LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ sggsvd()

subroutine sggsvd ( character  JOBU,
character  JOBV,
character  JOBQ,
integer  M,
integer  N,
integer  P,
integer  K,
integer  L,
real, dimension( lda, * )  A,
integer  LDA,
real, dimension( ldb, * )  B,
integer  LDB,
real, dimension( * )  ALPHA,
real, dimension( * )  BETA,
real, dimension( ldu, * )  U,
integer  LDU,
real, dimension( ldv, * )  V,
integer  LDV,
real, dimension( ldq, * )  Q,
integer  LDQ,
real, dimension( * )  WORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

SGGSVD computes the singular value decomposition (SVD) for OTHER matrices

Download SGGSVD + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 This routine is deprecated and has been replaced by routine SGGSVD3.

 SGGSVD computes the generalized singular value decomposition (GSVD)
 of an M-by-N real matrix A and P-by-N real matrix B:

       U**T*A*Q = D1*( 0 R ),    V**T*B*Q = D2*( 0 R )

 where U, V and Q are orthogonal matrices.
 Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
 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**T.
 If ( A**T,B**T)**T  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**T*A x = lambda* B**T*B x.
 In some literature, the GSVD of A and B is presented in the form
                  U**T*A*X = ( 0 D1 ),   V**T*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) ).
Parameters
[in]JOBU
          JOBU is CHARACTER*1
          = 'U':  Orthogonal matrix U is computed;
          = 'N':  U is not computed.
[in]JOBV
          JOBV is CHARACTER*1
          = 'V':  Orthogonal matrix V is computed;
          = 'N':  V is not computed.
[in]JOBQ
          JOBQ is CHARACTER*1
          = 'Q':  Orthogonal matrix Q is computed;
          = 'N':  Q is not computed.
[in]M
          M is INTEGER
          The number of rows of the matrix A.  M >= 0.
[in]N
          N is INTEGER
          The number of columns of the matrices A and B.  N >= 0.
[in]P
          P is INTEGER
          The number of rows of the matrix B.  P >= 0.
[out]K
          K is INTEGER
[out]L
          L is INTEGER

          On exit, K and L specify the dimension of the subblocks
          described in Purpose.
          K + L = effective numerical rank of (A**T,B**T)**T.
[in,out]A
          A is 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.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).
[in,out]B
          B is 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.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,P).
[out]ALPHA
          ALPHA is REAL array, dimension (N)
[out]BETA
          BETA is 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
[out]U
          U is REAL array, dimension (LDU,M)
          If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
          If JOBU = 'N', U is not referenced.
[in]LDU
          LDU is INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.
[out]V
          V is REAL array, dimension (LDV,P)
          If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
          If JOBV = 'N', V is not referenced.
[in]LDV
          LDV is INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.
[out]Q
          Q is REAL array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
          If JOBQ = 'N', Q is not referenced.
[in]LDQ
          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.
[out]WORK
          WORK is REAL array,
                      dimension (max(3*N,M,P)+N)
[out]IWORK
          IWORK is 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).
[out]INFO
          INFO is 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**T,B**T)**T. 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
December 2016
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 336 of file sggsvd.f.

336 *
337 * -- LAPACK driver routine (version 3.7.0) --
338 * -- LAPACK is a software package provided by Univ. of Tennessee, --
339 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
340 * December 2016
341 *
342 * .. Scalar Arguments ..
343  CHARACTER jobq, jobu, jobv
344  INTEGER info, k, l, lda, ldb, ldq, ldu, ldv, m, n, p
345 * ..
346 * .. Array Arguments ..
347  INTEGER iwork( * )
348  REAL a( lda, * ), alpha( * ), b( ldb, * ),
349  $ beta( * ), q( ldq, * ), u( ldu, * ),
350  $ v( ldv, * ), work( * )
351 * ..
352 *
353 * =====================================================================
354 *
355 * .. Local Scalars ..
356  LOGICAL wantq, wantu, wantv
357  INTEGER i, ibnd, isub, j, ncycle
358  REAL anorm, bnorm, smax, temp, tola, tolb, ulp, unfl
359 * ..
360 * .. External Functions ..
361  LOGICAL lsame
362  REAL slamch, slange
363  EXTERNAL lsame, slamch, slange
364 * ..
365 * .. External Subroutines ..
366  EXTERNAL scopy, sggsvp, stgsja, xerbla
367 * ..
368 * .. Intrinsic Functions ..
369  INTRINSIC max, min
370 * ..
371 * .. Executable Statements ..
372 *
373 * Test the input parameters
374 *
375  wantu = lsame( jobu, 'U' )
376  wantv = lsame( jobv, 'V' )
377  wantq = lsame( jobq, 'Q' )
378 *
379  info = 0
380  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
381  info = -1
382  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
383  info = -2
384  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
385  info = -3
386  ELSE IF( m.LT.0 ) THEN
387  info = -4
388  ELSE IF( n.LT.0 ) THEN
389  info = -5
390  ELSE IF( p.LT.0 ) THEN
391  info = -6
392  ELSE IF( lda.LT.max( 1, m ) ) THEN
393  info = -10
394  ELSE IF( ldb.LT.max( 1, p ) ) THEN
395  info = -12
396  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
397  info = -16
398  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
399  info = -18
400  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
401  info = -20
402  END IF
403  IF( info.NE.0 ) THEN
404  CALL xerbla( 'SGGSVD', -info )
405  RETURN
406  END IF
407 *
408 * Compute the Frobenius norm of matrices A and B
409 *
410  anorm = slange( '1', m, n, a, lda, work )
411  bnorm = slange( '1', p, n, b, ldb, work )
412 *
413 * Get machine precision and set up threshold for determining
414 * the effective numerical rank of the matrices A and B.
415 *
416  ulp = slamch( 'Precision' )
417  unfl = slamch( 'Safe Minimum' )
418  tola = max( m, n )*max( anorm, unfl )*ulp
419  tolb = max( p, n )*max( bnorm, unfl )*ulp
420 *
421 * Preprocessing
422 *
423  CALL sggsvp( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
424  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
425  $ work( n+1 ), info )
426 *
427 * Compute the GSVD of two upper "triangular" matrices
428 *
429  CALL stgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
430  $ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
431  $ work, ncycle, info )
432 *
433 * Sort the singular values and store the pivot indices in IWORK
434 * Copy ALPHA to WORK, then sort ALPHA in WORK
435 *
436  CALL scopy( n, alpha, 1, work, 1 )
437  ibnd = min( l, m-k )
438  DO 20 i = 1, ibnd
439 *
440 * Scan for largest ALPHA(K+I)
441 *
442  isub = i
443  smax = work( k+i )
444  DO 10 j = i + 1, ibnd
445  temp = work( k+j )
446  IF( temp.GT.smax ) THEN
447  isub = j
448  smax = temp
449  END IF
450  10 CONTINUE
451  IF( isub.NE.i ) THEN
452  work( k+isub ) = work( k+i )
453  work( k+i ) = smax
454  iwork( k+i ) = k + isub
455  ELSE
456  iwork( k+i ) = k + i
457  END IF
458  20 CONTINUE
459 *
460  RETURN
461 *
462 * End of SGGSVD
463 *
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:84
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine stgsja(JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, INFO)
STGSJA
Definition: stgsja.f:380
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:116
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine sggsvp(JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, INFO)
SGGSVP
Definition: sggsvp.f:258
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