text: /* lapack/single/sggsvd.f -- translated by f2c (version 20050501).
     You must link the resulting object file with libf2c:
          on Microsoft Windows system, link with libf2c.lib;
          on Linux or Unix systems, link with .../path/to/libf2c.a -lm
          or, if you install libf2c.a in a standard place, with -lf2c -lm
          -- in that order, at the end of the command line, as in
                  cc *.o -lf2c -lm
          Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
  
                  http://www.netlib.org/f2c/libf2c.zip
  */
  
  #ifdef __cplusplus
  extern "C" {
  #endif
  #include "v3p_netlib.h"
  
  /* Table of constant values */
  
  static integer c__1 = 1;
  
  /*<    >*/
  /* Subroutine */ int sggsvd_(const char *jobu, const char *jobv, const char *jobq, integer *m,
          integer *n, integer *p, integer *k, integer *l, real *a, integer *lda,
           real *b, integer *ldb, real *alpha, real *beta, real *u, integer *
          ldu, real *v, integer *ldv, real *q, integer *ldq, real *work,
          integer *iwork, integer *info, ftnlen jobu_len, ftnlen jobv_len,
          ftnlen jobq_len)
  {
      /* System generated locals */
      integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, u_dim1,
              u_offset, v_dim1, v_offset, i__1, i__2;
  
      /* Local variables */
      integer i__, j;
      real ulp;
      integer ibnd;
      real tola;
      integer isub;
      real tolb, unfl, temp, smax;
      extern logical lsame_(const char *, const char *, ftnlen, ftnlen);
      real anorm, bnorm;
      logical wantq;
      extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
              integer *);
      logical wantu, wantv;
      extern doublereal slamch_(char *, ftnlen), slange_(char *, integer *,
              integer *, real *, integer *, real *, ftnlen);
      integer ncycle;
      extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen), stgsja_(
              const char *, const char *, const char *, integer *, integer *, integer *, integer *
              , integer *, real *, integer *, real *, integer *, real *, real *,
               real *, real *, real *, integer *, real *, integer *, real *,
              integer *, real *, integer *, integer *, ftnlen, ftnlen, ftnlen),
              sggsvp_(const char *, const char *, const char *, integer *, integer *, integer *,
              real *, integer *, real *, integer *, real *, real *, integer *,
              integer *, real *, integer *, real *, integer *, real *, integer *
              , integer *, real *, real *, integer *, ftnlen, ftnlen, ftnlen);
      (void)jobu_len;
      (void)jobv_len;
      (void)jobq_len;
  
  /*  -- LAPACK driver routine (version 3.0) -- */
  /*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
  /*     Courant Institute, Argonne National Lab, and Rice University */
  /*     June 30, 1999 */
  
  /*     .. Scalar Arguments .. */
  /*<       CHARACTER          JOBQ, JOBU, JOBV >*/
  /*<       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P >*/
  /*     .. */
  /*     .. Array Arguments .. */
  /*<       INTEGER            IWORK( * ) >*/
  /*<    >*/
  /*     .. */
  
  /*  Purpose */
  /*  ======= */
  
  /*  SGGSVD computes the generalized singular value decomposition (GSVD) */
  /*  of an M-by-N real matrix A and P-by-N real matrix B: */
  
  /*      U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R ) */
  
  /*  where U, V and Q are orthogonal matrices, and Z' is the transpose */
  /*  of Z.  Let K+L = the effective numerical rank of the matrix (A',B')', */
  /*  then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and */
  /*  D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the */
  /*  following structures, respectively: */
  
  /*  If M-K-L >= 0, */
  
  /*                      K  L */
  /*         D1 =     K ( I  0 ) */
  /*                  L ( 0  C ) */
  /*              M-K-L ( 0  0 ) */
  
  /*                    K  L */
  /*         D2 =   L ( 0  S ) */
  /*              P-L ( 0  0 ) */
  
  /*                  N-K-L  K    L */
  /*    ( 0 R ) = K (  0   R11  R12 ) */
  /*              L (  0    0   R22 ) */
  
  /*  where */
  
  /*    C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), */
  /*    S = diag( BETA(K+1),  ... , BETA(K+L) ), */
  /*    C**2 + S**2 = I. */
  
  /*    R is stored in A(1:K+L,N-K-L+1:N) on exit. */
  
  /*  If M-K-L < 0, */
  
  /*                    K M-K K+L-M */
  /*         D1 =   K ( I  0    0   ) */
  /*              M-K ( 0  C    0   ) */
  
  /*                      K M-K K+L-M */
  /*         D2 =   M-K ( 0  S    0  ) */
  /*              K+L-M ( 0  0    I  ) */
  /*                P-L ( 0  0    0  ) */
  
  /*                     N-K-L  K   M-K  K+L-M */
  /*    ( 0 R ) =     K ( 0    R11  R12  R13  ) */
  /*                M-K ( 0     0   R22  R23  ) */
  /*              K+L-M ( 0     0    0   R33  ) */
  
  /*  where */
  
  /*    C = diag( ALPHA(K+1), ... , ALPHA(M) ), */
  /*    S = diag( BETA(K+1),  ... , BETA(M) ), */
  /*    C**2 + S**2 = I. */
  
  /*    (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored */
  /*    ( 0  R22 R23 ) */
  /*    in B(M-K+1:L,N+M-K-L+1:N) on exit. */
  
  /*  The routine computes C, S, R, and optionally the orthogonal */
  /*  transformation matrices U, V and Q. */
  
  /*  In particular, if B is an N-by-N nonsingular matrix, then the GSVD of */
  /*  A and B implicitly gives the SVD of A*inv(B): */
  /*                       A*inv(B) = U*(D1*inv(D2))*V'. */
  /*  If ( A',B')' has orthonormal columns, then the GSVD of A and B is */
  /*  also equal to the CS decomposition of A and B. Furthermore, the GSVD */
  /*  can be used to derive the solution of the eigenvalue problem: */
  /*                       A'*A x = lambda* B'*B x. */
  /*  In some literature, the GSVD of A and B is presented in the form */
  /*                   U'*A*X = ( 0 D1 ),   V'*B*X = ( 0 D2 ) */
  /*  where U and V are orthogonal and X is nonsingular, D1 and D2 are */
  /*  ``diagonal''.  The former GSVD form can be converted to the latter */
  /*  form by taking the nonsingular matrix X as */
  
  /*                       X = Q*( I   0    ) */
  /*                             ( 0 inv(R) ). */
  
  /*  Arguments */
  /*  ========= */
  
  /*  JOBU    (input) CHARACTER*1 */
  /*          = 'U':  Orthogonal matrix U is computed; */
  /*          = 'N':  U is not computed. */
  
  /*  JOBV    (input) CHARACTER*1 */
  /*          = 'V':  Orthogonal matrix V is computed; */
  /*          = 'N':  V is not computed. */
  
  /*  JOBQ    (input) CHARACTER*1 */
  /*          = 'Q':  Orthogonal matrix Q is computed; */
  /*          = 'N':  Q is not computed. */
  
  /*  M       (input) INTEGER */
  /*          The number of rows of the matrix A.  M >= 0. */
  
  /*  N       (input) INTEGER */
  /*          The number of columns of the matrices A and B.  N >= 0. */
  
  /*  P       (input) INTEGER */
  /*          The number of rows of the matrix B.  P >= 0. */
  
  /*  K       (output) INTEGER */
  /*  L       (output) INTEGER */
  /*          On exit, K and L specify the dimension of the subblocks */
  /*          described in the Purpose section. */
  /*          K + L = effective numerical rank of (A',B')'. */
  
  /*  A       (input/output) REAL array, dimension (LDA,N) */
  /*          On entry, the M-by-N matrix A. */
  /*          On exit, A contains the triangular matrix R, or part of R. */
  /*          See Purpose for details. */
  
  /*  LDA     (input) INTEGER */
  /*          The leading dimension of the array A. LDA >= max(1,M). */
  
  /*  B       (input/output) REAL array, dimension (LDB,N) */
  /*          On entry, the P-by-N matrix B. */
  /*          On exit, B contains the triangular matrix R if M-K-L < 0. */
  /*          See Purpose for details. */
  
  /*  LDB     (input) INTEGER */
  /*          The leading dimension of the array B. LDA >= max(1,P). */
  
  /*  ALPHA   (output) REAL array, dimension (N) */
  /*  BETA    (output) REAL array, dimension (N) */
  /*          On exit, ALPHA and BETA contain the generalized singular */
  /*          value pairs of A and B; */
  /*            ALPHA(1:K) = 1, */
  /*            BETA(1:K)  = 0, */
  /*          and if M-K-L >= 0, */
  /*            ALPHA(K+1:K+L) = C, */
  /*            BETA(K+1:K+L)  = S, */
  /*          or if M-K-L < 0, */
  /*            ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 */
  /*            BETA(K+1:M) =S, BETA(M+1:K+L) =1 */
  /*          and */
  /*            ALPHA(K+L+1:N) = 0 */
  /*            BETA(K+L+1:N)  = 0 */
  
  /*  U       (output) REAL array, dimension (LDU,M) */
  /*          If JOBU = 'U', U contains the M-by-M orthogonal matrix U. */
  /*          If JOBU = 'N', U is not referenced. */
  
  /*  LDU     (input) INTEGER */
  /*          The leading dimension of the array U. LDU >= max(1,M) if */
  /*          JOBU = 'U'; LDU >= 1 otherwise. */
  
  /*  V       (output) REAL array, dimension (LDV,P) */
  /*          If JOBV = 'V', V contains the P-by-P orthogonal matrix V. */
  /*          If JOBV = 'N', V is not referenced. */
  
  /*  LDV     (input) INTEGER */
  /*          The leading dimension of the array V. LDV >= max(1,P) if */
  /*          JOBV = 'V'; LDV >= 1 otherwise. */
  
  /*  Q       (output) REAL array, dimension (LDQ,N) */
  /*          If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. */
  /*          If JOBQ = 'N', Q is not referenced. */
  
  /*  LDQ     (input) INTEGER */
  /*          The leading dimension of the array Q. LDQ >= max(1,N) if */
  /*          JOBQ = 'Q'; LDQ >= 1 otherwise. */
  
  /*  WORK    (workspace) REAL array, */
  /*                      dimension (max(3*N,M,P)+N) */
  
  /*  IWORK   (workspace/output) INTEGER array, dimension (N) */
  /*          On exit, IWORK stores the sorting information. More */
  /*          precisely, the following loop will sort ALPHA */
  /*             for I = K+1, min(M,K+L) */
  /*                 swap ALPHA(I) and ALPHA(IWORK(I)) */
  /*             endfor */
  /*          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). */
  
  /*  INFO    (output)INTEGER */
  /*          = 0:  successful exit */
  /*          < 0:  if INFO = -i, the i-th argument had an illegal value. */
  /*          > 0:  if INFO = 1, the Jacobi-type procedure failed to */
  /*                converge.  For further details, see subroutine STGSJA. */
  
  /*  Internal Parameters */
  /*  =================== */
  
  /*  TOLA    REAL */
  /*  TOLB    REAL */
  /*          TOLA and TOLB are the thresholds to determine the effective */
  /*          rank of (A',B')'. Generally, they are set to */
  /*                   TOLA = MAX(M,N)*norm(A)*MACHEPS, */
  /*                   TOLB = MAX(P,N)*norm(B)*MACHEPS. */
  /*          The size of TOLA and TOLB may affect the size of backward */
  /*          errors of the decomposition. */
  
  /*  Further Details */
  /*  =============== */
  
  /*  2-96 Based on modifications by */
  /*     Ming Gu and Huan Ren, Computer Science Division, University of */
  /*     California at Berkeley, USA */
  
  /*  ===================================================================== */
  
  /*     .. Local Scalars .. */
  /*<       LOGICAL            WANTQ, WANTU, WANTV >*/
  /*<       INTEGER            I, IBND, ISUB, J, NCYCLE >*/
  /*<       REAL               ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL >*/
  /*     .. */
  /*     .. External Functions .. */
  /*<       LOGICAL            LSAME >*/
  /*<       REAL               SLAMCH, SLANGE >*/
  /*<       EXTERNAL           LSAME, SLAMCH, SLANGE >*/
  /*     .. */
  /*     .. External Subroutines .. */
  /*<       EXTERNAL           SCOPY, SGGSVP, STGSJA, XERBLA >*/
  /*     .. */
  /*     .. Intrinsic Functions .. */
  /*<       INTRINSIC          MAX, MIN >*/
  /*     .. */
  /*     .. Executable Statements .. */
  
  /*     Test the input parameters */
  
  /*<       WANTU = LSAME( JOBU, 'U' ) >*/
      /* Parameter adjustments */
      a_dim1 = *lda;
      a_offset = 1 + a_dim1;
      a -= a_offset;
      b_dim1 = *ldb;
      b_offset = 1 + b_dim1;
      b -= b_offset;
      --alpha;
      --beta;
      u_dim1 = *ldu;
      u_offset = 1 + u_dim1;
      u -= u_offset;
      v_dim1 = *ldv;
      v_offset = 1 + v_dim1;
      v -= v_offset;
      q_dim1 = *ldq;
      q_offset = 1 + q_dim1;
      q -= q_offset;
      --work;
      --iwork;
  
      /* Function Body */
      wantu = lsame_(jobu, "U", (ftnlen)1, (ftnlen)1);
  /*<       WANTV = LSAME( JOBV, 'V' ) >*/
      wantv = lsame_(jobv, "V", (ftnlen)1, (ftnlen)1);
  /*<       WANTQ = LSAME( JOBQ, 'Q' ) >*/
      wantq = lsame_(jobq, "Q", (ftnlen)1, (ftnlen)1);
  
  /*<       INFO = 0 >*/
      *info = 0;
  /*<       IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN >*/
      if (! (wantu || lsame_(jobu, "N", (ftnlen)1, (ftnlen)1))) {
  /*<          INFO = -1 >*/
          *info = -1;
  /*<       ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN >*/
      } else if (! (wantv || lsame_(jobv, "N", (ftnlen)1, (ftnlen)1))) {
  /*<          INFO = -2 >*/
          *info = -2;
  /*<       ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN >*/
      } else if (! (wantq || lsame_(jobq, "N", (ftnlen)1, (ftnlen)1))) {
  /*<          INFO = -3 >*/
          *info = -3;
  /*<       ELSE IF( M.LT.0 ) THEN >*/
      } else if (*m < 0) {
  /*<          INFO = -4 >*/
          *info = -4;
  /*<       ELSE IF( N.LT.0 ) THEN >*/
      } else if (*n < 0) {
  /*<          INFO = -5 >*/
          *info = -5;
  /*<       ELSE IF( P.LT.0 ) THEN >*/
      } else if (*p < 0) {
  /*<          INFO = -6 >*/
          *info = -6;
  /*<       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN >*/
      } else if (*lda < max(1,*m)) {
  /*<          INFO = -10 >*/
          *info = -10;
  /*<       ELSE IF( LDB.LT.MAX( 1, P ) ) THEN >*/
      } else if (*ldb < max(1,*p)) {
  /*<          INFO = -12 >*/
          *info = -12;
  /*<       ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN >*/
      } else if (*ldu < 1 || (wantu && *ldu < *m)) {
  /*<          INFO = -16 >*/
          *info = -16;
  /*<       ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN >*/
      } else if (*ldv < 1 || (wantv && *ldv < *p)) {
  /*<          INFO = -18 >*/
          *info = -18;
  /*<       ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN >*/
      } else if (*ldq < 1 || (wantq && *ldq < *n)) {
  /*<          INFO = -20 >*/
          *info = -20;
  /*<       END IF >*/
      }
  /*<       IF( INFO.NE.0 ) THEN >*/
      if (*info != 0) {
  /*<          CALL XERBLA( 'SGGSVD', -INFO ) >*/
          i__1 = -(*info);
          xerbla_("SGGSVD", &i__1, (ftnlen)6);
  /*<          RETURN >*/
          return 0;
  /*<       END IF >*/
      }
  
  /*     Compute the Frobenius norm of matrices A and B */
  
  /*<       ANORM = SLANGE( '1', M, N, A, LDA, WORK ) >*/
      anorm = slange_("1", m, n, &a[a_offset], lda, &work[1], (ftnlen)1);
  /*<       BNORM = SLANGE( '1', P, N, B, LDB, WORK ) >*/
      bnorm = slange_("1", p, n, &b[b_offset], ldb, &work[1], (ftnlen)1);
  
  /*     Get machine precision and set up threshold for determining */
  /*     the effective numerical rank of the matrices A and B. */
  
  /*<       ULP = SLAMCH( 'Precision' ) >*/
      ulp = slamch_("Precision", (ftnlen)9);
  /*<       UNFL = SLAMCH( 'Safe Minimum' ) >*/
      unfl = slamch_("Safe Minimum", (ftnlen)12);
  /*<       TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP >*/
      tola = max(*m,*n) * dmax(anorm,unfl) * ulp;
  /*<       TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP >*/
      tolb = max(*p,*n) * dmax(bnorm,unfl) * ulp;
  
  /*     Preprocessing */
  
  /*<    >*/
      sggsvp_(jobu, jobv, jobq, m, p, n, &a[a_offset], lda, &b[b_offset], ldb, &
              tola, &tolb, k, l, &u[u_offset], ldu, &v[v_offset], ldv, &q[
              q_offset], ldq, &iwork[1], &work[1], &work[*n + 1], info, (ftnlen)
              1, (ftnlen)1, (ftnlen)1);
  
  /*     Compute the GSVD of two upper "triangular" matrices */
  
  /*<    >*/
      stgsja_(jobu, jobv, jobq, m, p, n, k, l, &a[a_offset], lda, &b[b_offset],
              ldb, &tola, &tolb, &alpha[1], &beta[1], &u[u_offset], ldu, &v[
              v_offset], ldv, &q[q_offset], ldq, &work[1], &ncycle, info, (
              ftnlen)1, (ftnlen)1, (ftnlen)1);
  
  /*     Sort the singular values and store the pivot indices in IWORK */
  /*     Copy ALPHA to WORK, then sort ALPHA in WORK */
  
  /*<       CALL SCOPY( N, ALPHA, 1, WORK, 1 ) >*/
      scopy_(n, &alpha[1], &c__1, &work[1], &c__1);
  /*<       IBND = MIN( L, M-K ) >*/
  /* Computing MIN */
      i__1 = *l, i__2 = *m - *k;
      ibnd = min(i__1,i__2);
  /*<       DO 20 I = 1, IBND >*/
      i__1 = ibnd;
      for (i__ = 1; i__ <= i__1; ++i__) {
  
  /*        Scan for largest ALPHA(K+I) */
  
  /*<          ISUB = I >*/
          isub = i__;
  /*<          SMAX = WORK( K+I ) >*/
          smax = work[*k + i__];
  /*<          DO 10 J = I + 1, IBND >*/
          i__2 = ibnd;
          for (j = i__ + 1; j <= i__2; ++j) {
  /*<             TEMP = WORK( K+J ) >*/
              temp = work[*k + j];
  /*<             IF( TEMP.GT.SMAX ) THEN >*/
              if (temp > smax) {
  /*<                ISUB = J >*/
                  isub = j;
  /*<                SMAX = TEMP >*/
                  smax = temp;
  /*<             END IF >*/
              }
  /*<    10    CONTINUE >*/
  /* L10: */
          }
  /*<          IF( ISUB.NE.I ) THEN >*/
          if (isub != i__) {
  /*<             WORK( K+ISUB ) = WORK( K+I ) >*/
              work[*k + isub] = work[*k + i__];
  /*<             WORK( K+I ) = SMAX >*/
              work[*k + i__] = smax;
  /*<             IWORK( K+I ) = K + ISUB >*/
              iwork[*k + i__] = *k + isub;
  /*<          ELSE >*/
          } else {
  /*<             IWORK( K+I ) = K + I >*/
              iwork[*k + i__] = *k + i__;
  /*<          END IF >*/
          }
  /*<    20 CONTINUE >*/
  /* L20: */
      }
  
  /*<       RETURN >*/
      return 0;
  
  /*     End of SGGSVD */
  
  /*<       END >*/
  } /* sggsvd_ */
  
  #ifdef __cplusplus
          }
  #endif
  