 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ zggsvd3()

 subroutine zggsvd3 ( 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, integer LWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO )

ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Purpose:
``` ZGGSVD3 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 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**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(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] 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)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.```
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA
Further Details:
ZGGSVD3 replaces the deprecated subroutine ZGGSVD.

Definition at line 350 of file zggsvd3.f.

353*
354* -- LAPACK driver routine --
355* -- LAPACK is a software package provided by Univ. of Tennessee, --
356* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
357*
358* .. Scalar Arguments ..
359 CHARACTER JOBQ, JOBU, JOBV
360 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
361 \$ LWORK
362* ..
363* .. Array Arguments ..
364 INTEGER IWORK( * )
365 DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )
366 COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
367 \$ U( LDU, * ), V( LDV, * ), WORK( * )
368* ..
369*
370* =====================================================================
371*
372* .. Local Scalars ..
373 LOGICAL WANTQ, WANTU, WANTV, LQUERY
374 INTEGER I, IBND, ISUB, J, NCYCLE, LWKOPT
375 DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
376* ..
377* .. External Functions ..
378 LOGICAL LSAME
379 DOUBLE PRECISION DLAMCH, ZLANGE
380 EXTERNAL lsame, dlamch, zlange
381* ..
382* .. External Subroutines ..
383 EXTERNAL dcopy, xerbla, zggsvp3, ztgsja
384* ..
385* .. Intrinsic Functions ..
386 INTRINSIC max, min
387* ..
388* .. Executable Statements ..
389*
390* Decode and test the input parameters
391*
392 wantu = lsame( jobu, 'U' )
393 wantv = lsame( jobv, 'V' )
394 wantq = lsame( jobq, 'Q' )
395 lquery = ( lwork.EQ.-1 )
396 lwkopt = 1
397*
398* Test the input arguments
399*
400 info = 0
401 IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
402 info = -1
403 ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
404 info = -2
405 ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
406 info = -3
407 ELSE IF( m.LT.0 ) THEN
408 info = -4
409 ELSE IF( n.LT.0 ) THEN
410 info = -5
411 ELSE IF( p.LT.0 ) THEN
412 info = -6
413 ELSE IF( lda.LT.max( 1, m ) ) THEN
414 info = -10
415 ELSE IF( ldb.LT.max( 1, p ) ) THEN
416 info = -12
417 ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
418 info = -16
419 ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
420 info = -18
421 ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
422 info = -20
423 ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
424 info = -24
425 END IF
426*
427* Compute workspace
428*
429 IF( info.EQ.0 ) THEN
430 CALL zggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
431 \$ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
432 \$ work, work, -1, info )
433 lwkopt = n + int( work( 1 ) )
434 lwkopt = max( 2*n, lwkopt )
435 lwkopt = max( 1, lwkopt )
436 work( 1 ) = dcmplx( lwkopt )
437 END IF
438*
439 IF( info.NE.0 ) THEN
440 CALL xerbla( 'ZGGSVD3', -info )
441 RETURN
442 END IF
443 IF( lquery ) THEN
444 RETURN
445 ENDIF
446*
447* Compute the Frobenius norm of matrices A and B
448*
449 anorm = zlange( '1', m, n, a, lda, rwork )
450 bnorm = zlange( '1', p, n, b, ldb, rwork )
451*
452* Get machine precision and set up threshold for determining
453* the effective numerical rank of the matrices A and B.
454*
455 ulp = dlamch( 'Precision' )
456 unfl = dlamch( 'Safe Minimum' )
457 tola = max( m, n )*max( anorm, unfl )*ulp
458 tolb = max( p, n )*max( bnorm, unfl )*ulp
459*
460 CALL zggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
461 \$ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
462 \$ work, work( n+1 ), lwork-n, info )
463*
464* Compute the GSVD of two upper "triangular" matrices
465*
466 CALL ztgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
467 \$ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
468 \$ work, ncycle, info )
469*
470* Sort the singular values and store the pivot indices in IWORK
471* Copy ALPHA to RWORK, then sort ALPHA in RWORK
472*
473 CALL dcopy( n, alpha, 1, rwork, 1 )
474 ibnd = min( l, m-k )
475 DO 20 i = 1, ibnd
476*
477* Scan for largest ALPHA(K+I)
478*
479 isub = i
480 smax = rwork( k+i )
481 DO 10 j = i + 1, ibnd
482 temp = rwork( k+j )
483 IF( temp.GT.smax ) THEN
484 isub = j
485 smax = temp
486 END IF
487 10 CONTINUE
488 IF( isub.NE.i ) THEN
489 rwork( k+isub ) = rwork( k+i )
490 rwork( k+i ) = smax
491 iwork( k+i ) = k + isub
492 ELSE
493 iwork( k+i ) = k + i
494 END IF
495 20 CONTINUE
496*
497 work( 1 ) = dcmplx( lwkopt )
498 RETURN
499*
500* End of ZGGSVD3
501*
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 zggsvp3(JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK, TAU, WORK, LWORK, INFO)
ZGGSVP3
Definition: zggsvp3.f:278
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 dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
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