign_UINT8  ,   \
                        uint8_t    : GxB_Vector_subassign_UINT8  ,   \
                  const int16_t    : GxB_Vector_subassign_INT16  ,   \
                        int16_t    : GxB_Vector_subassign_INT16  ,   \
                  const uint16_t   : GxB_Vector_subassign_UINT16 ,   \
                        uint16_t   : GxB_Vector_subassign_UINT16 ,   \
                  const int32_t    : GxB_Vector_subassign_INT32  ,   \
                        int32_t    : GxB_Vector_subassign_INT32  ,   \
                  const uint32_t   : GxB_Vector_subassign_UINT32 ,   \
                        uint32_t   : GxB_Vector_subassign_UINT32 ,   \
                  const int64_t    : GxB_Vector_subassign_INT64  ,   \
                        int64_t    : GxB_Vector_subassign_INT64  ,   \
                  const uint64_t   : GxB_Vector_subassign_UINT64 ,   \
                        uint64_t   : GxB_Vector_subassign_UINT64 ,   \
                  const float      : GxB_Vector_subassign_FP32   ,   \
                        float      : GxB_Vector_subassign_FP32   ,   \
                  const double     : GxB_Vector_subassign_FP64   ,   \
                        double     : GxB_Vector_subassign_FP64   ,   \
                  const void *     : GxB_Vector_subassign_UDT    ,   \
                        void *     : GxB_Vector_subassign_UDT    ,   \
                  default          : GxB_Vector_subassign            \
              ),                                                  \
          default :                                               \
              _Generic                                            \
              (                                                   \
                  (arg4),                                         \
                  const bool       : GxB_Matrix_subassign_BOOL   ,   \
                        bool       : GxB_Matrix_subassign_BOOL   ,   \
                  const int8_t     : GxB_Matrix_subassign_INT8   ,   \
                        int8_t     : GxB_Matrix_subassign_INT8   ,   \
                  const uint8_t    : GxB_Matrix_subassign_UINT8  ,   \
                        uint8_t    : GxB_Matrix_subassign_UINT8  ,   \
                  const int16_t    : GxB_Matrix_subassign_INT16  ,   \
                        int16_t    : GxB_Matrix_subassign_INT16  ,   \
                  const uint16_t   : GxB_Matrix_subassign_UINT16 ,   \
                        uint16_t   : GxB_Matrix_subassign_UINT16 ,   \
                  const int32_t    : GxB_Matrix_subassign_INT32  ,   \
                        int32_t    : GxB_Matrix_subassign_INT32  ,   \
                  const uint32_t   : GxB_Matrix_subassign_UINT32 ,   \
                        uint32_t   : GxB_Matrix_subassign_UINT32 ,   \
                  const int64_t    : GxB_Matrix_subassign_INT64  ,   \
                        int64_t    : GxB_Matrix_subassign_INT64  ,   \
                  const uint64_t   : GxB_Matrix_subassign_UINT64 ,   \
                        uint64_t   : GxB_Matrix_subassign_UINT64 ,   \
                  const float      : GxB_Matrix_subassign_FP32   ,   \
                        float      : GxB_Matrix_subassign_FP32   ,   \
                  const double     : GxB_Matrix_subassign_FP64   ,   \
                        double     : GxB_Matrix_subassign_FP64   ,   \
                  const void *     : GxB_Matrix_subassign_UDT    ,   \
                        void *     : GxB_Matrix_subassign_UDT    ,   \
                  const GrB_Vector :                              \
                      _Generic                                    \
                      (                                           \
                          (arg5),                                 \
                          const GrB_Index *: GxB_Col_subassign ,     \
                                GrB_Index *: GxB_Col_subassign ,     \
                          default          : GxB_Row_subassign       \
                      ),                                          \
                  GrB_Vector :                                    \
                      _Generic                                    \
                      (                                           \
                          (arg5),                                 \
                          const GrB_Index *: GxB_Col_subassign ,     \
                                GrB_Index *: GxB_Col_subassign ,     \
                          default          : GxB_Row_subassign       \
                      ),                                          \
                  default    : GxB_Matrix_subassign                  \
              )                                                   \
      )                                                           \
      (arg1, Mask, accum, arg4, arg5, __VA_ARGS__)
  
  //------------------------------------------------------------------------------
  // matrix and vector assign: C<Mask>(I,J) = accum (C(I,J), A)
  //------------------------------------------------------------------------------
  
  // Assign entries in a matrix or vector; C(I,J) = A in MATLAB notation.
  // Each of these can be used with their generic name, GrB_assign.
  
  GrB_Info GrB_Vector_assign          // w<mask>(I) = accum (w(I),u)
  (
      GrB_Vector w,                   // input/output matrix for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w(I),t)
      const GrB_Vector u,             // first input:  vector u
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Matrix_assign          // C<Mask>(I,J) = accum (C(I,J),A)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),T)
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C, Mask, and A
  ) ;
  
  GrB_Info GrB_Col_assign             // C<mask>(I,j) = accum (C(I,j),u)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Vector mask,          // optional mask for C(:,j), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(C(I,j),t)
      const GrB_Vector u,             // input vector
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index j,              // column index
      const GrB_Descriptor desc       // descriptor for C(:,j) and mask
  ) ;
  
  GrB_Info GrB_Row_assign             // C<mask'>(i,J) = accum (C(i,J),u')
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Vector mask,          // optional mask for C(i,:), unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(C(i,J),t)
      const GrB_Vector u,             // input vector
      const GrB_Index i,              // row index
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C(i,:) and mask
  ) ;
  
  //------------------------------------------------------------------------------
  // GrB_Vector_assign_[SCALAR]:  scalar expansion assignment to subvector
  //------------------------------------------------------------------------------
  
  // Assigns a single scalar to a subvector, w<mask>(I) = accum(w(I),x).  The
  // scalar x is implicitly expanded into a vector u of size ni-by-1, with each
  // entry in u equal to x, and then w<mask>(I) = accum(w(I),u) is done.
  
  // Each of these can be used with their generic name, GrB_assign.
  
  GrB_Info GrB_Vector_assign_BOOL     // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w(I),x)
      const bool x,                   // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Vector_assign_INT8     // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const int8_t x,                 // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Vector_assign_UINT8    // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const uint8_t x,                // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Vector_assign_INT16    // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const int16_t x,                // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Vector_assign_UINT16   // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const uint16_t x,               // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Vector_assign_INT32    // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const int32_t x,                // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Vector_assign_UINT32   // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const uint32_t x,               // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Vector_assign_INT64    // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const int64_t x,                // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Vector_assign_UINT64   // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const uint64_t x,               // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Vector_assign_FP32     // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const float x,                  // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Vector_assign_FP64     // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const double x,                 // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Vector_assign_UDT      // w<mask>(I) = accum (w(I),x)
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(w(I),x)
      const void *x,                  // scalar to assign to w(I)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  //------------------------------------------------------------------------------
  // GrB_Matrix_assign_[SCALAR]:  scalar expansion assignment to submatrix
  //------------------------------------------------------------------------------
  
  // Assigns a single scalar to a submatrix, C<Mask>(I,J) = accum(C(I,J),x).  The
  // scalar x is implicitly expanded into a matrix A of size ni-by-nj, with each
  // entry in A equal to x, and then C<Mask>(I,J) = accum(C(I,J),A) is done.
  
  // Each of these can be used with their generic name, GrB_assign.
  
  GrB_Info GrB_Matrix_assign_BOOL     // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const bool x,                   // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  GrB_Info GrB_Matrix_assign_INT8     // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const int8_t x,                 // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  GrB_Info GrB_Matrix_assign_UINT8    // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const uint8_t x,                // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  GrB_Info GrB_Matrix_assign_INT16    // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const int16_t x,                // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  GrB_Info GrB_Matrix_assign_UINT16   // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const uint16_t x,               // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  GrB_Info GrB_Matrix_assign_INT32    // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const int32_t x,                // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  GrB_Info GrB_Matrix_assign_UINT32   // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const uint32_t x,               // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  GrB_Info GrB_Matrix_assign_INT64    // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const int64_t x,                // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  GrB_Info GrB_Matrix_assign_UINT64   // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const uint64_t x,               // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  GrB_Info GrB_Matrix_assign_FP32     // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const float x,                  // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  GrB_Info GrB_Matrix_assign_FP64     // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const double x,                 // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  GrB_Info GrB_Matrix_assign_UDT      // C<Mask>(I,J) = accum (C(I,J),x)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C(I,J),x)
      const void *x,                  // scalar to assign to C(I,J)
      const GrB_Index *I,             // row indices
      const GrB_Index ni,             // number of row indices
      const GrB_Index *J,             // column indices
      const GrB_Index nj,             // number of column indices
      const GrB_Descriptor desc       // descriptor for C and Mask
  ) ;
  
  //------------------------------------------------------------------------------
  // GrB_assign: generic submatrix/subvector assignment
  //------------------------------------------------------------------------------
  
  // GrB_assign is a generic function that provides access to all specific
  // GrB_*_assign* functions:
  
  // GrB_Vector_assign   (w,mask,acc,u,I,ni,d)     // w<mask>(I)   =acc(w(I),u)
  // GrB_Matrix_assign   (C,Mask,acc,A,I,ni,J,nj,d)// C<Mask>(I,J) =acc(C(I,J),A)
  // GrB_Col_assign      (C,mask,acc,u,I,ni,j,d)   // C<mask>(I,j) =acc(C(I,j),u)
  // GrB_Row_assign      (C,mask,acc,u,i,J,nj,d)   // C<mask'>(i,J)=acc(C(i,J),u')
  // GrB_Vector_assign_T (w,mask,acc,x,I,ni,d)     // w<mask>(I)   =acc(w(I),x)
  // GrB_Matrix_assign_T (C,Mask,acc,x,I,ni,J,nj,d)// C<Mask>(I,J) =acc(C(I,J),x)
  
  #define GrB_assign(arg1,Mask,accum,arg4,arg5,...)               \
      _Generic                                                    \
      (                                                           \
          (arg1),                                                 \
          GrB_Vector :                                            \
              _Generic                                            \
              (                                                   \
                  (arg4),                                         \
                  const bool       : GrB_Vector_assign_BOOL   ,   \
                        bool       : GrB_Vector_assign_BOOL   ,   \
                  const int8_t     : GrB_Vector_assign_INT8   ,   \
                        int8_t     : GrB_Vector_assign_INT8   ,   \
                  const uint8_t    : GrB_Vector_assign_UINT8  ,   \
                        uint8_t    : GrB_Vector_assign_UINT8  ,   \
                  const int16_t    : GrB_Vector_assign_INT16  ,   \
                        int16_t    : GrB_Vector_assign_INT16  ,   \
                  const uint16_t   : GrB_Vector_assign_UINT16 ,   \
                        uint16_t   : GrB_Vector_assign_UINT16 ,   \
                  const int32_t    : GrB_Vector_assign_INT32  ,   \
                        int32_t    : GrB_Vector_assign_INT32  ,   \
                  const uint32_t   : GrB_Vector_assign_UINT32 ,   \
                        uint32_t   : GrB_Vector_assign_UINT32 ,   \
                  const int64_t    : GrB_Vector_assign_INT64  ,   \
                        int64_t    : GrB_Vector_assign_INT64  ,   \
                  const uint64_t   : GrB_Vector_assign_UINT64 ,   \
                        uint64_t   : GrB_Vector_assign_UINT64 ,   \
                  const float      : GrB_Vector_assign_FP32   ,   \
                        float      : GrB_Vector_assign_FP32   ,   \
                  const double     : GrB_Vector_assign_FP64   ,   \
                        double     : GrB_Vector_assign_FP64   ,   \
                  const void *     : GrB_Vector_assign_UDT    ,   \
                        void *     : GrB_Vector_assign_UDT    ,   \
                  default          : GrB_Vector_assign            \
              ),                                                  \
          default :                                               \
              _Generic                                            \
              (                                                   \
                  (arg4),                                         \
                  const bool       : GrB_Matrix_assign_BOOL   ,   \
                        bool       : GrB_Matrix_assign_BOOL   ,   \
                  const int8_t     : GrB_Matrix_assign_INT8   ,   \
                        int8_t     : GrB_Matrix_assign_INT8   ,   \
                  const uint8_t    : GrB_Matrix_assign_UINT8  ,   \
                        uint8_t    : GrB_Matrix_assign_UINT8  ,   \
                  const int16_t    : GrB_Matrix_assign_INT16  ,   \
                        int16_t    : GrB_Matrix_assign_INT16  ,   \
                  const uint16_t   : GrB_Matrix_assign_UINT16 ,   \
                        uint16_t   : GrB_Matrix_assign_UINT16 ,   \
                  const int32_t    : GrB_Matrix_assign_INT32  ,   \
                        int32_t    : GrB_Matrix_assign_INT32  ,   \
                  const uint32_t   : GrB_Matrix_assign_UINT32 ,   \
                        uint32_t   : GrB_Matrix_assign_UINT32 ,   \
                  const int64_t    : GrB_Matrix_assign_INT64  ,   \
                        int64_t    : GrB_Matrix_assign_INT64  ,   \
                  const uint64_t   : GrB_Matrix_assign_UINT64 ,   \
                        uint64_t   : GrB_Matrix_assign_UINT64 ,   \
                  const float      : GrB_Matrix_assign_FP32   ,   \
                        float      : GrB_Matrix_assign_FP32   ,   \
                  const double     : GrB_Matrix_assign_FP64   ,   \
                        double     : GrB_Matrix_assign_FP64   ,   \
                  const void *     : GrB_Matrix_assign_UDT    ,   \
                        void *     : GrB_Matrix_assign_UDT    ,   \
                  const GrB_Vector :                              \
                      _Generic                                    \
                      (                                           \
                          (arg5),                                 \
                          const GrB_Index *: GrB_Col_assign ,     \
                                GrB_Index *: GrB_Col_assign ,     \
                          default          : GrB_Row_assign       \
                      ),                                          \
                  GrB_Vector :                                    \
                      _Generic                                    \
                      (                                           \
                          (arg5),                                 \
                          const GrB_Index *: GrB_Col_assign ,     \
                                GrB_Index *: GrB_Col_assign ,     \
                          default          : GrB_Row_assign       \
                      ),                                          \
                  default    : GrB_Matrix_assign                  \
              )                                                   \
      )                                                           \
      (arg1, Mask, accum, arg4, arg5, __VA_ARGS__)
  
  
  //------------------------------------------------------------------------------
  // matrix and vector apply
  //------------------------------------------------------------------------------
  
  // Apply a unary operator to the entries in a matrix or vector,
  // C<Mask> = accum (C, op (A)).
  
  // The input matrix A may be optionally transposed first, via the descriptor.
  
  GrB_Info GrB_Vector_apply           // w<mask> = accum (w, op(u))
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_UnaryOp op,           // operator to apply to the entries
      const GrB_Vector u,             // first input:  vector u
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GrB_Matrix_apply           // C<Mask> = accum (C, op(A)) or op(A')
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GrB_UnaryOp op,           // operator to apply to the entries
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Descriptor desc       // descriptor for C, mask, and A
  ) ;
  
  //------------------------------------------------------------------------------
  // GrB_apply: generic matrix/vector apply
  //------------------------------------------------------------------------------
  
  // GrB_apply is a generic function for applying a unary operator to a matrix
  // or vector and provides access to these functions:
  
  // GrB_Vector_apply (w,mask,acc,op,u,d)  // w<mask> = accum (w, op(u))
  // GrB_Matrix_apply (C,Mask,acc,op,A,d)  // C<Mask> = accum (C, op(A))
  
  #define GrB_apply(C,Mask,accum,op,A,desc)       \
      _Generic                                    \
      (                                           \
          (C),                                    \
          GrB_Vector   : GrB_Vector_apply ,       \
          GrB_Matrix   : GrB_Matrix_apply         \
      )                                           \
      (C, Mask, accum, op, A, desc)
  
  //------------------------------------------------------------------------------
  // matrix and vector selection
  //------------------------------------------------------------------------------
  
  // Select a subset of entries from a matrix or vector.
  // C<Mask> = accum (C, op (A,k)), where the entries of op(A,k) are a subset of
  // the entries of A.
  
  // The input matrix A may be optionally transposed first, via the descriptor.
  
  GrB_Info GxB_Vector_select          // w<mask> = accum (w, op(u,k))
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GxB_SelectOp op,          // operator to apply to the entries
      const GrB_Vector u,             // first input:  vector u
      const void *k,                  // optional input for the select operator
      const GrB_Descriptor desc       // descriptor for w and mask
  ) ;
  
  GrB_Info GxB_Matrix_select          // C<Mask> = accum (C, op(A,k)) or op(A',k)
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GxB_SelectOp op,          // operator to apply to the entries
      const GrB_Matrix A,             // first input:  matrix A
      const void *k,                  // optional input for the select operator
      const GrB_Descriptor desc       // descriptor for C, mask, and A
  ) ;
  
  //------------------------------------------------------------------------------
  // GxB_select: generic matrix/vector select
  //------------------------------------------------------------------------------
  
  // GxB_select is a generic function for applying a select operator to a matrix
  // or vector and provides access to these functions:
  
  // GrB_Vector_select (w,mask,acc,op,u,k,d)  // w<mask> = accum (w, op(u,k))
  // GrB_Matrix_select (C,Mask,acc,op,A,k,d)  // C<Mask> = accum (C, op(A,k))
  
  #define GxB_select(C,Mask,accum,op,A,k,desc)    \
      _Generic                                    \
      (                                           \
          (C),                                    \
          GrB_Vector   : GxB_Vector_select ,      \
          GrB_Matrix   : GxB_Matrix_select        \
      )                                           \
      (C, Mask, accum, op, A, k, desc)
  
  //------------------------------------------------------------------------------
  // matrix and vector reduction
  //------------------------------------------------------------------------------
  
  // Reduce the entries in a matrix to a vector.  By default these methods
  // compute a column vector t such that t(i) = sum (A (i,:)), and where "sum" is
  // a commutative and associative monoid with an identity value.  A can be
  // transposed, which reduces down the columns instead of the rows.  This
  // behavior is the transpose of the MATLAB convention, where r=sum(A) produces
  // a row vector and sums each column.
  
  GrB_Info GrB_Matrix_reduce_Monoid   // w<mask> = accum (w,reduce(A))
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_Monoid monoid,        // reduce operator for t=reduce(A)
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Descriptor desc       // descriptor for w, mask, and A
  ) ;
  
  GrB_Info GrB_Matrix_reduce_BinaryOp // w<mask> = accum (w,reduce(A))
  (
      GrB_Vector w,                   // input/output vector for results
      const GrB_Vector mask,          // optional mask for w, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for z=accum(w,t)
      const GrB_BinaryOp op,          // reduce operator for t=reduce(A)
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Descriptor desc       // descriptor for w, mask, and A
  ) ;
  
  //------------------------------------------------------------------------------
  // reduce a vector to a scalar
  //------------------------------------------------------------------------------
  
  // Reduce entries in a vector to a scalar, c = accum (c, reduce_to_scalar(u))
  
  // All entries in the vector are "summed" to a single scalar t using the reduce
  // monoid, which must be associative (otherwise the results are undefined).
  // The result is either assigned to the output scalar c (if accum is NULL), or
  // it accumulated in the result c via c = accum(c,t).  If the vector has no
  // entries, the result t is the identity value of the monoid.  Unlike most
  // other GraphBLAS operations, this operation uses an accum operator but no
  // mask.
  
  // Like all GraphBLAS operations, these take a last argument of a GraphBLAS
  // descriptor.  However, it is unused in the current GraphBLAS spec.  It may be
  // used in the future.
  
  GrB_Info GrB_Vector_reduce_BOOL     // c = accum (c, reduce_to_scalar (u))
  (
      bool *c,                        // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Vector_reduce_INT8     // c = accum (c, reduce_to_scalar (u))
  (
      int8_t *c,                      // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Vector_reduce_UINT8    // c = accum (c, reduce_to_scalar (u))
  (
      uint8_t *c,                     // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Vector_reduce_INT16    // c = accum (c, reduce_to_scalar (u))
  (
      int16_t *c,                     // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Vector_reduce_UINT16   // c = accum (c, reduce_to_scalar (u))
  (
      uint16_t *c,                    // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Vector_reduce_INT32    // c = accum (c, reduce_to_scalar (u))
  (
      int32_t *c,                     // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Vector_reduce_UINT32   // c = accum (c, reduce_to_scalar (u))
  (
      uint32_t *c,                    // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Vector_reduce_INT64    // c = accum (c, reduce_to_scalar (u))
  (
      int64_t *c,                     // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Vector_reduce_UINT64   // c = accum (c, reduce_to_scalar (u))
  (
      uint64_t *c,                    // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Vector_reduce_FP32     // c = accum (c, reduce_to_scalar (u))
  (
      float *c,                       // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Vector_reduce_FP64     // c = accum (c, reduce_to_scalar (u))
  (
      double *c,                      // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Vector_reduce_UDT      // c = accum (c, reduce_to_scalar (u))
  (
      void *c,                        // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Vector u,             // vector to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  //------------------------------------------------------------------------------
  // reduce a matrix to a scalar
  //------------------------------------------------------------------------------
  
  // Reduce entries in a matrix to a scalar, c = accum (c, reduce_to_scalar(A))
  
  // All entries in the matrix are "summed" to a single scalar t using the reduce
  // monoid, which must be associative (otherwise the results are undefined).
  // The result is either assigned to the output scalar c (if accum is NULL), or
  // it accumulated in the result c via c = accum(c,t).  If the matrix has no
  // entries, the result t is the identity value of the monoid.  Unlike most
  // other GraphBLAS operations, this operation uses an accum operator but no
  // mask.
  
  GrB_Info GrB_Matrix_reduce_BOOL     // c = accum (c, reduce_to_scalar (A))
  (
      bool *c,                        // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Matrix_reduce_INT8     // c = accum (c, reduce_to_scalar (A))
  (
      int8_t *c,                      // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Matrix_reduce_UINT8    // c = accum (c, reduce_to_scalar (A))
  (
      uint8_t *c,                     // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Matrix_reduce_INT16    // c = accum (c, reduce_to_scalar (A))
  (
      int16_t *c,                     // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Matrix_reduce_UINT16   // c = accum (c, reduce_to_scalar (A))
  (
      uint16_t *c,                    // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Matrix_reduce_INT32    // c = accum (c, reduce_to_scalar (A))
  (
      int32_t *c,                     // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Matrix_reduce_UINT32   // c = accum (c, reduce_to_scalar (A))
  (
      uint32_t *c,                    // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Matrix_reduce_INT64    // c = accum (c, reduce_to_scalar (A))
  (
      int64_t *c,                     // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Matrix_reduce_UINT64   // c = accum (c, reduce_to_scalar (A))
  (
      uint64_t *c,                    // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Matrix_reduce_FP32     // c = accum (c, reduce_to_scalar (A))
  (
      float *c,                       // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Matrix_reduce_FP64     // c = accum (c, reduce_to_scalar (A))
  (
      double *c,                      // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  GrB_Info GrB_Matrix_reduce_UDT      // c = accum (c, reduce_to_scalar (A))
  (
      void *c,                        // result scalar
      const GrB_BinaryOp accum,       // optional accum for c=accum(c,t)
      const GrB_Monoid monoid,        // monoid to do the reduction
      const GrB_Matrix A,             // matrix to reduce
      const GrB_Descriptor desc       // descriptor (currently unused)
  ) ;
  
  //------------------------------------------------------------------------------
  // GrB_reduce: generic matrix/vector reduction to a vector or scalar
  //------------------------------------------------------------------------------
  
  // GrB_reduce is a generic function that provides access to all GrB_*reduce*
  // functions:
  
  // reduce matrix to vector:
  // GrB_Matrix_reduce_Monoid   (w,mask,acc,mo,A,d) // w<mask> = acc (w,reduce(A))
  // GrB_Matrix_reduce_BinaryOp (w,mask,acc,op,A,d) // w<mask> = acc (w,reduce(A))
  // GrB_Vector_reduce_[SCALAR] (c, acc,monoid,u, d)
  // GrB_Matrix_reduce_[SCALAR] (c, acc,monoid,A, d)
  
  #define GrB_reduce(arg1,arg2,arg3,arg4,...)                 \
      _Generic                                                \
      (                                                       \
          (arg4),                                             \
          const GrB_Vector :                                  \
              _Generic                                        \
              (                                               \
                  (arg1),                                     \
                  bool     * : GrB_Vector_reduce_BOOL   ,     \
                  int8_t   * : GrB_Vector_reduce_INT8   ,     \
                  uint8_t  * : GrB_Vector_reduce_UINT8  ,     \
                  int16_t  * : GrB_Vector_reduce_INT16  ,     \
                  uint16_t * : GrB_Vector_reduce_UINT16 ,     \
                  int32_t  * : GrB_Vector_reduce_INT32  ,     \
                  uint32_t * : GrB_Vector_reduce_UINT32 ,     \
                  int64_t  * : GrB_Vector_reduce_INT64  ,     \
                  uint64_t * : GrB_Vector_reduce_UINT64 ,     \
                  float    * : GrB_Vector_reduce_FP32   ,     \
                  double   * : GrB_Vector_reduce_FP64   ,     \
                  default    : GrB_Vector_reduce_UDT          \
              ),                                              \
          GrB_Vector :                                        \
              _Generic                                        \
              (                                               \
                  (arg1),                                     \
                  bool     * : GrB_Vector_reduce_BOOL   ,     \
                  int8_t   * : GrB_Vector_reduce_INT8   ,     \
                  uint8_t  * : GrB_Vector_reduce_UINT8  ,     \
                  int16_t  * : GrB_Vector_reduce_INT16  ,     \
                  uint16_t * : GrB_Vector_reduce_UINT16 ,     \
                  int32_t  * : GrB_Vector_reduce_INT32  ,     \
                  uint32_t * : GrB_Vector_reduce_UINT32 ,     \
                  int64_t  * : GrB_Vector_reduce_INT64  ,     \
                  uint64_t * : GrB_Vector_reduce_UINT64 ,     \
                  float    * : GrB_Vector_reduce_FP32   ,     \
                  double   * : GrB_Vector_reduce_FP64   ,     \
                  default    : GrB_Vector_reduce_UDT          \
              ),                                              \
          const GrB_Matrix :                                  \
              _Generic                                        \
              (                                               \
                  (arg1),                                     \
                  bool     * : GrB_Matrix_reduce_BOOL   ,     \
                  int8_t   * : GrB_Matrix_reduce_INT8   ,     \
                  uint8_t  * : GrB_Matrix_reduce_UINT8  ,     \
                  int16_t  * : GrB_Matrix_reduce_INT16  ,     \
                  uint16_t * : GrB_Matrix_reduce_UINT16 ,     \
                  int32_t  * : GrB_Matrix_reduce_INT32  ,     \
                  uint32_t * : GrB_Matrix_reduce_UINT32 ,     \
                  int64_t  * : GrB_Matrix_reduce_INT64  ,     \
                  uint64_t * : GrB_Matrix_reduce_UINT64 ,     \
                  float    * : GrB_Matrix_reduce_FP32   ,     \
                  double   * : GrB_Matrix_reduce_FP64   ,     \
                  default    : GrB_Matrix_reduce_UDT          \
              ),                                              \
          GrB_Matrix :                                        \
              _Generic                                        \
              (                                               \
                  (arg1),                                     \
                  bool     * : GrB_Matrix_reduce_BOOL   ,     \
                  int8_t   * : GrB_Matrix_reduce_INT8   ,     \
                  uint8_t  * : GrB_Matrix_reduce_UINT8  ,     \
                  int16_t  * : GrB_Matrix_reduce_INT16  ,     \
                  uint16_t * : GrB_Matrix_reduce_UINT16 ,     \
                  int32_t  * : GrB_Matrix_reduce_INT32  ,     \
                  uint32_t * : GrB_Matrix_reduce_UINT32 ,     \
                  int64_t  * : GrB_Matrix_reduce_INT64  ,     \
                  uint64_t * : GrB_Matrix_reduce_UINT64 ,     \
                  float    * : GrB_Matrix_reduce_FP32   ,     \
                  double   * : GrB_Matrix_reduce_FP64   ,     \
                  default    : GrB_Matrix_reduce_UDT          \
              ),                                              \
          const GrB_Monoid   : GrB_Matrix_reduce_Monoid   ,   \
                GrB_Monoid   : GrB_Matrix_reduce_Monoid   ,   \
          const GrB_BinaryOp : GrB_Matrix_reduce_BinaryOp ,   \
                GrB_BinaryOp : GrB_Matrix_reduce_BinaryOp     \
      )                                                       \
      (arg1, arg2, arg3, arg4, __VA_ARGS__)
  
  //------------------------------------------------------------------------------
  // matrix transpose
  //------------------------------------------------------------------------------
  
  // T = A' is computed by default, but A can also be transposed via the
  // descriptor.  In this case A is not transposed at all, and T = A.  The result
  // is then passed through the Mask and accum, like almost all other GraphBLAS
  // operations.  This makes GrB_transpose a direct interface to the accum/mask
  // operation, C<Mask> = accum (C,A), or C<Mask> = accum (C,A') by default.
  
  GrB_Info GrB_transpose              // C<Mask> = accum (C, A')
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Descriptor desc       // descriptor for C, Mask, and A
  ) ;
  
  
  
  //==============================================================================
  // additional predefined objects
  //==============================================================================
  
  // SPEC: predefined monoids and semirings are extensions to the spec
  
  //------------------------------------------------------------------------------
  // built-in monoids
  //------------------------------------------------------------------------------
  
  // 44 unique monoids can be constructed using built-in types and operators, all
  // of which are defined below.  Four operators (min, max, plus, times) are
  // available for each of the 10 non-Boolean types, and four purely Boolean
  // monoids are available.
  
  extern GrB_Monoid
  
      // MIN monoids:
      GxB_MIN_INT8_MONOID,          // identity: INT8_MAX
      GxB_MIN_UINT8_MONOID,         // identity: UINT8_MAX
      GxB_MIN_INT16_MONOID,         // identity: INT16_MAX
      GxB_MIN_UINT16_MONOID,        // identity: UINT16_MAX
      GxB_MIN_INT32_MONOID,         // identity: INT32_MAX
      GxB_MIN_UINT32_MONOID,        // identity: UINT32_MAX
      GxB_MIN_INT64_MONOID,         // identity: INT64_MAX
      GxB_MIN_UINT64_MONOID,        // identity: UINT64_MAX
      GxB_MIN_FP32_MONOID,          // identity: INFINITY
      GxB_MIN_FP64_MONOID,          // identity: INFINITY
  
      // MAX monoids:
      GxB_MAX_INT8_MONOID,          // identity: INT8_MIN
      GxB_MAX_UINT8_MONOID,         // identity: 0
      GxB_MAX_INT16_MONOID,         // identity: INT16_MIN
      GxB_MAX_UINT16_MONOID,        // identity: 0
      GxB_MAX_INT32_MONOID,         // identity: INT32_MIN
      GxB_MAX_UINT32_MONOID,        // identity: 0
      GxB_MAX_INT64_MONOID,         // identity: INT64_MIN
      GxB_MAX_UINT64_MONOID,        // identity: 0
      GxB_MAX_FP32_MONOID,          // identity: -INFINITY
      GxB_MAX_FP64_MONOID,          // identity: -INFINITY
  
      // PLUS monoids:
      GxB_PLUS_INT8_MONOID,         // identity: 0
      GxB_PLUS_UINT8_MONOID,        // identity: 0
      GxB_PLUS_INT16_MONOID,        // identity: 0
      GxB_PLUS_UINT16_MONOID,       // identity: 0
      GxB_PLUS_INT32_MONOID,        // identity: 0
      GxB_PLUS_UINT32_MONOID,       // identity: 0
      GxB_PLUS_INT64_MONOID,        // identity: 0
      GxB_PLUS_UINT64_MONOID,       // identity: 0
      GxB_PLUS_FP32_MONOID,         // identity: 0
      GxB_PLUS_FP64_MONOID,         // identity: 0
  
      // TIMES monoids:
      GxB_TIMES_INT8_MONOID,        // identity: 1
      GxB_TIMES_UINT8_MONOID,       // identity: 1
      GxB_TIMES_INT16_MONOID,       // identity: 1
      GxB_TIMES_UINT16_MONOID,      // identity: 1
      GxB_TIMES_INT32_MONOID,       // identity: 1
      GxB_TIMES_UINT32_MONOID,      // identity: 1
      GxB_TIMES_INT64_MONOID,       // identity: 1
      GxB_TIMES_UINT64_MONOID,      // identity: 1
      GxB_TIMES_FP32_MONOID,        // identity: 1
      GxB_TIMES_FP64_MONOID,        // identity: 1
  
      // Boolean monoids:
      GxB_LOR_BOOL_MONOID,          // identity: false
      GxB_LAND_BOOL_MONOID,         // identity: true
      GxB_LXOR_BOOL_MONOID,         // identity: false
      GxB_EQ_BOOL_MONOID ;          // identity: true
  
  //------------------------------------------------------------------------------
  // built-in semirings
  //------------------------------------------------------------------------------
  
  // Using built-in types and operators, 960 unique semirings can be built.  This
  // count excludes redundant Boolean operators (for example GxB_TIMES_BOOL and
  // GxB_LAND_BOOL are different operators but they are redundant since they
  // always return the same result):
  
  // 680 semirings with a multiply operator TxT -> T where T is non-Boolean, from
  // the complete cross product of:
  
  //      4 add monoids (MIN, MAX, PLUS, TIMES)
  //      17 multiply operators:
  //          (FIRST, SECOND, MIN, MAX, PLUS, MINUS, TIMES, DIV,
  //           ISEQ, ISNE, ISGT, ISLT, ISGE, ISLE,
  //           LOR, LAND, LXOR)
  //      10 non-Boolean types, T
  
  // 240 semirings with a comparison operator TxT -> bool, where T is
  // non-Boolean, from the complete cross product of:
  
  //      4 Boolean add monoids: (LAND, LOR, LXOR, EQ)
  //      6 multiply operators: (EQ, NE, GT, LT, GE, LE)
  //      10 non-Boolean types, T
  
  // 40 semirings with purely Boolean types, bool x bool -> bool, from the
  // complete cross product of:
  
  //      4 Boolean add monoids (LAND, LOR, LXOR, EQ)
  //      10 multiply operators:
  //          (FIRST, SECOND, LOR, LAND, LXOR, EQ, GT, LT, GE, LE)
  
  // In the names below, each semiring has a name of the form GxB_add_mult_T
  // where add is the additive monoid, mult is the multiply operator, and T is
  // the type.  The type T is always the type of x and y for the z=mult(x,y)
  // operator.  The monoid's three types and the ztype of the mult operator are
  // always the same.  This is the type T for the first set, and Boolean for
  // the second and third sets of semirngs.
  
  extern GrB_Semiring
  
  //------------------------------------------------------------------------------
  // 680 non-Boolean semirings where all types are the same, given by suffix _T
  //------------------------------------------------------------------------------
  
  // semirings with multiply op: z = FIRST (x,y), all types x,y,z the same:
  GxB_MIN_FIRST_INT8     , GxB_MAX_FIRST_INT8     , GxB_PLUS_FIRST_INT8    , GxB_TIMES_FIRST_INT8   ,
  GxB_MIN_FIRST_UINT8    , GxB_MAX_FIRST_UINT8    , GxB_PLUS_FIRST_UINT8   , GxB_TIMES_FIRST_UINT8  ,
  GxB_MIN_FIRST_INT16    , GxB_MAX_FIRST_INT16    , GxB_PLUS_FIRST_INT16   , GxB_TIMES_FIRST_INT16  ,
  GxB_MIN_FIRST_UINT16   , GxB_MAX_FIRST_UINT16   , GxB_PLUS_FIRST_UINT16  , GxB_TIMES_FIRST_UINT16 ,
  GxB_MIN_FIRST_INT32    , GxB_MAX_FIRST_INT32    , GxB_PLUS_FIRST_INT32   , GxB_TIMES_FIRST_INT32  ,
  GxB_MIN_FIRST_UINT32   , GxB_MAX_FIRST_UINT32   , GxB_PLUS_FIRST_UINT32  , GxB_TIMES_FIRST_UINT32 ,
  GxB_MIN_FIRST_INT64    , GxB_MAX_FIRST_INT64    , GxB_PLUS_FIRST_INT64   , GxB_TIMES_FIRST_INT64  ,
  GxB_MIN_FIRST_UINT64   , GxB_MAX_FIRST_UINT64   , GxB_PLUS_FIRST_UINT64  , GxB_TIMES_FIRST_UINT64 ,
  GxB_MIN_FIRST_FP32     , GxB_MAX_FIRST_FP32     , GxB_PLUS_FIRST_FP32    , GxB_TIMES_FIRST_FP32   ,
  GxB_MIN_FIRST_FP64     , GxB_MAX_FIRST_FP64     , GxB_PLUS_FIRST_FP64    , GxB_TIMES_FIRST_FP64   ,
  
  // semirings with multiply op: z = SECOND (x,y), all types x,y,z the same:
  GxB_MIN_SECOND_INT8    , GxB_MAX_SECOND_INT8    , GxB_PLUS_SECOND_INT8   , GxB_TIMES_SECOND_INT8  ,
  GxB_MIN_SECOND_UINT8   , GxB_MAX_SECOND_UINT8   , GxB_PLUS_SECOND_UINT8  , GxB_TIMES_SECOND_UINT8 ,
  GxB_MIN_SECOND_INT16   , GxB_MAX_SECOND_INT16   , GxB_PLUS_SECOND_INT16  , GxB_TIMES_SECOND_INT16 ,
  GxB_MIN_SECOND_UINT16  , GxB_MAX_SECOND_UINT16  , GxB_PLUS_SECOND_UINT16 , GxB_TIMES_SECOND_UINT16,
  GxB_MIN_SECOND_INT32   , GxB_MAX_SECOND_INT32   , GxB_PLUS_SECOND_INT32  , GxB_TIMES_SECOND_INT32 ,
  GxB_MIN_SECOND_UINT32  , GxB_MAX_SECOND_UINT32  , GxB_PLUS_SECOND_UINT32 , GxB_TIMES_SECOND_UINT32,
  GxB_MIN_SECOND_INT64   , GxB_MAX_SECOND_INT64   , GxB_PLUS_SECOND_INT64  , GxB_TIMES_SECOND_INT64 ,
  GxB_MIN_SECOND_UINT64  , GxB_MAX_SECOND_UINT64  , GxB_PLUS_SECOND_UINT64 , GxB_TIMES_SECOND_UINT64,
  GxB_MIN_SECOND_FP32    , GxB_MAX_SECOND_FP32    , GxB_PLUS_SECOND_FP32   , GxB_TIMES_SECOND_FP32  ,
  GxB_MIN_SECOND_FP64    , GxB_MAX_SECOND_FP64    , GxB_PLUS_SECOND_FP64   , GxB_TIMES_SECOND_FP64  ,
  
  // semirings with multiply op: z = MIN (x,y), all types x,y,z the same:
  GxB_MIN_MIN_INT8       , GxB_MAX_MIN_INT8       , GxB_PLUS_MIN_INT8      , GxB_TIMES_MIN_INT8     ,
  GxB_MIN_MIN_UINT8      , GxB_MAX_MIN_UINT8      , GxB_PLUS_MIN_UINT8     , GxB_TIMES_MIN_UINT8    ,
  GxB_MIN_MIN_INT16      , GxB_MAX_MIN_INT16      , GxB_PLUS_MIN_INT16     , GxB_TIMES_MIN_INT16    ,
  GxB_MIN_MIN_UINT16     , GxB_MAX_MIN_UINT16     , GxB_PLUS_MIN_UINT16    , GxB_TIMES_MIN_UINT16   ,
  GxB_MIN_MIN_INT32      , GxB_MAX_MIN_INT32      , GxB_PLUS_MIN_INT32     , GxB_TIMES_MIN_INT32    ,
  GxB_MIN_MIN_UINT32     , GxB_MAX_MIN_UINT32     , GxB_PLUS_MIN_UINT32    , GxB_TIMES_MIN_UINT32   ,
  GxB_MIN_MIN_INT64      , GxB_MAX_MIN_INT64      , GxB_PLUS_MIN_INT64     , GxB_TIMES_MIN_INT64    ,
  GxB_MIN_MIN_UINT64     , GxB_MAX_MIN_UINT64     , GxB_PLUS_MIN_UINT64    , GxB_TIMES_MIN_UINT64   ,
  GxB_MIN_MIN_FP32       , GxB_MAX_MIN_FP32       , GxB_PLUS_MIN_FP32      , GxB_TIMES_MIN_FP32     ,
  GxB_MIN_MIN_FP64       , GxB_MAX_MIN_FP64       , GxB_PLUS_MIN_FP64      , GxB_TIMES_MIN_FP64     ,
  
  // semirings with multiply op: z = MAX (x,y), all types x,y,z the same:
  GxB_MIN_MAX_INT8       , GxB_MAX_MAX_INT8       , GxB_PLUS_MAX_INT8      , GxB_TIMES_MAX_INT8     ,
  GxB_MIN_MAX_UINT8      , GxB_MAX_MAX_UINT8      , GxB_PLUS_MAX_UINT8     , GxB_TIMES_MAX_UINT8    ,
  GxB_MIN_MAX_INT16      , GxB_MAX_MAX_INT16      , GxB_PLUS_MAX_INT16     , GxB_TIMES_MAX_INT16    ,
  GxB_MIN_MAX_UINT16     , GxB_MAX_MAX_UINT16     , GxB_PLUS_MAX_UINT16    , GxB_TIMES_MAX_UINT16   ,
  GxB_MIN_MAX_INT32      , GxB_MAX_MAX_INT32      , GxB_PLUS_MAX_INT32     , GxB_TIMES_MAX_INT32    ,
  GxB_MIN_MAX_UINT32     , GxB_MAX_MAX_UINT32     , GxB_PLUS_MAX_UINT32    , GxB_TIMES_MAX_UINT32   ,
  GxB_MIN_MAX_INT64      , GxB_MAX_MAX_INT64      , GxB_PLUS_MAX_INT64     , GxB_TIMES_MAX_INT64    ,
  GxB_MIN_MAX_UINT64     , GxB_MAX_MAX_UINT64     , GxB_PLUS_MAX_UINT64    , GxB_TIMES_MAX_UINT64   ,
  GxB_MIN_MAX_FP32       , GxB_MAX_MAX_FP32       , GxB_PLUS_MAX_FP32      , GxB_TIMES_MAX_FP32     ,
  GxB_MIN_MAX_FP64       , GxB_MAX_MAX_FP64       , GxB_PLUS_MAX_FP64      , GxB_TIMES_MAX_FP64     ,
  
  // semirings with multiply op: z = PLUS (x,y), all types x,y,z the same:
  GxB_MIN_PLUS_INT8      , GxB_MAX_PLUS_INT8      , GxB_PLUS_PLUS_INT8     , GxB_TIMES_PLUS_INT8    ,
  GxB_MIN_PLUS_UINT8     , GxB_MAX_PLUS_UINT8     , GxB_PLUS_PLUS_UINT8    , GxB_TIMES_PLUS_UINT8   ,
  GxB_MIN_PLUS_INT16     , GxB_MAX_PLUS_INT16     , GxB_PLUS_PLUS_INT16    , GxB_TIMES_PLUS_INT16   ,
  GxB_MIN_PLUS_UINT16    , GxB_MAX_PLUS_UINT16    , GxB_PLUS_PLUS_UINT16   , GxB_TIMES_PLUS_UINT16  ,
  GxB_MIN_PLUS_INT32     , GxB_MAX_PLUS_INT32     , GxB_PLUS_PLUS_INT32    , GxB_TIMES_PLUS_INT32   ,
  GxB_MIN_PLUS_UINT32    , GxB_MAX_PLUS_UINT32    , GxB_PLUS_PLUS_UINT32   , GxB_TIMES_PLUS_UINT32  ,
  GxB_MIN_PLUS_INT64     , GxB_MAX_PLUS_INT64     , GxB_PLUS_PLUS_INT64    , GxB_TIMES_PLUS_INT64   ,
  GxB_MIN_PLUS_UINT64    , GxB_MAX_PLUS_UINT64    , GxB_PLUS_PLUS_UINT64   , GxB_TIMES_PLUS_UINT64  ,
  GxB_MIN_PLUS_FP32      , GxB_MAX_PLUS_FP32      , GxB_PLUS_PLUS_FP32     , GxB_TIMES_PLUS_FP32    ,
  GxB_MIN_PLUS_FP64      , GxB_MAX_PLUS_FP64      , GxB_PLUS_PLUS_FP64     , GxB_TIMES_PLUS_FP64    ,
  
  // semirings with multiply op: z = MINUS (x,y), all types x,y,z the same:
  GxB_MIN_MINUS_INT8     , GxB_MAX_MINUS_INT8     , GxB_PLUS_MINUS_INT8    , GxB_TIMES_MINUS_INT8   ,
  GxB_MIN_MINUS_UINT8    , GxB_MAX_MINUS_UINT8    , GxB_PLUS_MINUS_UINT8   , GxB_TIMES_MINUS_UINT8  ,
  GxB_MIN_MINUS_INT16    , GxB_MAX_MINUS_INT16    , GxB_PLUS_MINUS_INT16   , GxB_TIMES_MINUS_INT16  ,
  GxB_MIN_MINUS_UINT16   , GxB_MAX_MINUS_UINT16   , GxB_PLUS_MINUS_UINT16  , GxB_TIMES_MINUS_UINT16 ,
  GxB_MIN_MINUS_INT32    , GxB_MAX_MINUS_INT32    , GxB_PLUS_MINUS_INT32   , GxB_TIMES_MINUS_INT32  ,
  GxB_MIN_MINUS_UINT32   , GxB_MAX_MINUS_UINT32   , GxB_PLUS_MINUS_UINT32  , GxB_TIMES_MINUS_UINT32 ,
  GxB_MIN_MINUS_INT64    , GxB_MAX_MINUS_INT64    , GxB_PLUS_MINUS_INT64   , GxB_TIMES_MINUS_INT64  ,
  GxB_MIN_MINUS_UINT64   , GxB_MAX_MINUS_UINT64   , GxB_PLUS_MINUS_UINT64  , GxB_TIMES_MINUS_UINT64 ,
  GxB_MIN_MINUS_FP32     , GxB_MAX_MINUS_FP32     , GxB_PLUS_MINUS_FP32    , GxB_TIMES_MINUS_FP32   ,
  GxB_MIN_MINUS_FP64     , GxB_MAX_MINUS_FP64     , GxB_PLUS_MINUS_FP64    , GxB_TIMES_MINUS_FP64   ,
  
  // semirings with multiply op: z = TIMES (x,y), all types x,y,z the same:
  GxB_MIN_TIMES_INT8     , GxB_MAX_TIMES_INT8     , GxB_PLUS_TIMES_INT8    , GxB_TIMES_TIMES_INT8   ,
  GxB_MIN_TIMES_UINT8    , GxB_MAX_TIMES_UINT8    , GxB_PLUS_TIMES_UINT8   , GxB_TIMES_TIMES_UINT8  ,
  GxB_MIN_TIMES_INT16    , GxB_MAX_TIMES_INT16    , GxB_PLUS_TIMES_INT16   , GxB_TIMES_TIMES_INT16  ,
  GxB_MIN_TIMES_UINT16   , GxB_MAX_TIMES_UINT16   , GxB_PLUS_TIMES_UINT16  , GxB_TIMES_TIMES_UINT16 ,
  GxB_MIN_TIMES_INT32    , GxB_MAX_TIMES_INT32    , GxB_PLUS_TIMES_INT32   , GxB_TIMES_TIMES_INT32  ,
  GxB_MIN_TIMES_UINT32   , GxB_MAX_TIMES_UINT32   , GxB_PLUS_TIMES_UINT32  , GxB_TIMES_TIMES_UINT32 ,
  GxB_MIN_TIMES_INT64    , GxB_MAX_TIMES_INT64    , GxB_PLUS_TIMES_INT64   , GxB_TIMES_TIMES_INT64  ,
  GxB_MIN_TIMES_UINT64   , GxB_MAX_TIMES_UINT64   , GxB_PLUS_TIMES_UINT64  , GxB_TIMES_TIMES_UINT64 ,
  GxB_MIN_TIMES_FP32     , GxB_MAX_TIMES_FP32     , GxB_PLUS_TIMES_FP32    , GxB_TIMES_TIMES_FP32   ,
  GxB_MIN_TIMES_FP64     , GxB_MAX_TIMES_FP64     , GxB_PLUS_TIMES_FP64    , GxB_TIMES_TIMES_FP64   ,
  
  // semirings with multiply op: z = DIV (x,y), all types x,y,z the same:
  GxB_MIN_DIV_INT8       , GxB_MAX_DIV_INT8       , GxB_PLUS_DIV_INT8      , GxB_TIMES_DIV_INT8     ,
  GxB_MIN_DIV_UINT8      , GxB_MAX_DIV_UINT8      , GxB_PLUS_DIV_UINT8     , GxB_TIMES_DIV_UINT8    ,
  GxB_MIN_DIV_INT16      , GxB_MAX_DIV_INT16      , GxB_PLUS_DIV_INT16     , GxB_TIMES_DIV_INT16    ,
  GxB_MIN_DIV_UINT16     , GxB_MAX_DIV_UINT16     , GxB_PLUS_DIV_UINT16    , GxB_TIMES_DIV_UINT16   ,
  GxB_MIN_DIV_INT32      , GxB_MAX_DIV_INT32      , GxB_PLUS_DIV_INT32     , GxB_TIMES_DIV_INT32    ,
  GxB_MIN_DIV_UINT32     , GxB_MAX_DIV_UINT32     , GxB_PLUS_DIV_UINT32    , GxB_TIMES_DIV_UINT32   ,
  GxB_MIN_DIV_INT64      , GxB_MAX_DIV_INT64      , GxB_PLUS_DIV_INT64     , GxB_TIMES_DIV_INT64    ,
  GxB_MIN_DIV_UINT64     , GxB_MAX_DIV_UINT64     , GxB_PLUS_DIV_UINT64    , GxB_TIMES_DIV_UINT64   ,
  GxB_MIN_DIV_FP32       , GxB_MAX_DIV_FP32       , GxB_PLUS_DIV_FP32      , GxB_TIMES_DIV_FP32     ,
  GxB_MIN_DIV_FP64       , GxB_MAX_DIV_FP64       , GxB_PLUS_DIV_FP64      , GxB_TIMES_DIV_FP64     ,
  
  // semirings with multiply op: z = ISEQ (x,y), all types x,y,z the same:
  GxB_MIN_ISEQ_INT8      , GxB_MAX_ISEQ_INT8      , GxB_PLUS_ISEQ_INT8     , GxB_TIMES_ISEQ_INT8    ,
  GxB_MIN_ISEQ_UINT8     , GxB_MAX_ISEQ_UINT8     , GxB_PLUS_ISEQ_UINT8    , GxB_TIMES_ISEQ_UINT8   ,
  GxB_MIN_ISEQ_INT16     , GxB_MAX_ISEQ_INT16     , GxB_PLUS_ISEQ_INT16    , GxB_TIMES_ISEQ_INT16   ,
  GxB_MIN_ISEQ_UINT16    , GxB_MAX_ISEQ_UINT16    , GxB_PLUS_ISEQ_UINT16   , GxB_TIMES_ISEQ_UINT16  ,
  GxB_MIN_ISEQ_INT32     , GxB_MAX_ISEQ_INT32     , GxB_PLUS_ISEQ_INT32    , GxB_TIMES_ISEQ_INT32   ,
  GxB_MIN_ISEQ_UINT32    , GxB_MAX_ISEQ_UINT32    , GxB_PLUS_ISEQ_UINT32   , GxB_TIMES_ISEQ_UINT32  ,
  GxB_MIN_ISEQ_INT64     , GxB_MAX_ISEQ_INT64     , GxB_PLUS_ISEQ_INT64    , GxB_TIMES_ISEQ_INT64   ,
  GxB_MIN_ISEQ_UINT64    , GxB_MAX_ISEQ_UINT64    , GxB_PLUS_ISEQ_UINT64   , GxB_TIMES_ISEQ_UINT64  ,
  GxB_MIN_ISEQ_FP32      , GxB_MAX_ISEQ_FP32      , GxB_PLUS_ISEQ_FP32     , GxB_TIMES_ISEQ_FP32    ,
  GxB_MIN_ISEQ_FP64      , GxB_MAX_ISEQ_FP64      , GxB_PLUS_ISEQ_FP64     , GxB_TIMES_ISEQ_FP64    ,
  
  // semirings with multiply op: z = ISNE (x,y), all types x,y,z the same:
  GxB_MIN_ISNE_INT8      , GxB_MAX_ISNE_INT8      , GxB_PLUS_ISNE_INT8     , GxB_TIMES_ISNE_INT8    ,
  GxB_MIN_ISNE_UINT8     , GxB_MAX_ISNE_UINT8     , GxB_PLUS_ISNE_UINT8    , GxB_TIMES_ISNE_UINT8   ,
  GxB_MIN_ISNE_INT16     , GxB_MAX_ISNE_INT16     , GxB_PLUS_ISNE_INT16    , GxB_TIMES_ISNE_INT16   ,
  GxB_MIN_ISNE_UINT16    , GxB_MAX_ISNE_UINT16    , GxB_PLUS_ISNE_UINT16   , GxB_TIMES_ISNE_UINT16  ,
  GxB_MIN_ISNE_INT32     , GxB_MAX_ISNE_INT32     , GxB_PLUS_ISNE_INT32    , GxB_TIMES_ISNE_INT32   ,
  GxB_MIN_ISNE_UINT32    , GxB_MAX_ISNE_UINT32    , GxB_PLUS_ISNE_UINT32   , GxB_TIMES_ISNE_UINT32  ,
  GxB_MIN_ISNE_INT64     , GxB_MAX_ISNE_INT64     , GxB_PLUS_ISNE_INT64    , GxB_TIMES_ISNE_INT64   ,
  GxB_MIN_ISNE_UINT64    , GxB_MAX_ISNE_UINT64    , GxB_PLUS_ISNE_UINT64   , GxB_TIMES_ISNE_UINT64  ,
  GxB_MIN_ISNE_FP32      , GxB_MAX_ISNE_FP32      , GxB_PLUS_ISNE_FP32     , GxB_TIMES_ISNE_FP32    ,
  GxB_MIN_ISNE_FP64      , GxB_MAX_ISNE_FP64      , GxB_PLUS_ISNE_FP64     , GxB_TIMES_ISNE_FP64    ,
  
  // semirings with multiply op: z = ISGT (x,y), all types x,y,z the same:
  GxB_MIN_ISGT_INT8      , GxB_MAX_ISGT_INT8      , GxB_PLUS_ISGT_INT8     , GxB_TIMES_ISGT_INT8    ,
  GxB_MIN_ISGT_UINT8     , GxB_MAX_ISGT_UINT8     , GxB_PLUS_ISGT_UINT8    , GxB_TIMES_ISGT_UINT8   ,
  GxB_MIN_ISGT_INT16     , GxB_MAX_ISGT_INT16     , GxB_PLUS_ISGT_INT16    , GxB_TIMES_ISGT_INT16   ,
  GxB_MIN_ISGT_UINT16    , GxB_MAX_ISGT_UINT16    , GxB_PLUS_ISGT_UINT16   , GxB_TIMES_ISGT_UINT16  ,
  GxB_MIN_ISGT_INT32     , GxB_MAX_ISGT_INT32     , GxB_PLUS_ISGT_INT32    , GxB_TIMES_ISGT_INT32   ,
  GxB_MIN_ISGT_UINT32    , GxB_MAX_ISGT_UINT32    , GxB_PLUS_ISGT_UINT32   , GxB_TIMES_ISGT_UINT32  ,
  GxB_MIN_ISGT_INT64     , GxB_MAX_ISGT_INT64     , GxB_PLUS_ISGT_INT64    , GxB_TIMES_ISGT_INT64   ,
  GxB_MIN_ISGT_UINT64    , GxB_MAX_ISGT_UINT64    , GxB_PLUS_ISGT_UINT64   , GxB_TIMES_ISGT_UINT64  ,
  GxB_MIN_ISGT_FP32      , GxB_MAX_ISGT_FP32      , GxB_PLUS_ISGT_FP32     , GxB_TIMES_ISGT_FP32    ,
  GxB_MIN_ISGT_FP64      , GxB_MAX_ISGT_FP64      , GxB_PLUS_ISGT_FP64     , GxB_TIMES_ISGT_FP64    ,
  
  // semirings with multiply op: z = ISLT (x,y), all types x,y,z the same:
  GxB_MIN_ISLT_INT8      , GxB_MAX_ISLT_INT8      , GxB_PLUS_ISLT_INT8     , GxB_TIMES_ISLT_INT8    ,
  GxB_MIN_ISLT_UINT8     , GxB_MAX_ISLT_UINT8     , GxB_PLUS_ISLT_UINT8    , GxB_TIMES_ISLT_UINT8   ,
  GxB_MIN_ISLT_INT16     , GxB_MAX_ISLT_INT16     , GxB_PLUS_ISLT_INT16    , GxB_TIMES_ISLT_INT16   ,
  GxB_MIN_ISLT_UINT16    , GxB_MAX_ISLT_UINT16    , GxB_PLUS_ISLT_UINT16   , GxB_TIMES_ISLT_UINT16  ,
  GxB_MIN_ISLT_INT32     , GxB_MAX_ISLT_INT32     , GxB_PLUS_ISLT_INT32    , GxB_TIMES_ISLT_INT32   ,
  GxB_MIN_ISLT_UINT32    , GxB_MAX_ISLT_UINT32    , GxB_PLUS_ISLT_UINT32   , GxB_TIMES_ISLT_UINT32  ,
  GxB_MIN_ISLT_INT64     , GxB_MAX_ISLT_INT64     , GxB_PLUS_ISLT_INT64    , GxB_TIMES_ISLT_INT64   ,
  GxB_MIN_ISLT_UINT64    , GxB_MAX_ISLT_UINT64    , GxB_PLUS_ISLT_UINT64   , GxB_TIMES_ISLT_UINT64  ,
  GxB_MIN_ISLT_FP32      , GxB_MAX_ISLT_FP32      , GxB_PLUS_ISLT_FP32     , GxB_TIMES_ISLT_FP32    ,
  GxB_MIN_ISLT_FP64      , GxB_MAX_ISLT_FP64      , GxB_PLUS_ISLT_FP64     , GxB_TIMES_ISLT_FP64    ,
  
  // semirings with multiply op: z = ISGE (x,y), all types x,y,z the same:
  GxB_MIN_ISGE_INT8      , GxB_MAX_ISGE_INT8      , GxB_PLUS_ISGE_INT8     , GxB_TIMES_ISGE_INT8    ,
  GxB_MIN_ISGE_UINT8     , GxB_MAX_ISGE_UINT8     , GxB_PLUS_ISGE_UINT8    , GxB_TIMES_ISGE_UINT8   ,
  GxB_MIN_ISGE_INT16     , GxB_MAX_ISGE_INT16     , GxB_PLUS_ISGE_INT16    , GxB_TIMES_ISGE_INT16   ,
  GxB_MIN_ISGE_UINT16    , GxB_MAX_ISGE_UINT16    , GxB_PLUS_ISGE_UINT16   , GxB_TIMES_ISGE_UINT16  ,
  GxB_MIN_ISGE_INT32     , GxB_MAX_ISGE_INT32     , GxB_PLUS_ISGE_INT32    , GxB_TIMES_ISGE_INT32   ,
  GxB_MIN_ISGE_UINT32    , GxB_MAX_ISGE_UINT32    , GxB_PLUS_ISGE_UINT32   , GxB_TIMES_ISGE_UINT32  ,
  GxB_MIN_ISGE_INT64     , GxB_MAX_ISGE_INT64     , GxB_PLUS_ISGE_INT64    , GxB_TIMES_ISGE_INT64   ,
  GxB_MIN_ISGE_UINT64    , GxB_MAX_ISGE_UINT64    , GxB_PLUS_ISGE_UINT64   , GxB_TIMES_ISGE_UINT64  ,
  GxB_MIN_ISGE_FP32      , GxB_MAX_ISGE_FP32      , GxB_PLUS_ISGE_FP32     , GxB_TIMES_ISGE_FP32    ,
  GxB_MIN_ISGE_FP64      , GxB_MAX_ISGE_FP64      , GxB_PLUS_ISGE_FP64     , GxB_TIMES_ISGE_FP64    ,
  
  // semirings with multiply op: z = ISLE (x,y), all types x,y,z the same:
  GxB_MIN_ISLE_INT8      , GxB_MAX_ISLE_INT8      , GxB_PLUS_ISLE_INT8     , GxB_TIMES_ISLE_INT8    ,
  GxB_MIN_ISLE_UINT8     , GxB_MAX_ISLE_UINT8     , GxB_PLUS_ISLE_UINT8    , GxB_TIMES_ISLE_UINT8   ,
  GxB_MIN_ISLE_INT16     , GxB_MAX_ISLE_INT16     , GxB_PLUS_ISLE_INT16    , GxB_TIMES_ISLE_INT16   ,
  GxB_MIN_ISLE_UINT16    , GxB_MAX_ISLE_UINT16    , GxB_PLUS_ISLE_UINT16   , GxB_TIMES_ISLE_UINT16  ,
  GxB_MIN_ISLE_INT32     , GxB_MAX_ISLE_INT32     , GxB_PLUS_ISLE_INT32    , GxB_TIMES_ISLE_INT32   ,
  GxB_MIN_ISLE_UINT32    , GxB_MAX_ISLE_UINT32    , GxB_PLUS_ISLE_UINT32   , GxB_TIMES_ISLE_UINT32  ,
  GxB_MIN_ISLE_INT64     , GxB_MAX_ISLE_INT64     , GxB_PLUS_ISLE_INT64    , GxB_TIMES_ISLE_INT64   ,
  GxB_MIN_ISLE_UINT64    , GxB_MAX_ISLE_UINT64    , GxB_PLUS_ISLE_UINT64   , GxB_TIMES_ISLE_UINT64  ,
  GxB_MIN_ISLE_FP32      , GxB_MAX_ISLE_FP32      , GxB_PLUS_ISLE_FP32     , GxB_TIMES_ISLE_FP32    ,
  GxB_MIN_ISLE_FP64      , GxB_MAX_ISLE_FP64      , GxB_PLUS_ISLE_FP64     , GxB_TIMES_ISLE_FP64    ,
  
  // semirings with multiply op: z = LOR (x,y), all types x,y,z the same:
  GxB_MIN_LOR_INT8       , GxB_MAX_LOR_INT8       , GxB_PLUS_LOR_INT8      , GxB_TIMES_LOR_INT8     ,
  GxB_MIN_LOR_UINT8      , GxB_MAX_LOR_UINT8      , GxB_PLUS_LOR_UINT8     , GxB_TIMES_LOR_UINT8    ,
  GxB_MIN_LOR_INT16      , GxB_MAX_LOR_INT16      , GxB_PLUS_LOR_INT16     , GxB_TIMES_LOR_INT16    ,
  GxB_MIN_LOR_UINT16     , GxB_MAX_LOR_UINT16     , GxB_PLUS_LOR_UINT16    , GxB_TIMES_LOR_UINT16   ,
  GxB_MIN_LOR_INT32      , GxB_MAX_LOR_INT32      , GxB_PLUS_LOR_INT32     , GxB_TIMES_LOR_INT32    ,
  GxB_MIN_LOR_UINT32     , GxB_MAX_LOR_UINT32     , GxB_PLUS_LOR_UINT32    , GxB_TIMES_LOR_UINT32   ,
  GxB_MIN_LOR_INT64      , GxB_MAX_LOR_INT64      , GxB_PLUS_LOR_INT64     , GxB_TIMES_LOR_INT64    ,
  GxB_MIN_LOR_UINT64     , GxB_MAX_LOR_UINT64     , GxB_PLUS_LOR_UINT64    , GxB_TIMES_LOR_UINT64   ,
  GxB_MIN_LOR_FP32       , GxB_MAX_LOR_FP32       , GxB_PLUS_LOR_FP32      , GxB_TIMES_LOR_FP32     ,
  GxB_MIN_LOR_FP64       , GxB_MAX_LOR_FP64       , GxB_PLUS_LOR_FP64      , GxB_TIMES_LOR_FP64     ,
  
  // semirings with multiply op: z = LAND (x,y), all types x,y,z the same:
  GxB_MIN_LAND_INT8      , GxB_MAX_LAND_INT8      , GxB_PLUS_LAND_INT8     , GxB_TIMES_LAND_INT8    ,
  GxB_MIN_LAND_UINT8     , GxB_MAX_LAND_UINT8     , GxB_PLUS_LAND_UINT8    , GxB_TIMES_LAND_UINT8   ,
  GxB_MIN_LAND_INT16     , GxB_MAX_LAND_INT16     , GxB_PLUS_LAND_INT16    , GxB_TIMES_LAND_INT16   ,
  GxB_MIN_LAND_UINT16    , GxB_MAX_LAND_UINT16    , GxB_PLUS_LAND_UINT16   , GxB_TIMES_LAND_UINT16  ,
  GxB_MIN_LAND_INT32     , GxB_MAX_LAND_INT32     , GxB_PLUS_LAND_INT32    , GxB_TIMES_LAND_INT32   ,
  GxB_MIN_LAND_UINT32    , GxB_MAX_LAND_UINT32    , GxB_PLUS_LAND_UINT32   , GxB_TIMES_LAND_UINT32  ,
  GxB_MIN_LAND_INT64     , GxB_MAX_LAND_INT64     , GxB_PLUS_LAND_INT64    , GxB_TIMES_LAND_INT64   ,
  GxB_MIN_LAND_UINT64    , GxB_MAX_LAND_UINT64    , GxB_PLUS_LAND_UINT64   , GxB_TIMES_LAND_UINT64  ,
  GxB_MIN_LAND_FP32      , GxB_MAX_LAND_FP32      , GxB_PLUS_LAND_FP32     , GxB_TIMES_LAND_FP32    ,
  GxB_MIN_LAND_FP64      , GxB_MAX_LAND_FP64      , GxB_PLUS_LAND_FP64     , GxB_TIMES_LAND_FP64    ,
  
  // semirings with multiply op: z = LXOR (x,y), all types x,y,z the same:
  GxB_MIN_LXOR_INT8      , GxB_MAX_LXOR_INT8      , GxB_PLUS_LXOR_INT8     , GxB_TIMES_LXOR_INT8    ,
  GxB_MIN_LXOR_UINT8     , GxB_MAX_LXOR_UINT8     , GxB_PLUS_LXOR_UINT8    , GxB_TIMES_LXOR_UINT8   ,
  GxB_MIN_LXOR_INT16     , GxB_MAX_LXOR_INT16     , GxB_PLUS_LXOR_INT16    , GxB_TIMES_LXOR_INT16   ,
  GxB_MIN_LXOR_UINT16    , GxB_MAX_LXOR_UINT16    , GxB_PLUS_LXOR_UINT16   , GxB_TIMES_LXOR_UINT16  ,
  GxB_MIN_LXOR_INT32     , GxB_MAX_LXOR_INT32     , GxB_PLUS_LXOR_INT32    , GxB_TIMES_LXOR_INT32   ,
  GxB_MIN_LXOR_UINT32    , GxB_MAX_LXOR_UINT32    , GxB_PLUS_LXOR_UINT32   , GxB_TIMES_LXOR_UINT32  ,
  GxB_MIN_LXOR_INT64     , GxB_MAX_LXOR_INT64     , GxB_PLUS_LXOR_INT64    , GxB_TIMES_LXOR_INT64   ,
  GxB_MIN_LXOR_UINT64    , GxB_MAX_LXOR_UINT64    , GxB_PLUS_LXOR_UINT64   , GxB_TIMES_LXOR_UINT64  ,
  GxB_MIN_LXOR_FP32      , GxB_MAX_LXOR_FP32      , GxB_PLUS_LXOR_FP32     , GxB_TIMES_LXOR_FP32    ,
  GxB_MIN_LXOR_FP64      , GxB_MAX_LXOR_FP64      , GxB_PLUS_LXOR_FP64     , GxB_TIMES_LXOR_FP64    ,
  
  //------------------------------------------------------------------------------
  // 240 semirings with comparison ops of the form TxT->bool, and Boolean monoids
  //------------------------------------------------------------------------------
  
  // semirings with multiply op: z = EQ (x,y), where z is Boolean and x,y are given by the suffix:
  GxB_LOR_EQ_INT8        , GxB_LAND_EQ_INT8       , GxB_LXOR_EQ_INT8       , GxB_EQ_EQ_INT8         ,
  GxB_LOR_EQ_UINT8       , GxB_LAND_EQ_UINT8      , GxB_LXOR_EQ_UINT8      , GxB_EQ_EQ_UINT8        ,
  GxB_LOR_EQ_INT16       , GxB_LAND_EQ_INT16      , GxB_LXOR_EQ_INT16      , GxB_EQ_EQ_INT16        ,
  GxB_LOR_EQ_UINT16      , GxB_LAND_EQ_UINT16     , GxB_LXOR_EQ_UINT16     , GxB_EQ_EQ_UINT16       ,
  GxB_LOR_EQ_INT32       , GxB_LAND_EQ_INT32      , GxB_LXOR_EQ_INT32      , GxB_EQ_EQ_INT32        ,
  GxB_LOR_EQ_UINT32      , GxB_LAND_EQ_UINT32     , GxB_LXOR_EQ_UINT32     , GxB_EQ_EQ_UINT32       ,
  GxB_LOR_EQ_INT64       , GxB_LAND_EQ_INT64      , GxB_LXOR_EQ_INT64      , GxB_EQ_EQ_INT64        ,
  GxB_LOR_EQ_UINT64      , GxB_LAND_EQ_UINT64     , GxB_LXOR_EQ_UINT64     , GxB_EQ_EQ_UINT64       ,
  GxB_LOR_EQ_FP32        , GxB_LAND_EQ_FP32       , GxB_LXOR_EQ_FP32       , GxB_EQ_EQ_FP32         ,
  GxB_LOR_EQ_FP64        , GxB_LAND_EQ_FP64       , GxB_LXOR_EQ_FP64       , GxB_EQ_EQ_FP64         ,
  
  // semirings with multiply op: z = NE (x,y), where z is Boolean and x,y are given by the suffix:
  GxB_LOR_NE_INT8        , GxB_LAND_NE_INT8       , GxB_LXOR_NE_INT8       , GxB_EQ_NE_INT8         ,
  GxB_LOR_NE_UINT8       , GxB_LAND_NE_UINT8      , GxB_LXOR_NE_UINT8      , GxB_EQ_NE_UINT8        ,
  GxB_LOR_NE_INT16       , GxB_LAND_NE_INT16      , GxB_LXOR_NE_INT16      , GxB_EQ_NE_INT16        ,
  GxB_LOR_NE_UINT16      , GxB_LAND_NE_UINT16     , GxB_LXOR_NE_UINT16     , GxB_EQ_NE_UINT16       ,
  GxB_LOR_NE_INT32       , GxB_LAND_NE_INT32      , GxB_LXOR_NE_INT32      , GxB_EQ_NE_INT32        ,
  GxB_LOR_NE_UINT32      , GxB_LAND_NE_UINT32     , GxB_LXOR_NE_UINT32     , GxB_EQ_NE_UINT32       ,
  GxB_LOR_NE_INT64       , GxB_LAND_NE_INT64      , GxB_LXOR_NE_INT64      , GxB_EQ_NE_INT64        ,
  GxB_LOR_NE_UINT64      , GxB_LAND_NE_UINT64     , GxB_LXOR_NE_UINT64     , GxB_EQ_NE_UINT64       ,
  GxB_LOR_NE_FP32        , GxB_LAND_NE_FP32       , GxB_LXOR_NE_FP32       , GxB_EQ_NE_FP32         ,
  GxB_LOR_NE_FP64        , GxB_LAND_NE_FP64       , GxB_LXOR_NE_FP64       , GxB_EQ_NE_FP64         ,
  
  // semirings with multiply op: z = GT (x,y), where z is Boolean and x,y are given by the suffix:
  GxB_LOR_GT_INT8        , GxB_LAND_GT_INT8       , GxB_LXOR_GT_INT8       , GxB_EQ_GT_INT8         ,
  GxB_LOR_GT_UINT8       , GxB_LAND_GT_UINT8      , GxB_LXOR_GT_UINT8      , GxB_EQ_GT_UINT8        ,
  GxB_LOR_GT_INT16       , GxB_LAND_GT_INT16      , GxB_LXOR_GT_INT16      , GxB_EQ_GT_INT16        ,
  GxB_LOR_GT_UINT16      , GxB_LAND_GT_UINT16     , GxB_LXOR_GT_UINT16     , GxB_EQ_GT_UINT16       ,
  GxB_LOR_GT_INT32       , GxB_LAND_GT_INT32      , GxB_LXOR_GT_INT32      , GxB_EQ_GT_INT32        ,
  GxB_LOR_GT_UINT32      , GxB_LAND_GT_UINT32     , GxB_LXOR_GT_UINT32     , GxB_EQ_GT_UINT32       ,
  GxB_LOR_GT_INT64       , GxB_LAND_GT_INT64      , GxB_LXOR_GT_INT64      , GxB_EQ_GT_INT64        ,
  GxB_LOR_GT_UINT64      , GxB_LAND_GT_UINT64     , GxB_LXOR_GT_UINT64     , GxB_EQ_GT_UINT64       ,
  GxB_LOR_GT_FP32        , GxB_LAND_GT_FP32       , GxB_LXOR_GT_FP32       , GxB_EQ_GT_FP32         ,
  GxB_LOR_GT_FP64        , GxB_LAND_GT_FP64       , GxB_LXOR_GT_FP64       , GxB_EQ_GT_FP64         ,
  
  // semirings with multiply op: z = LT (x,y), where z is Boolean and x,y are given by the suffix:
  GxB_LOR_LT_INT8        , GxB_LAND_LT_INT8       , GxB_LXOR_LT_INT8       , GxB_EQ_LT_INT8         ,
  GxB_LOR_LT_UINT8       , GxB_LAND_LT_UINT8      , GxB_LXOR_LT_UINT8      , GxB_EQ_LT_UINT8        ,
  GxB_LOR_LT_INT16       , GxB_LAND_LT_INT16      , GxB_LXOR_LT_INT16      , GxB_EQ_LT_INT16        ,
  GxB_LOR_LT_UINT16      , GxB_LAND_LT_UINT16     , GxB_LXOR_LT_UINT16     , GxB_EQ_LT_UINT16       ,
  GxB_LOR_LT_INT32       , GxB_LAND_LT_INT32      , GxB_LXOR_LT_INT32      , GxB_EQ_LT_INT32        ,
  GxB_LOR_LT_UINT32      , GxB_LAND_LT_UINT32     , GxB_LXOR_LT_UINT32     , GxB_EQ_LT_UINT32       ,
  GxB_LOR_LT_INT64       , GxB_LAND_LT_INT64      , GxB_LXOR_LT_INT64      , GxB_EQ_LT_INT64        ,
  GxB_LOR_LT_UINT64      , GxB_LAND_LT_UINT64     , GxB_LXOR_LT_UINT64     , GxB_EQ_LT_UINT64       ,
  GxB_LOR_LT_FP32        , GxB_LAND_LT_FP32       , GxB_LXOR_LT_FP32       , GxB_EQ_LT_FP32         ,
  GxB_LOR_LT_FP64        , GxB_LAND_LT_FP64       , GxB_LXOR_LT_FP64       , GxB_EQ_LT_FP64         ,
  
  // semirings with multiply op: z = GE (x,y), where z is Boolean and x,y are given by the suffix:
  GxB_LOR_GE_INT8        , GxB_LAND_GE_INT8       , GxB_LXOR_GE_INT8       , GxB_EQ_GE_INT8         ,
  GxB_LOR_GE_UINT8       , GxB_LAND_GE_UINT8      , GxB_LXOR_GE_UINT8      , GxB_EQ_GE_UINT8        ,
  GxB_LOR_GE_INT16       , GxB_LAND_GE_INT16      , GxB_LXOR_GE_INT16      , GxB_EQ_GE_INT16        ,
  GxB_LOR_GE_UINT16      , GxB_LAND_GE_UINT16     , GxB_LXOR_GE_UINT16     , GxB_EQ_GE_UINT16       ,
  GxB_LOR_GE_INT32       , GxB_LAND_GE_INT32      , GxB_LXOR_GE_INT32      , GxB_EQ_GE_INT32        ,
  GxB_LOR_GE_UINT32      , GxB_LAND_GE_UINT32     , GxB_LXOR_GE_UINT32     , GxB_EQ_GE_UINT32       ,
  GxB_LOR_GE_INT64       , GxB_LAND_GE_INT64      , GxB_LXOR_GE_INT64      , GxB_EQ_GE_INT64        ,
  GxB_LOR_GE_UINT64      , GxB_LAND_GE_UINT64     , GxB_LXOR_GE_UINT64     , GxB_EQ_GE_UINT64       ,
  GxB_LOR_GE_FP32        , GxB_LAND_GE_FP32       , GxB_LXOR_GE_FP32       , GxB_EQ_GE_FP32         ,
  GxB_LOR_GE_FP64        , GxB_LAND_GE_FP64       , GxB_LXOR_GE_FP64       , GxB_EQ_GE_FP64         ,
  
  // semirings with multiply op: z = LE (x,y), where z is Boolean and x,y are given by the suffix:
  GxB_LOR_LE_INT8        , GxB_LAND_LE_INT8       , GxB_LXOR_LE_INT8       , GxB_EQ_LE_INT8         ,
  GxB_LOR_LE_UINT8       , GxB_LAND_LE_UINT8      , GxB_LXOR_LE_UINT8      , GxB_EQ_LE_UINT8        ,
  GxB_LOR_LE_INT16       , GxB_LAND_LE_INT16      , GxB_LXOR_LE_INT16      , GxB_EQ_LE_INT16        ,
  GxB_LOR_LE_UINT16      , GxB_LAND_LE_UINT16     , GxB_LXOR_LE_UINT16     , GxB_EQ_LE_UINT16       ,
  GxB_LOR_LE_INT32       , GxB_LAND_LE_INT32      , GxB_LXOR_LE_INT32      , GxB_EQ_LE_INT32        ,
  GxB_LOR_LE_UINT32      , GxB_LAND_LE_UINT32     , GxB_LXOR_LE_UINT32     , GxB_EQ_LE_UINT32       ,
  GxB_LOR_LE_INT64       , GxB_LAND_LE_INT64      , GxB_LXOR_LE_INT64      , GxB_EQ_LE_INT64        ,
  GxB_LOR_LE_UINT64      , GxB_LAND_LE_UINT64     , GxB_LXOR_LE_UINT64     , GxB_EQ_LE_UINT64       ,
  GxB_LOR_LE_FP32        , GxB_LAND_LE_FP32       , GxB_LXOR_LE_FP32       , GxB_EQ_LE_FP32         ,
  GxB_LOR_LE_FP64        , GxB_LAND_LE_FP64       , GxB_LXOR_LE_FP64       , GxB_EQ_LE_FP64         ,
  
  //------------------------------------------------------------------------------
  // 40 purely Boolean semirings
  //------------------------------------------------------------------------------
  
  // purely boolean semirings (in the form GxB_(add monoid)_(multipy operator)_BOOL:
  GxB_LOR_FIRST_BOOL     , GxB_LAND_FIRST_BOOL    , GxB_LXOR_FIRST_BOOL    , GxB_EQ_FIRST_BOOL      , 
  GxB_LOR_SECOND_BOOL    , GxB_LAND_SECOND_BOOL   , GxB_LXOR_SECOND_BOOL   , GxB_EQ_SECOND_BOOL     , 
  GxB_LOR_LOR_BOOL       , GxB_LAND_LOR_BOOL      , GxB_LXOR_LOR_BOOL      , GxB_EQ_LOR_BOOL        , 
  GxB_LOR_LAND_BOOL      , GxB_LAND_LAND_BOOL     , GxB_LXOR_LAND_BOOL     , GxB_EQ_LAND_BOOL       , 
  GxB_LOR_LXOR_BOOL      , GxB_LAND_LXOR_BOOL     , GxB_LXOR_LXOR_BOOL     , GxB_EQ_LXOR_BOOL       , 
  GxB_LOR_EQ_BOOL        , GxB_LAND_EQ_BOOL       , GxB_LXOR_EQ_BOOL       , GxB_EQ_EQ_BOOL         , 
  GxB_LOR_GT_BOOL        , GxB_LAND_GT_BOOL       , GxB_LXOR_GT_BOOL       , GxB_EQ_GT_BOOL         , 
  GxB_LOR_LT_BOOL        , GxB_LAND_LT_BOOL       , GxB_LXOR_LT_BOOL       , GxB_EQ_LT_BOOL         , 
  GxB_LOR_GE_BOOL        , GxB_LAND_GE_BOOL       , GxB_LXOR_GE_BOOL       , GxB_EQ_GE_BOOL         , 
  GxB_LOR_LE_BOOL        , GxB_LAND_LE_BOOL       , GxB_LXOR_LE_BOOL       , GxB_EQ_LE_BOOL         ; 
  
  //------------------------------------------------------------------------------
  // GxB_stats: memory usage and other statistics
  //------------------------------------------------------------------------------
  
  typedef struct
  {
      int64_t nmalloc ;       // # of objects malloc'ed but not yet freed
      int64_t inuse ;         // memory in use (in bytes)
      int64_t maxused ;       // max memory used since last call to GxB_stats
      int64_t future [20] ;   // not used, reserved for future use
      double xfuture [20] ;   // not used, reserved for future use
  }
  GxB_Statistics ;
  
  GrB_Info GxB_stats
  (
      GxB_Statistics *stats
  ) ;
  
  //------------------------------------------------------------------------------
  // GxB_resize:  change the size of a matrix or vector
  //------------------------------------------------------------------------------
  
  // If the dimensions decrease, entries that fall outside the resized matrix or
  // vector are deleted
  
  GrB_Info GxB_Matrix_resize      // change the size of a matrix
  (
      GrB_Matrix A,               // matrix to modify
      const GrB_Index nrows_new,  // new number of rows in matrix
      const GrB_Index ncols_new   // new number of columns in matrix
  ) ;
  
  GrB_Info GxB_Vector_resize      // change the size of a vector
  (
      GrB_Vector u,               // vector to modify
      const GrB_Index nrows_new   // new number of rows in vector
  ) ;
  
  // GxB_resize is a generic function for resizing a matrix or vector
  
  // GrB_Vector_resize (u,nrows_new)
  // GrB_Matrix_resize (A,nrows_new,ncols_new)
  
  #define GxB_resize(arg1,...)                                \
      _Generic                                                \
      (                                                       \
          (arg1),                                             \
                GrB_Vector : GxB_Vector_resize ,              \
                GrB_Matrix : GxB_Matrix_resize                \
      )                                                       \
      (arg1, __VA_ARGS__)
  
  //------------------------------------------------------------------------------
  // GxB_kron:  Kronecker product
  //------------------------------------------------------------------------------
  
  GrB_Info GxB_kron                   // C<Mask> = accum (C, kron(A,B))
  (
      GrB_Matrix C,                   // input/output matrix for results
      const GrB_Matrix Mask,          // optional mask for C, unused if NULL
      const GrB_BinaryOp accum,       // optional accum for Z=accum(C,T)
      const GrB_BinaryOp op,          // defines '*' for T=kron(A,B)
      const GrB_Matrix A,             // first input:  matrix A
      const GrB_Matrix B,             // second input: matrix B
      const GrB_Descriptor desc       // descriptor for C, Mask, A, and B
  ) ;
  
  #endif
  
  