 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ ztgsja()

 subroutine ztgsja ( character JOBU, character JOBV, character JOBQ, integer M, integer P, integer N, integer K, integer L, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, double precision TOLA, double precision TOLB, 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 NCYCLE, integer INFO )

ZTGSJA

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Purpose:
``` ZTGSJA computes the generalized singular value decomposition (GSVD)
of two complex upper triangular (or trapezoidal) matrices A and B.

On entry, it is assumed that matrices A and B have the following
forms, which may be obtained by the preprocessing subroutine ZGGSVP
from a general M-by-N matrix A and P-by-N matrix B:

N-K-L  K    L
A =    K ( 0    A12  A13 ) if M-K-L >= 0;
L ( 0     0   A23 )
M-K-L ( 0     0    0  )

N-K-L  K    L
A =  K ( 0    A12  A13 ) if M-K-L < 0;
M-K ( 0     0   A23 )

N-K-L  K    L
B =  L ( 0     0   B13 )
P-L ( 0     0    0  )

where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal.

On exit,

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

where U, V and Q are unitary matrices.
R is a nonsingular upper triangular matrix, and D1
and D2 are ``diagonal'' matrices, which are of the following
structures:

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 ) K
L (  0    0   R22 ) L

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.

R = ( 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 computation of the unitary transformation matrices U, V or Q
is optional.  These matrices may either be formed explicitly, or they
may be postmultiplied into input matrices U1, V1, or Q1.```
Parameters
 [in] JOBU ``` JOBU is CHARACTER*1 = 'U': U must contain a unitary matrix U1 on entry, and the product U1*U is returned; = 'I': U is initialized to the unit matrix, and the unitary matrix U is returned; = 'N': U is not computed.``` [in] JOBV ``` JOBV is CHARACTER*1 = 'V': V must contain a unitary matrix V1 on entry, and the product V1*V is returned; = 'I': V is initialized to the unit matrix, and the unitary matrix V is returned; = 'N': V is not computed.``` [in] JOBQ ``` JOBQ is CHARACTER*1 = 'Q': Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned; = 'I': Q is initialized to the unit matrix, and the unitary matrix Q is returned; = 'N': Q is not computed.``` [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] P ``` P is INTEGER The number of rows of the matrix B. P >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrices A and B. N >= 0.``` [in] K ` K is INTEGER` [in] L ``` L is INTEGER K and L specify the subblocks in the input matrices A and B: A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N) of A and B, whose GSVD is going to be computed by ZTGSJA. See Further Details.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, A(N-K+1:N,1:MIN(K+L,M) ) 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, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains a part of R. See Purpose for details.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P).``` [in] TOLA ` TOLA is DOUBLE PRECISION` [in] TOLB ``` TOLB is DOUBLE PRECISION TOLA and TOLB are the convergence criteria for the Jacobi- Kogbetliantz iteration procedure. Generally, they are the same as used in the preprocessing step, say TOLA = MAX(M,N)*norm(A)*MAZHEPS, TOLB = MAX(P,N)*norm(B)*MAZHEPS.``` [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) = diag(C), BETA(K+1:K+L) = diag(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. Furthermore, if K+L < N, ALPHA(K+L+1:N) = 0 and BETA(K+L+1:N) = 0.``` [in,out] U ``` U is COMPLEX*16 array, dimension (LDU,M) On entry, if JOBU = 'U', U must contain a matrix U1 (usually the unitary matrix returned by ZGGSVP). On exit, if JOBU = 'I', U contains the unitary matrix U; if JOBU = 'U', U contains the product U1*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.``` [in,out] V ``` V is COMPLEX*16 array, dimension (LDV,P) On entry, if JOBV = 'V', V must contain a matrix V1 (usually the unitary matrix returned by ZGGSVP). On exit, if JOBV = 'I', V contains the unitary matrix V; if JOBV = 'V', V contains the product V1*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.``` [in,out] Q ``` Q is COMPLEX*16 array, dimension (LDQ,N) On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the unitary matrix returned by ZGGSVP). On exit, if JOBQ = 'I', Q contains the unitary matrix Q; if JOBQ = 'Q', Q contains the product Q1*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 (2*N)` [out] NCYCLE ``` NCYCLE is INTEGER The number of cycles required for convergence.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. = 1: the procedure does not converge after MAXIT cycles.```
Internal Parameters:
```  MAXIT   INTEGER
MAXIT specifies the total loops that the iterative procedure
may take. If after MAXIT cycles, the routine fails to
converge, we return INFO = 1.```
Date
December 2016
Further Details:
```  ZTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
matrix B13 to the form:

U1**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,

where U1, V1 and Q1 are unitary matrix.
C1 and S1 are diagonal matrices satisfying

C1**2 + S1**2 = I,

and R1 is an L-by-L nonsingular upper triangular matrix.```

Definition at line 381 of file ztgsja.f.

381 *
382 * -- LAPACK computational routine (version 3.7.0) --
383 * -- LAPACK is a software package provided by Univ. of Tennessee, --
384 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
385 * December 2016
386 *
387 * .. Scalar Arguments ..
388  CHARACTER jobq, jobu, jobv
389  INTEGER info, k, l, lda, ldb, ldq, ldu, ldv, m, n,
390  \$ ncycle, p
391  DOUBLE PRECISION tola, tolb
392 * ..
393 * .. Array Arguments ..
394  DOUBLE PRECISION alpha( * ), beta( * )
395  COMPLEX*16 a( lda, * ), b( ldb, * ), q( ldq, * ),
396  \$ u( ldu, * ), v( ldv, * ), work( * )
397 * ..
398 *
399 * =====================================================================
400 *
401 * .. Parameters ..
402  INTEGER maxit
403  parameter( maxit = 40 )
404  DOUBLE PRECISION zero, one
405  parameter( zero = 0.0d+0, one = 1.0d+0 )
406  COMPLEX*16 czero, cone
407  parameter( czero = ( 0.0d+0, 0.0d+0 ),
408  \$ cone = ( 1.0d+0, 0.0d+0 ) )
409 * ..
410 * .. Local Scalars ..
411 *
412  LOGICAL initq, initu, initv, upper, wantq, wantu, wantv
413  INTEGER i, j, kcycle
414  DOUBLE PRECISION a1, a3, b1, b3, csq, csu, csv, error, gamma,
415  \$ rwk, ssmin
416  COMPLEX*16 a2, b2, snq, snu, snv
417 * ..
418 * .. External Functions ..
419  LOGICAL lsame
420  EXTERNAL lsame
421 * ..
422 * .. External Subroutines ..
423  EXTERNAL dlartg, xerbla, zcopy, zdscal, zlags2, zlapll,
424  \$ zlaset, zrot
425 * ..
426 * .. Intrinsic Functions ..
427  INTRINSIC abs, dble, dconjg, max, min
428 * ..
429 * .. Executable Statements ..
430 *
431 * Decode and test the input parameters
432 *
433  initu = lsame( jobu, 'I' )
434  wantu = initu .OR. lsame( jobu, 'U' )
435 *
436  initv = lsame( jobv, 'I' )
437  wantv = initv .OR. lsame( jobv, 'V' )
438 *
439  initq = lsame( jobq, 'I' )
440  wantq = initq .OR. lsame( jobq, 'Q' )
441 *
442  info = 0
443  IF( .NOT.( initu .OR. wantu .OR. lsame( jobu, 'N' ) ) ) THEN
444  info = -1
445  ELSE IF( .NOT.( initv .OR. wantv .OR. lsame( jobv, 'N' ) ) ) THEN
446  info = -2
447  ELSE IF( .NOT.( initq .OR. wantq .OR. lsame( jobq, 'N' ) ) ) THEN
448  info = -3
449  ELSE IF( m.LT.0 ) THEN
450  info = -4
451  ELSE IF( p.LT.0 ) THEN
452  info = -5
453  ELSE IF( n.LT.0 ) THEN
454  info = -6
455  ELSE IF( lda.LT.max( 1, m ) ) THEN
456  info = -10
457  ELSE IF( ldb.LT.max( 1, p ) ) THEN
458  info = -12
459  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
460  info = -18
461  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
462  info = -20
463  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
464  info = -22
465  END IF
466  IF( info.NE.0 ) THEN
467  CALL xerbla( 'ZTGSJA', -info )
468  RETURN
469  END IF
470 *
471 * Initialize U, V and Q, if necessary
472 *
473  IF( initu )
474  \$ CALL zlaset( 'Full', m, m, czero, cone, u, ldu )
475  IF( initv )
476  \$ CALL zlaset( 'Full', p, p, czero, cone, v, ldv )
477  IF( initq )
478  \$ CALL zlaset( 'Full', n, n, czero, cone, q, ldq )
479 *
480 * Loop until convergence
481 *
482  upper = .false.
483  DO 40 kcycle = 1, maxit
484 *
485  upper = .NOT.upper
486 *
487  DO 20 i = 1, l - 1
488  DO 10 j = i + 1, l
489 *
490  a1 = zero
491  a2 = czero
492  a3 = zero
493  IF( k+i.LE.m )
494  \$ a1 = dble( a( k+i, n-l+i ) )
495  IF( k+j.LE.m )
496  \$ a3 = dble( a( k+j, n-l+j ) )
497 *
498  b1 = dble( b( i, n-l+i ) )
499  b3 = dble( b( j, n-l+j ) )
500 *
501  IF( upper ) THEN
502  IF( k+i.LE.m )
503  \$ a2 = a( k+i, n-l+j )
504  b2 = b( i, n-l+j )
505  ELSE
506  IF( k+j.LE.m )
507  \$ a2 = a( k+j, n-l+i )
508  b2 = b( j, n-l+i )
509  END IF
510 *
511  CALL zlags2( upper, a1, a2, a3, b1, b2, b3, csu, snu,
512  \$ csv, snv, csq, snq )
513 *
514 * Update (K+I)-th and (K+J)-th rows of matrix A: U**H *A
515 *
516  IF( k+j.LE.m )
517  \$ CALL zrot( l, a( k+j, n-l+1 ), lda, a( k+i, n-l+1 ),
518  \$ lda, csu, dconjg( snu ) )
519 *
520 * Update I-th and J-th rows of matrix B: V**H *B
521 *
522  CALL zrot( l, b( j, n-l+1 ), ldb, b( i, n-l+1 ), ldb,
523  \$ csv, dconjg( snv ) )
524 *
525 * Update (N-L+I)-th and (N-L+J)-th columns of matrices
526 * A and B: A*Q and B*Q
527 *
528  CALL zrot( min( k+l, m ), a( 1, n-l+j ), 1,
529  \$ a( 1, n-l+i ), 1, csq, snq )
530 *
531  CALL zrot( l, b( 1, n-l+j ), 1, b( 1, n-l+i ), 1, csq,
532  \$ snq )
533 *
534  IF( upper ) THEN
535  IF( k+i.LE.m )
536  \$ a( k+i, n-l+j ) = czero
537  b( i, n-l+j ) = czero
538  ELSE
539  IF( k+j.LE.m )
540  \$ a( k+j, n-l+i ) = czero
541  b( j, n-l+i ) = czero
542  END IF
543 *
544 * Ensure that the diagonal elements of A and B are real.
545 *
546  IF( k+i.LE.m )
547  \$ a( k+i, n-l+i ) = dble( a( k+i, n-l+i ) )
548  IF( k+j.LE.m )
549  \$ a( k+j, n-l+j ) = dble( a( k+j, n-l+j ) )
550  b( i, n-l+i ) = dble( b( i, n-l+i ) )
551  b( j, n-l+j ) = dble( b( j, n-l+j ) )
552 *
553 * Update unitary matrices U, V, Q, if desired.
554 *
555  IF( wantu .AND. k+j.LE.m )
556  \$ CALL zrot( m, u( 1, k+j ), 1, u( 1, k+i ), 1, csu,
557  \$ snu )
558 *
559  IF( wantv )
560  \$ CALL zrot( p, v( 1, j ), 1, v( 1, i ), 1, csv, snv )
561 *
562  IF( wantq )
563  \$ CALL zrot( n, q( 1, n-l+j ), 1, q( 1, n-l+i ), 1, csq,
564  \$ snq )
565 *
566  10 CONTINUE
567  20 CONTINUE
568 *
569  IF( .NOT.upper ) THEN
570 *
571 * The matrices A13 and B13 were lower triangular at the start
572 * of the cycle, and are now upper triangular.
573 *
574 * Convergence test: test the parallelism of the corresponding
575 * rows of A and B.
576 *
577  error = zero
578  DO 30 i = 1, min( l, m-k )
579  CALL zcopy( l-i+1, a( k+i, n-l+i ), lda, work, 1 )
580  CALL zcopy( l-i+1, b( i, n-l+i ), ldb, work( l+1 ), 1 )
581  CALL zlapll( l-i+1, work, 1, work( l+1 ), 1, ssmin )
582  error = max( error, ssmin )
583  30 CONTINUE
584 *
585  IF( abs( error ).LE.min( tola, tolb ) )
586  \$ GO TO 50
587  END IF
588 *
589 * End of cycle loop
590 *
591  40 CONTINUE
592 *
593 * The algorithm has not converged after MAXIT cycles.
594 *
595  info = 1
596  GO TO 100
597 *
598  50 CONTINUE
599 *
600 * If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
601 * Compute the generalized singular value pairs (ALPHA, BETA), and
602 * set the triangular matrix R to array A.
603 *
604  DO 60 i = 1, k
605  alpha( i ) = one
606  beta( i ) = zero
607  60 CONTINUE
608 *
609  DO 70 i = 1, min( l, m-k )
610 *
611  a1 = dble( a( k+i, n-l+i ) )
612  b1 = dble( b( i, n-l+i ) )
613 *
614  IF( a1.NE.zero ) THEN
615  gamma = b1 / a1
616 *
617  IF( gamma.LT.zero ) THEN
618  CALL zdscal( l-i+1, -one, b( i, n-l+i ), ldb )
619  IF( wantv )
620  \$ CALL zdscal( p, -one, v( 1, i ), 1 )
621  END IF
622 *
623  CALL dlartg( abs( gamma ), one, beta( k+i ), alpha( k+i ),
624  \$ rwk )
625 *
626  IF( alpha( k+i ).GE.beta( k+i ) ) THEN
627  CALL zdscal( l-i+1, one / alpha( k+i ), a( k+i, n-l+i ),
628  \$ lda )
629  ELSE
630  CALL zdscal( l-i+1, one / beta( k+i ), b( i, n-l+i ),
631  \$ ldb )
632  CALL zcopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),
633  \$ lda )
634  END IF
635 *
636  ELSE
637 *
638  alpha( k+i ) = zero
639  beta( k+i ) = one
640  CALL zcopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),
641  \$ lda )
642  END IF
643  70 CONTINUE
644 *
645 * Post-assignment
646 *
647  DO 80 i = m + 1, k + l
648  alpha( i ) = zero
649  beta( i ) = one
650  80 CONTINUE
651 *
652  IF( k+l.LT.n ) THEN
653  DO 90 i = k + l + 1, n
654  alpha( i ) = zero
655  beta( i ) = zero
656  90 CONTINUE
657  END IF
658 *
659  100 CONTINUE
660  ncycle = kcycle
661 *
662  RETURN
663 *
664 * End of ZTGSJA
665 *
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:83
subroutine zrot(N, CX, INCX, CY, INCY, C, S)
ZROT applies a plane rotation with real cosine and complex sine to a pair of complex vectors...
Definition: zrot.f:105
subroutine dlartg(F, G, CS, SN, R)
DLARTG generates a plane rotation with real cosine and real sine.
Definition: dlartg.f:99
subroutine zlags2(UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
ZLAGS2
Definition: zlags2.f:160
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: zlaset.f:108
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:80
subroutine zlapll(N, X, INCX, Y, INCY, SSMIN)
ZLAPLL measures the linear dependence of two vectors.
Definition: zlapll.f:102
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