LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ dggsvd3()

subroutine dggsvd3 ( character  JOBU,
character  JOBV,
character  JOBQ,
integer  M,
integer  N,
integer  P,
integer  K,
integer  L,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( ldb, * )  B,
integer  LDB,
double precision, dimension( * )  ALPHA,
double precision, dimension( * )  BETA,
double precision, dimension( ldu, * )  U,
integer  LDU,
double precision, dimension( ldv, * )  V,
integer  LDV,
double precision, dimension( ldq, * )  Q,
integer  LDQ,
double precision, dimension( * )  WORK,
integer  LWORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Download DGGSVD3 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DGGSVD3 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)
[out]BETA
          BETA is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]LWORK
          LWORK is INTEGER
          The dimension of the array WORK.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.
[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 DTGSJA.
Internal Parameters:
  TOLA    DOUBLE PRECISION
  TOLB    DOUBLE PRECISION
          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
August 2015
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA
Further Details:
DGGSVD3 replaces the deprecated subroutine DGGSVD.

Definition at line 351 of file dggsvd3.f.

351 *
352 * -- LAPACK driver routine (version 3.7.0) --
353 * -- LAPACK is a software package provided by Univ. of Tennessee, --
354 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
355 * August 2015
356 *
357 * .. Scalar Arguments ..
358  CHARACTER jobq, jobu, jobv
359  INTEGER info, k, l, lda, ldb, ldq, ldu, ldv, m, n, p,
360  $ lwork
361 * ..
362 * .. Array Arguments ..
363  INTEGER iwork( * )
364  DOUBLE PRECISION a( lda, * ), alpha( * ), b( ldb, * ),
365  $ beta( * ), q( ldq, * ), u( ldu, * ),
366  $ v( ldv, * ), work( * )
367 * ..
368 *
369 * =====================================================================
370 *
371 * .. Local Scalars ..
372  LOGICAL wantq, wantu, wantv, lquery
373  INTEGER i, ibnd, isub, j, ncycle, lwkopt
374  DOUBLE PRECISION anorm, bnorm, smax, temp, tola, tolb, ulp, unfl
375 * ..
376 * .. External Functions ..
377  LOGICAL lsame
378  DOUBLE PRECISION dlamch, dlange
379  EXTERNAL lsame, dlamch, dlange
380 * ..
381 * .. External Subroutines ..
382  EXTERNAL dcopy, dggsvp3, dtgsja, xerbla
383 * ..
384 * .. Intrinsic Functions ..
385  INTRINSIC max, min
386 * ..
387 * .. Executable Statements ..
388 *
389 * Decode and test the input parameters
390 *
391  wantu = lsame( jobu, 'U' )
392  wantv = lsame( jobv, 'V' )
393  wantq = lsame( jobq, 'Q' )
394  lquery = ( lwork.EQ.-1 )
395  lwkopt = 1
396 *
397 * Test the input arguments
398 *
399  info = 0
400  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
401  info = -1
402  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
403  info = -2
404  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
405  info = -3
406  ELSE IF( m.LT.0 ) THEN
407  info = -4
408  ELSE IF( n.LT.0 ) THEN
409  info = -5
410  ELSE IF( p.LT.0 ) THEN
411  info = -6
412  ELSE IF( lda.LT.max( 1, m ) ) THEN
413  info = -10
414  ELSE IF( ldb.LT.max( 1, p ) ) THEN
415  info = -12
416  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
417  info = -16
418  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
419  info = -18
420  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
421  info = -20
422  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
423  info = -24
424  END IF
425 *
426 * Compute workspace
427 *
428  IF( info.EQ.0 ) THEN
429  CALL dggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
430  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
431  $ work, -1, info )
432  lwkopt = n + int( work( 1 ) )
433  lwkopt = max( 2*n, lwkopt )
434  lwkopt = max( 1, lwkopt )
435  work( 1 ) = dble( lwkopt )
436  END IF
437 *
438  IF( info.NE.0 ) THEN
439  CALL xerbla( 'DGGSVD3', -info )
440  RETURN
441  END IF
442  IF( lquery ) THEN
443  RETURN
444  ENDIF
445 *
446 * Compute the Frobenius norm of matrices A and B
447 *
448  anorm = dlange( '1', m, n, a, lda, work )
449  bnorm = dlange( '1', p, n, b, ldb, work )
450 *
451 * Get machine precision and set up threshold for determining
452 * the effective numerical rank of the matrices A and B.
453 *
454  ulp = dlamch( 'Precision' )
455  unfl = dlamch( 'Safe Minimum' )
456  tola = max( m, n )*max( anorm, unfl )*ulp
457  tolb = max( p, n )*max( bnorm, unfl )*ulp
458 *
459 * Preprocessing
460 *
461  CALL dggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
462  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
463  $ work( n+1 ), lwork-n, info )
464 *
465 * Compute the GSVD of two upper "triangular" matrices
466 *
467  CALL dtgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
468  $ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
469  $ work, ncycle, info )
470 *
471 * Sort the singular values and store the pivot indices in IWORK
472 * Copy ALPHA to WORK, then sort ALPHA in WORK
473 *
474  CALL dcopy( n, alpha, 1, work, 1 )
475  ibnd = min( l, m-k )
476  DO 20 i = 1, ibnd
477 *
478 * Scan for largest ALPHA(K+I)
479 *
480  isub = i
481  smax = work( k+i )
482  DO 10 j = i + 1, ibnd
483  temp = work( k+j )
484  IF( temp.GT.smax ) THEN
485  isub = j
486  smax = temp
487  END IF
488  10 CONTINUE
489  IF( isub.NE.i ) THEN
490  work( k+isub ) = work( k+i )
491  work( k+i ) = smax
492  iwork( k+i ) = k + isub
493  ELSE
494  iwork( k+i ) = k + i
495  END IF
496  20 CONTINUE
497 *
498  work( 1 ) = dble( lwkopt )
499  RETURN
500 *
501 * End of DGGSVD3
502 *
subroutine dtgsja(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)
DTGSJA
Definition: dtgsja.f:380
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:84
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:116
subroutine dggsvp3(JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, LWORK, INFO)
DGGSVP3
Definition: dggsvp3.f:274
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
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