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

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Functions/Subroutines

subroutine dtgsja (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)
 DTGSJA More...
 

Function/Subroutine Documentation

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

DTGSJA

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

Purpose:
 DTGSJA computes the generalized singular value decomposition (GSVD)
 of two real 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 DGGSVP
 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**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),

 where U, V and Q are orthogonal 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 orthogonal 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 an orthogonal matrix U1 on entry, and
                  the product U1*U is returned;
          = 'I':  U is initialized to the unit matrix, and the
                  orthogonal matrix U is returned;
          = 'N':  U is not computed.
[in]JOBV
          JOBV is CHARACTER*1
          = 'V':  V must contain an orthogonal matrix V1 on entry, and
                  the product V1*V is returned;
          = 'I':  V is initialized to the unit matrix, and the
                  orthogonal matrix V is returned;
          = 'N':  V is not computed.
[in]JOBQ
          JOBQ is CHARACTER*1
          = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and
                  the product Q1*Q is returned;
          = 'I':  Q is initialized to the unit matrix, and the
                  orthogonal 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 DTGSJA.
          See Further Details.
[in,out]A
          A is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDU,M)
          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
          the orthogonal matrix returned by DGGSVP).
          On exit,
          if JOBU = 'I', U contains the orthogonal 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 DOUBLE PRECISION array, dimension (LDV,P)
          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
          the orthogonal matrix returned by DGGSVP).
          On exit,
          if JOBV = 'I', V contains the orthogonal 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 DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
          the orthogonal matrix returned by DGGSVP).
          On exit,
          if JOBQ = 'I', Q contains the orthogonal 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 DOUBLE PRECISION 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
November 2011
Further Details:
  DTGSJA 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**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,

  where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
  of Z.  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 380 of file dtgsja.f.

380 *
381 * -- LAPACK computational routine (version 3.4.0) --
382 * -- LAPACK is a software package provided by Univ. of Tennessee, --
383 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
384 * November 2011
385 *
386 * .. Scalar Arguments ..
387  CHARACTER jobq, jobu, jobv
388  INTEGER info, k, l, lda, ldb, ldq, ldu, ldv, m, n,
389  $ ncycle, p
390  DOUBLE PRECISION tola, tolb
391 * ..
392 * .. Array Arguments ..
393  DOUBLE PRECISION a( lda, * ), alpha( * ), b( ldb, * ),
394  $ beta( * ), q( ldq, * ), u( ldu, * ),
395  $ v( ldv, * ), work( * )
396 * ..
397 *
398 * =====================================================================
399 *
400 * .. Parameters ..
401  INTEGER maxit
402  parameter( maxit = 40 )
403  DOUBLE PRECISION zero, one
404  parameter( zero = 0.0d+0, one = 1.0d+0 )
405 * ..
406 * .. Local Scalars ..
407 *
408  LOGICAL initq, initu, initv, upper, wantq, wantu, wantv
409  INTEGER i, j, kcycle
410  DOUBLE PRECISION a1, a2, a3, b1, b2, b3, csq, csu, csv, error,
411  $ gamma, rwk, snq, snu, snv, ssmin
412 * ..
413 * .. External Functions ..
414  LOGICAL lsame
415  EXTERNAL lsame
416 * ..
417 * .. External Subroutines ..
418  EXTERNAL dcopy, dlags2, dlapll, dlartg, dlaset, drot,
419  $ dscal, xerbla
420 * ..
421 * .. Intrinsic Functions ..
422  INTRINSIC abs, max, min
423 * ..
424 * .. Executable Statements ..
425 *
426 * Decode and test the input parameters
427 *
428  initu = lsame( jobu, 'I' )
429  wantu = initu .OR. lsame( jobu, 'U' )
430 *
431  initv = lsame( jobv, 'I' )
432  wantv = initv .OR. lsame( jobv, 'V' )
433 *
434  initq = lsame( jobq, 'I' )
435  wantq = initq .OR. lsame( jobq, 'Q' )
436 *
437  info = 0
438  IF( .NOT.( initu .OR. wantu .OR. lsame( jobu, 'N' ) ) ) THEN
439  info = -1
440  ELSE IF( .NOT.( initv .OR. wantv .OR. lsame( jobv, 'N' ) ) ) THEN
441  info = -2
442  ELSE IF( .NOT.( initq .OR. wantq .OR. lsame( jobq, 'N' ) ) ) THEN
443  info = -3
444  ELSE IF( m.LT.0 ) THEN
445  info = -4
446  ELSE IF( p.LT.0 ) THEN
447  info = -5
448  ELSE IF( n.LT.0 ) THEN
449  info = -6
450  ELSE IF( lda.LT.max( 1, m ) ) THEN
451  info = -10
452  ELSE IF( ldb.LT.max( 1, p ) ) THEN
453  info = -12
454  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
455  info = -18
456  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
457  info = -20
458  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
459  info = -22
460  END IF
461  IF( info.NE.0 ) THEN
462  CALL xerbla( 'DTGSJA', -info )
463  RETURN
464  END IF
465 *
466 * Initialize U, V and Q, if necessary
467 *
468  IF( initu )
469  $ CALL dlaset( 'Full', m, m, zero, one, u, ldu )
470  IF( initv )
471  $ CALL dlaset( 'Full', p, p, zero, one, v, ldv )
472  IF( initq )
473  $ CALL dlaset( 'Full', n, n, zero, one, q, ldq )
474 *
475 * Loop until convergence
476 *
477  upper = .false.
478  DO 40 kcycle = 1, maxit
479 *
480  upper = .NOT.upper
481 *
482  DO 20 i = 1, l - 1
483  DO 10 j = i + 1, l
484 *
485  a1 = zero
486  a2 = zero
487  a3 = zero
488  IF( k+i.LE.m )
489  $ a1 = a( k+i, n-l+i )
490  IF( k+j.LE.m )
491  $ a3 = a( k+j, n-l+j )
492 *
493  b1 = b( i, n-l+i )
494  b3 = b( j, n-l+j )
495 *
496  IF( upper ) THEN
497  IF( k+i.LE.m )
498  $ a2 = a( k+i, n-l+j )
499  b2 = b( i, n-l+j )
500  ELSE
501  IF( k+j.LE.m )
502  $ a2 = a( k+j, n-l+i )
503  b2 = b( j, n-l+i )
504  END IF
505 *
506  CALL dlags2( upper, a1, a2, a3, b1, b2, b3, csu, snu,
507  $ csv, snv, csq, snq )
508 *
509 * Update (K+I)-th and (K+J)-th rows of matrix A: U**T *A
510 *
511  IF( k+j.LE.m )
512  $ CALL drot( l, a( k+j, n-l+1 ), lda, a( k+i, n-l+1 ),
513  $ lda, csu, snu )
514 *
515 * Update I-th and J-th rows of matrix B: V**T *B
516 *
517  CALL drot( l, b( j, n-l+1 ), ldb, b( i, n-l+1 ), ldb,
518  $ csv, snv )
519 *
520 * Update (N-L+I)-th and (N-L+J)-th columns of matrices
521 * A and B: A*Q and B*Q
522 *
523  CALL drot( min( k+l, m ), a( 1, n-l+j ), 1,
524  $ a( 1, n-l+i ), 1, csq, snq )
525 *
526  CALL drot( l, b( 1, n-l+j ), 1, b( 1, n-l+i ), 1, csq,
527  $ snq )
528 *
529  IF( upper ) THEN
530  IF( k+i.LE.m )
531  $ a( k+i, n-l+j ) = zero
532  b( i, n-l+j ) = zero
533  ELSE
534  IF( k+j.LE.m )
535  $ a( k+j, n-l+i ) = zero
536  b( j, n-l+i ) = zero
537  END IF
538 *
539 * Update orthogonal matrices U, V, Q, if desired.
540 *
541  IF( wantu .AND. k+j.LE.m )
542  $ CALL drot( m, u( 1, k+j ), 1, u( 1, k+i ), 1, csu,
543  $ snu )
544 *
545  IF( wantv )
546  $ CALL drot( p, v( 1, j ), 1, v( 1, i ), 1, csv, snv )
547 *
548  IF( wantq )
549  $ CALL drot( n, q( 1, n-l+j ), 1, q( 1, n-l+i ), 1, csq,
550  $ snq )
551 *
552  10 CONTINUE
553  20 CONTINUE
554 *
555  IF( .NOT.upper ) THEN
556 *
557 * The matrices A13 and B13 were lower triangular at the start
558 * of the cycle, and are now upper triangular.
559 *
560 * Convergence test: test the parallelism of the corresponding
561 * rows of A and B.
562 *
563  error = zero
564  DO 30 i = 1, min( l, m-k )
565  CALL dcopy( l-i+1, a( k+i, n-l+i ), lda, work, 1 )
566  CALL dcopy( l-i+1, b( i, n-l+i ), ldb, work( l+1 ), 1 )
567  CALL dlapll( l-i+1, work, 1, work( l+1 ), 1, ssmin )
568  error = max( error, ssmin )
569  30 CONTINUE
570 *
571  IF( abs( error ).LE.min( tola, tolb ) )
572  $ GO TO 50
573  END IF
574 *
575 * End of cycle loop
576 *
577  40 CONTINUE
578 *
579 * The algorithm has not converged after MAXIT cycles.
580 *
581  info = 1
582  GO TO 100
583 *
584  50 CONTINUE
585 *
586 * If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
587 * Compute the generalized singular value pairs (ALPHA, BETA), and
588 * set the triangular matrix R to array A.
589 *
590  DO 60 i = 1, k
591  alpha( i ) = one
592  beta( i ) = zero
593  60 CONTINUE
594 *
595  DO 70 i = 1, min( l, m-k )
596 *
597  a1 = a( k+i, n-l+i )
598  b1 = b( i, n-l+i )
599 *
600  IF( a1.NE.zero ) THEN
601  gamma = b1 / a1
602 *
603 * change sign if necessary
604 *
605  IF( gamma.LT.zero ) THEN
606  CALL dscal( l-i+1, -one, b( i, n-l+i ), ldb )
607  IF( wantv )
608  $ CALL dscal( p, -one, v( 1, i ), 1 )
609  END IF
610 *
611  CALL dlartg( abs( gamma ), one, beta( k+i ), alpha( k+i ),
612  $ rwk )
613 *
614  IF( alpha( k+i ).GE.beta( k+i ) ) THEN
615  CALL dscal( l-i+1, one / alpha( k+i ), a( k+i, n-l+i ),
616  $ lda )
617  ELSE
618  CALL dscal( l-i+1, one / beta( k+i ), b( i, n-l+i ),
619  $ ldb )
620  CALL dcopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),
621  $ lda )
622  END IF
623 *
624  ELSE
625 *
626  alpha( k+i ) = zero
627  beta( k+i ) = one
628  CALL dcopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),
629  $ lda )
630 *
631  END IF
632 *
633  70 CONTINUE
634 *
635 * Post-assignment
636 *
637  DO 80 i = m + 1, k + l
638  alpha( i ) = zero
639  beta( i ) = one
640  80 CONTINUE
641 *
642  IF( k+l.LT.n ) THEN
643  DO 90 i = k + l + 1, n
644  alpha( i ) = zero
645  beta( i ) = zero
646  90 CONTINUE
647  END IF
648 *
649  100 CONTINUE
650  ncycle = kcycle
651  RETURN
652 *
653 * End of DTGSJA
654 *
subroutine dlapll(N, X, INCX, Y, INCY, SSMIN)
DLAPLL measures the linear dependence of two vectors.
Definition: dlapll.f:104
subroutine dlags2(UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
DLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such tha...
Definition: dlags2.f:154
subroutine dlartg(F, G, CS, SN, R)
DLARTG generates a plane rotation with real cosine and real sine.
Definition: dlartg.f:99
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
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:55
subroutine drot(N, DX, INCX, DY, INCY, C, S)
DROT
Definition: drot.f:53
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: dlaset.f:112

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