text: /*
      -- MAGMA (version 2.2.0) --
         Univ. of Tennessee, Knoxville
         Univ. of California, Berkeley
         Univ. of Colorado, Denver
         @date November 2016
  
         @author Raffaele Solca
         @author Stan Tomov
         @author Mark Gates
  
         @generated from src/zlatrd2.cpp, normal z -> s, Sun Nov 20 20:20:23 2016
  
  */
  #include "magma_internal.h"
  
  #define REAL
  
  /***************************************************************************//**
      Purpose
      -------
      SLATRD2 reduces NB rows and columns of a real symmetric matrix A to
      symmetric tridiagonal form by an orthogonal similarity
      transformation Q' * A * Q, and returns the matrices V and W which are
      needed to apply the transformation to the unreduced part of A.
  
      If UPLO = MagmaUpper, SLATRD reduces the last NB rows and columns of a
      matrix, of which the upper triangle is supplied;
      if UPLO = MagmaLower, SLATRD reduces the first NB rows and columns of a
      matrix, of which the lower triangle is supplied.
  
      This is an auxiliary routine called by SSYTRD2_GPU. It uses an
      accelerated HEMV that needs extra memory.
  
      Arguments
      ---------
      @param[in]
      uplo    magma_uplo_t
              Specifies whether the upper or lower triangular part of the
              symmetric matrix A is stored:
        -     = MagmaUpper: Upper triangular
        -     = MagmaLower: Lower triangular
  
      @param[in]
      n       INTEGER
              The order of the matrix A.
  
      @param[in]
      nb      INTEGER
              The number of rows and columns to be reduced.
  
      @param[in,out]
      A       REAL array, dimension (LDA,N)
              On entry, the symmetric matrix A.  If UPLO = MagmaUpper, the leading
              n-by-n upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = MagmaLower, the
              leading n-by-n lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit:
        -     if UPLO = MagmaUpper, the last NB columns have been reduced to
                tridiagonal form, with the diagonal elements overwriting
                the diagonal elements of A; the elements above the diagonal
                with the array TAU, represent the orthogonal matrix Q as a
                product of elementary reflectors;
        -     if UPLO = MagmaLower, the first NB columns have been reduced to
                tridiagonal form, with the diagonal elements overwriting
                the diagonal elements of A; the elements below the diagonal
                with the array TAU, represent the  orthogonal matrix Q as a
                product of elementary reflectors.
              See Further Details.
  
      @param[in]
      lda     INTEGER
              The leading dimension of the array A.  LDA >= (1,N).
  
      @param[out]
      e       REAL array, dimension (N-1)
              If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal
              elements of the last NB columns of the reduced matrix;
              if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of
              the first NB columns of the reduced matrix.
  
      @param[out]
      tau     REAL array, dimension (N-1)
              The scalar factors of the elementary reflectors, stored in
              TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower.
              See Further Details.
  
      @param[out]
      W       REAL array, dimension (LDW,NB)
              The n-by-nb matrix W required to update the unreduced part
              of A.
  
      @param[in]
      ldw     INTEGER
              The leading dimension of the array W. LDW >= max(1,N).
      
      @param
      work    TODO: dimension (lwork)?
      
      @param
      lwork   TODO: size??
      
      @param
      dA      TODO: dimension (ldda, n) ??
      
      @param
      ldda    TODO: ldda >= n ??
      
      @param
      dW      TODO: dimension (lddw, 2*nb) ??
      
      @param
      lddw    TODO: lddw >= n ??
      
      @param
      dwork   TODO: dimension (ldwork) ??
      
      @param
      ldwork  TODO: ldwork >= ceil(n/64)*ldda ??
  
      @param[in]
      queue   magma_queue_t
              Queue to execute in.
  
      Further Details
      ---------------
      If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary
      reflectors
  
          Q = H(n) H(n-1) . . . H(n-nb+1).
  
      Each H(i) has the form
  
          H(i) = I - tau * v * v'
  
      where tau is a real scalar, and v is a real vector with
      v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
      and tau in TAU(i-1).
  
      If UPLO = MagmaLower, the matrix Q is represented as a product of elementary
      reflectors
  
          Q = H(1) H(2) . . . H(nb).
  
      Each H(i) has the form
  
          H(i) = I - tau * v * v'
  
      where tau is a real scalar, and v is a real vector with
      v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
      and tau in TAU(i).
  
      The elements of the vectors v together form the n-by-nb matrix V
      which is needed, with W, to apply the transformation to the unreduced
      part of the matrix, using a symmetric rank-2k update of the form:
      A := A - V*W' - W*V'.
  
      The contents of A on exit are illustrated by the following examples
      with n = 5 and nb = 2:
  
      if UPLO = MagmaUpper:                       if UPLO = MagmaLower:
  
          (  a   a   a   v4  v5 )              (  d                  )
          (      a   a   v4  v5 )              (  1   d              )
          (          a   1   v5 )              (  v1  1   a          )
          (              d   1  )              (  v1  v2  a   a      )
          (                  d  )              (  v1  v2  a   a   a  )
  
      where d denotes a diagonal element of the reduced matrix, a denotes
      an element of the original matrix that is unchanged, and vi denotes
      an element of the vector defining H(i).
  
      @ingroup magma_latrd
  *******************************************************************************/
  extern "C" magma_int_t
  magma_slatrd2(
      magma_uplo_t uplo, magma_int_t n, magma_int_t nb,
      float *A,  magma_int_t lda,
      float *e, float *tau,
      float *W,  magma_int_t ldw,
      float *work, magma_int_t lwork,
      magmaFloat_ptr dA, magma_int_t ldda,
      magmaFloat_ptr dW, magma_int_t lddw,
      magmaFloat_ptr dwork, magma_int_t ldwork,
      magma_queue_t queue )
  {
      #define A(i_, j_) (A + (i_) + (j_)*lda)
      #define W(i_, j_) (W + (i_) + (j_)*ldw)
      
      #define dA(i_, j_) (dA + (i_) + (j_)*ldda)
      #define dW(i_, j_) (dW + (i_) + (j_)*lddw)
  
      /* Constants */
      const float c_neg_one = MAGMA_S_NEG_ONE;
      const float c_one     = MAGMA_S_ONE;
      const float c_zero    = MAGMA_S_ZERO;
      const magma_int_t ione = 1;
  
      /* Local variables */
      float alpha, value;
      magma_int_t i, i_n, i_1, iw;
  
      /* Check arguments */
      magma_int_t info = 0;
      if ( uplo != MagmaLower && uplo != MagmaUpper ) {
          info = -1;
      } else if ( n < 0 ) {
          info = -2;
      } else if ( nb < 1 ) {
          info = -3;
      } else if ( lda < max(1,n) ) {
          info = -5;
      } else if ( ldw < max(1,n) ) {
          info = -9;
      } else if ( lwork < max(1,n) ) {
          info = -11;
      } else if ( ldda < max(1,n) ) {
          info = -13;
      } else if ( lddw < max(1,n) ) {
          info = -15;
      } else if ( ldwork < ldda*magma_ceildiv(n,64) ) {
          info = -17;
      }
      
      if (info != 0) {
          magma_xerbla( __func__, -(info) );
          return info;
      }
      
      /* Quick return if possible */
      if (n == 0) {
          return info;
      }
      
      if (uplo == MagmaUpper) {
          /* Reduce last NB columns of upper triangle */
          for (i = n-1; i >= n - nb; --i) {
              i_1 = i + 1;
              i_n = n - i - 1;
              
              iw = i - n + nb;
              if (i < n-1) {
                  /* Update A(1:i,i) */
                  #ifdef COMPLEX
                  lapackf77_slacgv( &i_n, W(i, iw+1), &ldw );
                  #endif
                  blasf77_sgemv( "No transpose", &i_1, &i_n, &c_neg_one, A(0, i+1), &lda,
                                 W(i, iw+1), &ldw, &c_one, A(0, i), &ione );
                  #ifdef COMPLEX
                  lapackf77_slacgv( &i_n, W(i, iw+1), &ldw );
                  lapackf77_slacgv( &i_n, A(i, i+1),  &lda );
                  #endif
                  blasf77_sgemv( "No transpose", &i_1, &i_n, &c_neg_one, W(0, iw+1), &ldw,
                                 A(i, i+1), &lda, &c_one, A(0, i), &ione );
                  #ifdef COMPLEX
                  lapackf77_slacgv( &i_n, A(i, i+1), &lda );
                  #endif
              }
              if (i > 0) {
                  /* Generate elementary reflector H(i) to annihilate A(1:i-2,i) */
                  alpha = *A(i-1, i);
                  
                  lapackf77_slarfg( &i, &alpha, A(0, i), &ione, &tau[i - 1] );
                  
                  e[i-1] = MAGMA_S_REAL( alpha );
                  *A(i-1,i) = MAGMA_S_ONE;
                  
                  /* Compute W(1:i-1,i) */
                  // 1. Send the block reflector  A(0:n-i-1,i) to the GPU
                  magma_ssetvector_async( i, A(0, i), 1, dA(0, i), 1, queue );
                  
                  magmablas_ssymv_work( MagmaUpper, i, c_one, dA(0, 0), ldda,
                                        dA(0, i), ione, c_zero, dW(0, iw), ione,
                                        dwork, ldwork, queue );
                  
                  // 2. Start getting the result back (asynchronously)
                  magma_sgetmatrix_async( i, 1,
                                          dW(0, iw), lddw,
                                          W(0, iw),  ldw, queue );
                  
                  if (i < n-1) {
                      blasf77_sgemv( MagmaConjTransStr, &i, &i_n, &c_one, W(0, iw+1), &ldw,
                                     A(0, i), &ione, &c_zero, W(i+1, iw), &ione );
                  }
                  
                  // 3. Here we need ssymv result W(0, iw)
                  magma_queue_sync( queue );
                  
                  if (i < n-1) {
                      blasf77_sgemv( "No transpose", &i, &i_n, &c_neg_one, A(0, i+1), &lda,
                                     W(i+1, iw), &ione, &c_one, W(0, iw), &ione );
                      
                      blasf77_sgemv( MagmaConjTransStr, &i, &i_n, &c_one, A(0, i+1), &lda,
                                     A(0, i), &ione, &c_zero, W(i+1, iw), &ione );
                      
                      blasf77_sgemv( "No transpose", &i, &i_n, &c_neg_one, W(0, iw+1), &ldw,
                                     W(i+1, iw), &ione, &c_one, W(0, iw), &ione );
                  }
                  
                  blasf77_sscal( &i, &tau[i - 1], W(0, iw), &ione );
                  
                  value = magma_cblas_sdot( i, W(0,iw), ione, A(0,i), ione );
                  alpha = tau[i - 1] * -0.5f * value;
                  blasf77_saxpy( &i, &alpha, A(0, i), &ione,
                                 W(0, iw), &ione );
              }
          }
      }
      else {
          /*  Reduce first NB columns of lower triangle */
          for (i = 0; i < nb; ++i) {
              /* Update A(i:n,i) */
              i_n = n - i;
              #ifdef COMPLEX
              lapackf77_slacgv( &i, W(i, 0), &ldw );
              #endif
              blasf77_sgemv( "No transpose", &i_n, &i, &c_neg_one, A(i, 0), &lda,
                             W(i, 0), &ldw, &c_one, A(i, i), &ione );
              #ifdef COMPLEX
              lapackf77_slacgv( &i, W(i, 0), &ldw );
              lapackf77_slacgv( &i, A(i, 0), &lda );
              #endif
              blasf77_sgemv( "No transpose", &i_n, &i, &c_neg_one, W(i, 0), &ldw,
                             A(i, 0), &lda, &c_one, A(i, i), &ione );
              #ifdef COMPLEX
              lapackf77_slacgv( &i, A(i, 0), &lda );
              #endif
              
              if (i < n-1) {
                  /* Generate elementary reflector H(i) to annihilate A(i+2:n,i) */
                  i_n = n - i - 1;
                  alpha = *A(i+1, i);
                  lapackf77_slarfg( &i_n, &alpha, A(min(i+2,n-1), i), &ione, &tau[i] );
                  e[i] = MAGMA_S_REAL( alpha );
                  *A(i+1,i) = MAGMA_S_ONE;
                  
                  /* Compute W(i+1:n,i) */
                  // 1. Send the block reflector  A(i+1:n,i) to the GPU
                  magma_ssetvector_async( i_n, A(i+1, i), 1, dA(i+1, i), 1, queue );
                  
                  magmablas_ssymv_work( MagmaLower, i_n, c_one, dA(i+1, i+1), ldda,
                                        dA(i+1, i), ione, c_zero, dW(i+1, i), ione,
                                        dwork, ldwork, queue );
                  
                  // 2. Start getting the result back (asynchronously)
                  magma_sgetmatrix_async( i_n, 1,
                                          dW(i+1, i), lddw,
                                          W(i+1, i),  ldw, queue );
                  
                  blasf77_sgemv( MagmaConjTransStr, &i_n, &i, &c_one, W(i+1, 0), &ldw,
                                 A(i+1, i), &ione, &c_zero, W(0, i), &ione );
                  
                  blasf77_sgemv( "No transpose", &i_n, &i, &c_neg_one, A(i+1, 0), &lda,
                                 W(0, i), &ione, &c_zero, work, &ione );
                  
                  blasf77_sgemv( MagmaConjTransStr, &i_n, &i, &c_one, A(i+1, 0), &lda,
                                 A(i+1, i), &ione, &c_zero, W(0, i), &ione );
                  
                  // 3. Here we need ssymv result W(i+1, i)
                  magma_queue_sync( queue );
                  
                  if (i != 0)
                      blasf77_saxpy( &i_n, &c_one, work, &ione, W(i+1, i), &ione );
                  
                  blasf77_sgemv( "No transpose", &i_n, &i, &c_neg_one, W(i+1, 0), &ldw,
                                 W(0, i), &ione, &c_one, W(i+1, i), &ione );
                  blasf77_sscal( &i_n, &tau[i], W(i+1,i), &ione );
                  
                  value = magma_cblas_sdot( i_n, W(i+1,i), ione, A(i+1,i), ione );
                  alpha = tau[i] * -0.5f * value;
                  blasf77_saxpy( &i_n, &alpha, A(i+1, i), &ione, W(i+1,i), &ione );
              }
          }
      }
  
      return info;
  } /* magma_slatrd */
  