 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ ctgsja()

 subroutine ctgsja ( character JOBU, character JOBV, character JOBQ, integer M, integer P, integer N, integer K, integer L, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, real TOLA, real TOLB, real, dimension( * ) ALPHA, real, dimension( * ) BETA, complex, dimension( ldu, * ) U, integer LDU, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( * ) WORK, integer NCYCLE, integer INFO )

CTGSJA

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Purpose:
``` CTGSJA 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 CGGSVP
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 CTGSJA. See Further Details.``` [in,out] A ``` A is COMPLEX 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 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 REAL` [in] TOLB ``` TOLB is REAL 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)*MACHEPS, TOLB = MAX(P,N)*norm(B)*MACHEPS.``` [out] ALPHA ` ALPHA is REAL array, dimension (N)` [out] BETA ``` BETA is REAL 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 BETA(K+L+1:N) = 0.``` [in,out] U ``` U is COMPLEX array, dimension (LDU,M) On entry, if JOBU = 'U', U must contain a matrix U1 (usually the unitary matrix returned by CGGSVP). 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 array, dimension (LDV,P) On entry, if JOBV = 'V', V must contain a matrix V1 (usually the unitary matrix returned by CGGSVP). 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 array, dimension (LDQ,N) On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the unitary matrix returned by CGGSVP). 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 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.```
Further Details:
```  CTGSJA 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 376 of file ctgsja.f.

379 *
380 * -- LAPACK computational routine --
381 * -- LAPACK is a software package provided by Univ. of Tennessee, --
382 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
383 *
384 * .. Scalar Arguments ..
385  CHARACTER JOBQ, JOBU, JOBV
386  INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
387  \$ NCYCLE, P
388  REAL TOLA, TOLB
389 * ..
390 * .. Array Arguments ..
391  REAL ALPHA( * ), BETA( * )
392  COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
393  \$ U( LDU, * ), V( LDV, * ), WORK( * )
394 * ..
395 *
396 * =====================================================================
397 *
398 * .. Parameters ..
399  INTEGER MAXIT
400  parameter( maxit = 40 )
401  REAL ZERO, ONE, HUGENUM
402  parameter( zero = 0.0e+0, one = 1.0e+0 )
403  COMPLEX CZERO, CONE
404  parameter( czero = ( 0.0e+0, 0.0e+0 ),
405  \$ cone = ( 1.0e+0, 0.0e+0 ) )
406 * ..
407 * .. Local Scalars ..
408 *
409  LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
410  INTEGER I, J, KCYCLE
411  REAL A1, A3, B1, B3, CSQ, CSU, CSV, ERROR, GAMMA,
412  \$ RWK, SSMIN
413  COMPLEX A2, B2, SNQ, SNU, SNV
414 * ..
415 * .. External Functions ..
416  LOGICAL LSAME
417  EXTERNAL lsame
418 * ..
419 * .. External Subroutines ..
420  EXTERNAL ccopy, clags2, clapll, claset, crot, csscal,
421  \$ slartg, xerbla
422 * ..
423 * .. Intrinsic Functions ..
424  INTRINSIC abs, conjg, max, min, real, huge
425  parameter( hugenum = huge(zero) )
426 * ..
427 * .. Executable Statements ..
428 *
429 * Decode and test the input parameters
430 *
431  initu = lsame( jobu, 'I' )
432  wantu = initu .OR. lsame( jobu, 'U' )
433 *
434  initv = lsame( jobv, 'I' )
435  wantv = initv .OR. lsame( jobv, 'V' )
436 *
437  initq = lsame( jobq, 'I' )
438  wantq = initq .OR. lsame( jobq, 'Q' )
439 *
440  info = 0
441  IF( .NOT.( initu .OR. wantu .OR. lsame( jobu, 'N' ) ) ) THEN
442  info = -1
443  ELSE IF( .NOT.( initv .OR. wantv .OR. lsame( jobv, 'N' ) ) ) THEN
444  info = -2
445  ELSE IF( .NOT.( initq .OR. wantq .OR. lsame( jobq, 'N' ) ) ) THEN
446  info = -3
447  ELSE IF( m.LT.0 ) THEN
448  info = -4
449  ELSE IF( p.LT.0 ) THEN
450  info = -5
451  ELSE IF( n.LT.0 ) THEN
452  info = -6
453  ELSE IF( lda.LT.max( 1, m ) ) THEN
454  info = -10
455  ELSE IF( ldb.LT.max( 1, p ) ) THEN
456  info = -12
457  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
458  info = -18
459  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
460  info = -20
461  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
462  info = -22
463  END IF
464  IF( info.NE.0 ) THEN
465  CALL xerbla( 'CTGSJA', -info )
466  RETURN
467  END IF
468 *
469 * Initialize U, V and Q, if necessary
470 *
471  IF( initu )
472  \$ CALL claset( 'Full', m, m, czero, cone, u, ldu )
473  IF( initv )
474  \$ CALL claset( 'Full', p, p, czero, cone, v, ldv )
475  IF( initq )
476  \$ CALL claset( 'Full', n, n, czero, cone, q, ldq )
477 *
478 * Loop until convergence
479 *
480  upper = .false.
481  DO 40 kcycle = 1, maxit
482 *
483  upper = .NOT.upper
484 *
485  DO 20 i = 1, l - 1
486  DO 10 j = i + 1, l
487 *
488  a1 = zero
489  a2 = czero
490  a3 = zero
491  IF( k+i.LE.m )
492  \$ a1 = real( a( k+i, n-l+i ) )
493  IF( k+j.LE.m )
494  \$ a3 = real( a( k+j, n-l+j ) )
495 *
496  b1 = real( b( i, n-l+i ) )
497  b3 = real( b( j, n-l+j ) )
498 *
499  IF( upper ) THEN
500  IF( k+i.LE.m )
501  \$ a2 = a( k+i, n-l+j )
502  b2 = b( i, n-l+j )
503  ELSE
504  IF( k+j.LE.m )
505  \$ a2 = a( k+j, n-l+i )
506  b2 = b( j, n-l+i )
507  END IF
508 *
509  CALL clags2( upper, a1, a2, a3, b1, b2, b3, csu, snu,
510  \$ csv, snv, csq, snq )
511 *
512 * Update (K+I)-th and (K+J)-th rows of matrix A: U**H *A
513 *
514  IF( k+j.LE.m )
515  \$ CALL crot( l, a( k+j, n-l+1 ), lda, a( k+i, n-l+1 ),
516  \$ lda, csu, conjg( snu ) )
517 *
518 * Update I-th and J-th rows of matrix B: V**H *B
519 *
520  CALL crot( l, b( j, n-l+1 ), ldb, b( i, n-l+1 ), ldb,
521  \$ csv, conjg( snv ) )
522 *
523 * Update (N-L+I)-th and (N-L+J)-th columns of matrices
524 * A and B: A*Q and B*Q
525 *
526  CALL crot( min( k+l, m ), a( 1, n-l+j ), 1,
527  \$ a( 1, n-l+i ), 1, csq, snq )
528 *
529  CALL crot( l, b( 1, n-l+j ), 1, b( 1, n-l+i ), 1, csq,
530  \$ snq )
531 *
532  IF( upper ) THEN
533  IF( k+i.LE.m )
534  \$ a( k+i, n-l+j ) = czero
535  b( i, n-l+j ) = czero
536  ELSE
537  IF( k+j.LE.m )
538  \$ a( k+j, n-l+i ) = czero
539  b( j, n-l+i ) = czero
540  END IF
541 *
542 * Ensure that the diagonal elements of A and B are real.
543 *
544  IF( k+i.LE.m )
545  \$ a( k+i, n-l+i ) = real( a( k+i, n-l+i ) )
546  IF( k+j.LE.m )
547  \$ a( k+j, n-l+j ) = real( a( k+j, n-l+j ) )
548  b( i, n-l+i ) = real( b( i, n-l+i ) )
549  b( j, n-l+j ) = real( b( j, n-l+j ) )
550 *
551 * Update unitary matrices U, V, Q, if desired.
552 *
553  IF( wantu .AND. k+j.LE.m )
554  \$ CALL crot( m, u( 1, k+j ), 1, u( 1, k+i ), 1, csu,
555  \$ snu )
556 *
557  IF( wantv )
558  \$ CALL crot( p, v( 1, j ), 1, v( 1, i ), 1, csv, snv )
559 *
560  IF( wantq )
561  \$ CALL crot( n, q( 1, n-l+j ), 1, q( 1, n-l+i ), 1, csq,
562  \$ snq )
563 *
564  10 CONTINUE
565  20 CONTINUE
566 *
567  IF( .NOT.upper ) THEN
568 *
569 * The matrices A13 and B13 were lower triangular at the start
570 * of the cycle, and are now upper triangular.
571 *
572 * Convergence test: test the parallelism of the corresponding
573 * rows of A and B.
574 *
575  error = zero
576  DO 30 i = 1, min( l, m-k )
577  CALL ccopy( l-i+1, a( k+i, n-l+i ), lda, work, 1 )
578  CALL ccopy( l-i+1, b( i, n-l+i ), ldb, work( l+1 ), 1 )
579  CALL clapll( l-i+1, work, 1, work( l+1 ), 1, ssmin )
580  error = max( error, ssmin )
581  30 CONTINUE
582 *
583  IF( abs( error ).LE.min( tola, tolb ) )
584  \$ GO TO 50
585  END IF
586 *
587 * End of cycle loop
588 *
589  40 CONTINUE
590 *
591 * The algorithm has not converged after MAXIT cycles.
592 *
593  info = 1
594  GO TO 100
595 *
596  50 CONTINUE
597 *
598 * If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
599 * Compute the generalized singular value pairs (ALPHA, BETA), and
600 * set the triangular matrix R to array A.
601 *
602  DO 60 i = 1, k
603  alpha( i ) = one
604  beta( i ) = zero
605  60 CONTINUE
606 *
607  DO 70 i = 1, min( l, m-k )
608 *
609  a1 = real( a( k+i, n-l+i ) )
610  b1 = real( b( i, n-l+i ) )
611  gamma = b1 / a1
612 *
613  IF( (gamma.LE.hugenum).AND.(gamma.GE.-hugenum) ) THEN
614 *
615  IF( gamma.LT.zero ) THEN
616  CALL csscal( l-i+1, -one, b( i, n-l+i ), ldb )
617  IF( wantv )
618  \$ CALL csscal( p, -one, v( 1, i ), 1 )
619  END IF
620 *
621  CALL slartg( abs( gamma ), one, beta( k+i ), alpha( k+i ),
622  \$ rwk )
623 *
624  IF( alpha( k+i ).GE.beta( k+i ) ) THEN
625  CALL csscal( l-i+1, one / alpha( k+i ), a( k+i, n-l+i ),
626  \$ lda )
627  ELSE
628  CALL csscal( l-i+1, one / beta( k+i ), b( i, n-l+i ),
629  \$ ldb )
630  CALL ccopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),
631  \$ lda )
632  END IF
633 *
634  ELSE
635  alpha( k+i ) = zero
636  beta( k+i ) = one
637  CALL ccopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),
638  \$ lda )
639  END IF
640  70 CONTINUE
641 *
642 * Post-assignment
643 *
644  DO 80 i = m + 1, k + l
645  alpha( i ) = zero
646  beta( i ) = one
647  80 CONTINUE
648 *
649  IF( k+l.LT.n ) THEN
650  DO 90 i = k + l + 1, n
651  alpha( i ) = zero
652  beta( i ) = zero
653  90 CONTINUE
654  END IF
655 *
656  100 CONTINUE
657  ncycle = kcycle
658 *
659  RETURN
660 *
661 * End of CTGSJA
662 *
subroutine slartg(f, g, c, s, r)
SLARTG generates a plane rotation with real cosine and real sine.
Definition: slartg.f90:113
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:78
subroutine claset(UPLO, M, N, ALPHA, BETA, A, LDA)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: claset.f:106
subroutine clags2(UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
CLAGS2
Definition: clags2.f:158
subroutine crot(N, CX, INCX, CY, INCY, C, S)
CROT applies a plane rotation with real cosine and complex sine to a pair of complex vectors.
Definition: crot.f:103
subroutine clapll(N, X, INCX, Y, INCY, SSMIN)
CLAPLL measures the linear dependence of two vectors.
Definition: clapll.f:100
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