LAPACK  3.5.0
LAPACK: Linear Algebra PACKage
zggsvd.f File Reference

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Functions/Subroutines

subroutine zggsvd (JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, RWORK, IWORK, INFO)
  ZGGSVD computes the singular value decomposition (SVD) for OTHER matrices More...
 

Function/Subroutine Documentation

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:
 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.
Date
November 2011
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 337 of file zggsvd.f.

337 *
338 * -- LAPACK driver routine (version 3.4.0) --
339 * -- LAPACK is a software package provided by Univ. of Tennessee, --
340 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
341 * November 2011
342 *
343 * .. Scalar Arguments ..
344  CHARACTER jobq, jobu, jobv
345  INTEGER info, k, l, lda, ldb, ldq, ldu, ldv, m, n, p
346 * ..
347 * .. Array Arguments ..
348  INTEGER iwork( * )
349  DOUBLE PRECISION alpha( * ), beta( * ), rwork( * )
350  COMPLEX*16 a( lda, * ), b( ldb, * ), q( ldq, * ),
351  $ u( ldu, * ), v( ldv, * ), work( * )
352 * ..
353 *
354 * =====================================================================
355 *
356 * .. Local Scalars ..
357  LOGICAL wantq, wantu, wantv
358  INTEGER i, ibnd, isub, j, ncycle
359  DOUBLE PRECISION anorm, bnorm, smax, temp, tola, tolb, ulp, unfl
360 * ..
361 * .. External Functions ..
362  LOGICAL lsame
363  DOUBLE PRECISION dlamch, zlange
364  EXTERNAL lsame, dlamch, zlange
365 * ..
366 * .. External Subroutines ..
367  EXTERNAL dcopy, xerbla, zggsvp, ztgsja
368 * ..
369 * .. Intrinsic Functions ..
370  INTRINSIC max, min
371 * ..
372 * .. Executable Statements ..
373 *
374 * Decode and test the input parameters
375 *
376  wantu = lsame( jobu, 'U' )
377  wantv = lsame( jobv, 'V' )
378  wantq = lsame( jobq, 'Q' )
379 *
380  info = 0
381  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
382  info = -1
383  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
384  info = -2
385  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
386  info = -3
387  ELSE IF( m.LT.0 ) THEN
388  info = -4
389  ELSE IF( n.LT.0 ) THEN
390  info = -5
391  ELSE IF( p.LT.0 ) THEN
392  info = -6
393  ELSE IF( lda.LT.max( 1, m ) ) THEN
394  info = -10
395  ELSE IF( ldb.LT.max( 1, p ) ) THEN
396  info = -12
397  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
398  info = -16
399  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
400  info = -18
401  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
402  info = -20
403  END IF
404  IF( info.NE.0 ) THEN
405  CALL xerbla( 'ZGGSVD', -info )
406  RETURN
407  END IF
408 *
409 * Compute the Frobenius norm of matrices A and B
410 *
411  anorm = zlange( '1', m, n, a, lda, rwork )
412  bnorm = zlange( '1', p, n, b, ldb, rwork )
413 *
414 * Get machine precision and set up threshold for determining
415 * the effective numerical rank of the matrices A and B.
416 *
417  ulp = dlamch( 'Precision' )
418  unfl = dlamch( 'Safe Minimum' )
419  tola = max( m, n )*max( anorm, unfl )*ulp
420  tolb = max( p, n )*max( bnorm, unfl )*ulp
421 *
422  CALL zggsvp( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
423  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
424  $ work, work( n+1 ), info )
425 *
426 * Compute the GSVD of two upper "triangular" matrices
427 *
428  CALL ztgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
429  $ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
430  $ work, ncycle, info )
431 *
432 * Sort the singular values and store the pivot indices in IWORK
433 * Copy ALPHA to RWORK, then sort ALPHA in RWORK
434 *
435  CALL dcopy( n, alpha, 1, rwork, 1 )
436  ibnd = min( l, m-k )
437  DO 20 i = 1, ibnd
438 *
439 * Scan for largest ALPHA(K+I)
440 *
441  isub = i
442  smax = rwork( k+i )
443  DO 10 j = i + 1, ibnd
444  temp = rwork( k+j )
445  IF( temp.GT.smax ) THEN
446  isub = j
447  smax = temp
448  END IF
449  10 CONTINUE
450  IF( isub.NE.i ) THEN
451  rwork( k+isub ) = rwork( k+i )
452  rwork( k+i ) = smax
453  iwork( k+i ) = k + isub
454  ELSE
455  iwork( k+i ) = k + i
456  END IF
457  20 CONTINUE
458 *
459  RETURN
460 *
461 * End of ZGGSVD
462 *
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:53
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
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:117
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:381
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65

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