LAPACK  3.10.1
LAPACK: Linear Algebra PACKage

◆ zggsvd()

subroutine zggsvd ( character  JOBU,
character  JOBV,
character  JOBQ,
integer  M,
integer  N,
integer  P,
integer  K,
integer  L,
complex*16, dimension( lda, * )  A,
integer  LDA,
complex*16, dimension( ldb, * )  B,
integer  LDB,
double precision, dimension( * )  ALPHA,
double precision, dimension( * )  BETA,
complex*16, dimension( ldu, * )  U,
integer  LDU,
complex*16, dimension( ldv, * )  V,
integer  LDV,
complex*16, dimension( ldq, * )  Q,
integer  LDQ,
complex*16, dimension( * )  WORK,
double precision, dimension( * )  RWORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

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

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

Purpose:
 This routine is deprecated and has been replaced by routine ZGGSVD3.

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

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

 where U, V and Q are unitary matrices.
 Let K+L = the effective numerical rank of the
 matrix (A**H,B**H)**H, 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 unitary
 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**H.
 If ( A**H,B**H)**H has orthnormal 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**H*A x = lambda* B**H*B x.
 In some literature, the GSVD of A and B is presented in the form
                  U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
 where U and V are orthogonal and X is nonsingular, and 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':  Unitary matrix U is computed;
          = 'N':  U is not computed.
[in]JOBV
          JOBV is CHARACTER*1
          = 'V':  Unitary matrix V is computed;
          = 'N':  V is not computed.
[in]JOBQ
          JOBQ is CHARACTER*1
          = 'Q':  Unitary 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**H,B**H)**H.
[in,out]A
          A is COMPLEX*16 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 COMPLEX*16 array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, B contains part of 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 COMPLEX*16 array, dimension (LDU,M)
          If JOBU = 'U', U contains the M-by-M unitary 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 COMPLEX*16 array, dimension (LDV,P)
          If JOBV = 'V', V contains the P-by-P unitary 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 COMPLEX*16 array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the N-by-N unitary 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 COMPLEX*16 array, dimension (max(3*N,M,P)+N)
[out]RWORK
          RWORK is DOUBLE PRECISION array, dimension (2*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 ZTGSJA.
Internal Parameters:
  TOLA    DOUBLE PRECISION
  TOLB    DOUBLE PRECISION
          TOLA and TOLB are the thresholds to determine the effective
          rank of (A**H,B**H)**H. Generally, they are set to
                   TOLA = MAX(M,N)*norm(A)*MAZHEPS,
                   TOLB = MAX(P,N)*norm(B)*MAZHEPS.
          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.
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 334 of file zggsvd.f.

337 *
338 * -- LAPACK driver routine --
339 * -- LAPACK is a software package provided by Univ. of Tennessee, --
340 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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  DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )
349  COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
350  $ U( LDU, * ), V( LDV, * ), WORK( * )
351 * ..
352 *
353 * =====================================================================
354 *
355 * .. Local Scalars ..
356  LOGICAL WANTQ, WANTU, WANTV
357  INTEGER I, IBND, ISUB, J, NCYCLE
358  DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
359 * ..
360 * .. External Functions ..
361  LOGICAL LSAME
362  DOUBLE PRECISION DLAMCH, ZLANGE
363  EXTERNAL lsame, dlamch, zlange
364 * ..
365 * .. External Subroutines ..
366  EXTERNAL dcopy, xerbla, zggsvp, ztgsja
367 * ..
368 * .. Intrinsic Functions ..
369  INTRINSIC max, min
370 * ..
371 * .. Executable Statements ..
372 *
373 * Decode and 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( 'ZGGSVD', -info )
405  RETURN
406  END IF
407 *
408 * Compute the Frobenius norm of matrices A and B
409 *
410  anorm = zlange( '1', m, n, a, lda, rwork )
411  bnorm = zlange( '1', p, n, b, ldb, rwork )
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 = dlamch( 'Precision' )
417  unfl = dlamch( 'Safe Minimum' )
418  tola = max( m, n )*max( anorm, unfl )*ulp
419  tolb = max( p, n )*max( bnorm, unfl )*ulp
420 *
421  CALL zggsvp( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
422  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
423  $ work, work( n+1 ), info )
424 *
425 * Compute the GSVD of two upper "triangular" matrices
426 *
427  CALL ztgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
428  $ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
429  $ work, ncycle, info )
430 *
431 * Sort the singular values and store the pivot indices in IWORK
432 * Copy ALPHA to RWORK, then sort ALPHA in RWORK
433 *
434  CALL dcopy( n, alpha, 1, rwork, 1 )
435  ibnd = min( l, m-k )
436  DO 20 i = 1, ibnd
437 *
438 * Scan for largest ALPHA(K+I)
439 *
440  isub = i
441  smax = rwork( k+i )
442  DO 10 j = i + 1, ibnd
443  temp = rwork( k+j )
444  IF( temp.GT.smax ) THEN
445  isub = j
446  smax = temp
447  END IF
448  10 CONTINUE
449  IF( isub.NE.i ) THEN
450  rwork( k+isub ) = rwork( k+i )
451  rwork( k+i ) = smax
452  iwork( k+i ) = k + isub
453  ELSE
454  iwork( k+i ) = k + i
455  END IF
456  20 CONTINUE
457 *
458  RETURN
459 *
460 * End of ZGGSVD
461 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function zlange(NORM, M, N, A, LDA, WORK)
ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlange.f:115
subroutine ztgsja(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)
ZTGSJA
Definition: ztgsja.f:379
subroutine zggsvp(JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK, TAU, WORK, INFO)
ZGGSVP
Definition: zggsvp.f:265
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
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