LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ zgbsvx()

subroutine zgbsvx ( character  fact,
character  trans,
integer  n,
integer  kl,
integer  ku,
integer  nrhs,
complex*16, dimension( ldab, * )  ab,
integer  ldab,
complex*16, dimension( ldafb, * )  afb,
integer  ldafb,
integer, dimension( * )  ipiv,
character  equed,
double precision, dimension( * )  r,
double precision, dimension( * )  c,
complex*16, dimension( ldb, * )  b,
integer  ldb,
complex*16, dimension( ldx, * )  x,
integer  ldx,
double precision  rcond,
double precision, dimension( * )  ferr,
double precision, dimension( * )  berr,
complex*16, dimension( * )  work,
double precision, dimension( * )  rwork,
integer  info 
)

ZGBSVX computes the solution to system of linear equations A * X = B for GB matrices

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

Purpose:
 ZGBSVX uses the LU factorization to compute the solution to a complex
 system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
 where A is a band matrix of order N with KL subdiagonals and KU
 superdiagonals, and X and B are N-by-NRHS matrices.

 Error bounds on the solution and a condition estimate are also
 provided.
Description:
 The following steps are performed by this subroutine:

 1. If FACT = 'E', real scaling factors are computed to equilibrate
    the system:
       TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
       TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
       TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
    Whether or not the system will be equilibrated depends on the
    scaling of the matrix A, but if equilibration is used, A is
    overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
    or diag(C)*B (if TRANS = 'T' or 'C').

 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
    matrix A (after equilibration if FACT = 'E') as
       A = L * U,
    where L is a product of permutation and unit lower triangular
    matrices with KL subdiagonals, and U is upper triangular with
    KL+KU superdiagonals.

 3. If some U(i,i)=0, so that U is exactly singular, then the routine
    returns with INFO = i. Otherwise, the factored form of A is used
    to estimate the condition number of the matrix A.  If the
    reciprocal of the condition number is less than machine precision,
    INFO = N+1 is returned as a warning, but the routine still goes on
    to solve for X and compute error bounds as described below.

 4. The system of equations is solved for X using the factored form
    of A.

 5. Iterative refinement is applied to improve the computed solution
    matrix and calculate error bounds and backward error estimates
    for it.

 6. If equilibration was used, the matrix X is premultiplied by
    diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
    that it solves the original system before equilibration.
Parameters
[in]FACT
          FACT is CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.
          = 'F':  On entry, AFB and IPIV contain the factored form of
                  A.  If EQUED is not 'N', the matrix A has been
                  equilibrated with scaling factors given by R and C.
                  AB, AFB, and IPIV are not modified.
          = 'N':  The matrix A will be copied to AFB and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then
                  copied to AFB and factored.
[in]TRANS
          TRANS is CHARACTER*1
          Specifies the form of the system of equations.
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose)
[in]N
          N is INTEGER
          The number of linear equations, i.e., the order of the
          matrix A.  N >= 0.
[in]KL
          KL is INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.
[in]NRHS
          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.
[in,out]AB
          AB is COMPLEX*16 array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

          If FACT = 'F' and EQUED is not 'N', then A must have been
          equilibrated by the scaling factors in R and/or C.  AB is not
          modified if FACT = 'F' or 'N', or if FACT = 'E' and
          EQUED = 'N' on exit.

          On exit, if EQUED .ne. 'N', A is scaled as follows:
          EQUED = 'R':  A := diag(R) * A
          EQUED = 'C':  A := A * diag(C)
          EQUED = 'B':  A := diag(R) * A * diag(C).
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.
[in,out]AFB
          AFB is COMPLEX*16 array, dimension (LDAFB,N)
          If FACT = 'F', then AFB is an input argument and on entry
          contains details of the LU factorization of the band matrix
          A, as computed by ZGBTRF.  U is stored as an upper triangular
          band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
          and the multipliers used during the factorization are stored
          in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB is
          the factored form of the equilibrated matrix A.

          If FACT = 'N', then AFB is an output argument and on exit
          returns details of the LU factorization of A.

          If FACT = 'E', then AFB is an output argument and on exit
          returns details of the LU factorization of the equilibrated
          matrix A (see the description of AB for the form of the
          equilibrated matrix).
[in]LDAFB
          LDAFB is INTEGER
          The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
[in,out]IPIV
          IPIV is INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains the pivot indices from the factorization A = L*U
          as computed by ZGBTRF; row i of the matrix was interchanged
          with row IPIV(i).

          If FACT = 'N', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = L*U
          of the original matrix A.

          If FACT = 'E', then IPIV is an output argument and on exit
          contains the pivot indices from the factorization A = L*U
          of the equilibrated matrix A.
[in,out]EQUED
          EQUED is CHARACTER*1
          Specifies the form of equilibration that was done.
          = 'N':  No equilibration (always true if FACT = 'N').
          = 'R':  Row equilibration, i.e., A has been premultiplied by
                  diag(R).
          = 'C':  Column equilibration, i.e., A has been postmultiplied
                  by diag(C).
          = 'B':  Both row and column equilibration, i.e., A has been
                  replaced by diag(R) * A * diag(C).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.
[in,out]R
          R is DOUBLE PRECISION array, dimension (N)
          The row scale factors for A.  If EQUED = 'R' or 'B', A is
          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
          is not accessed.  R is an input argument if FACT = 'F';
          otherwise, R is an output argument.  If FACT = 'F' and
          EQUED = 'R' or 'B', each element of R must be positive.
[in,out]C
          C is DOUBLE PRECISION array, dimension (N)
          The column scale factors for A.  If EQUED = 'C' or 'B', A is
          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
          is not accessed.  C is an input argument if FACT = 'F';
          otherwise, C is an output argument.  If FACT = 'F' and
          EQUED = 'C' or 'B', each element of C must be positive.
[in,out]B
          B is COMPLEX*16 array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit,
          if EQUED = 'N', B is not modified;
          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
          diag(R)*B;
          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
          overwritten by diag(C)*B.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
[out]X
          X is COMPLEX*16 array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
          to the original system of equations.  Note that A and B are
          modified on exit if EQUED .ne. 'N', and the solution to the
          equilibrated system is inv(diag(C))*X if TRANS = 'N' and
          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
          and EQUED = 'R' or 'B'.
[in]LDX
          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).
[out]RCOND
          RCOND is DOUBLE PRECISION
          The estimate of the reciprocal condition number of the matrix
          A after equilibration (if done).  If RCOND is less than the
          machine precision (in particular, if RCOND = 0), the matrix
          is singular to working precision.  This condition is
          indicated by a return code of INFO > 0.
[out]FERR
          FERR is DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.
[out]BERR
          BERR is DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).
[out]WORK
          WORK is COMPLEX*16 array, dimension (2*N)
[out]RWORK
          RWORK is DOUBLE PRECISION array, dimension (MAX(1,N))
          On exit, RWORK(1) contains the reciprocal pivot growth
          factor norm(A)/norm(U). The "max absolute element" norm is
          used. If RWORK(1) is much less than 1, then the stability
          of the LU factorization of the (equilibrated) matrix A
          could be poor. This also means that the solution X, condition
          estimator RCOND, and forward error bound FERR could be
          unreliable. If factorization fails with 0<INFO<=N, then
          RWORK(1) contains the reciprocal pivot growth factor for the
          leading INFO columns of A.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  U(i,i) is exactly zero.  The factorization
                       has been completed, but the factor U is exactly
                       singular, so the solution and error bounds
                       could not be computed. RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the
                       value of RCOND would suggest.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 367 of file zgbsvx.f.

370*
371* -- LAPACK driver routine --
372* -- LAPACK is a software package provided by Univ. of Tennessee, --
373* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
374*
375* .. Scalar Arguments ..
376 CHARACTER EQUED, FACT, TRANS
377 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
378 DOUBLE PRECISION RCOND
379* ..
380* .. Array Arguments ..
381 INTEGER IPIV( * )
382 DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
383 $ RWORK( * )
384 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
385 $ WORK( * ), X( LDX, * )
386* ..
387*
388* =====================================================================
389* Moved setting of INFO = N+1 so INFO does not subsequently get
390* overwritten. Sven, 17 Mar 05.
391* =====================================================================
392*
393* .. Parameters ..
394 DOUBLE PRECISION ZERO, ONE
395 parameter( zero = 0.0d+0, one = 1.0d+0 )
396* ..
397* .. Local Scalars ..
398 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
399 CHARACTER NORM
400 INTEGER I, INFEQU, J, J1, J2
401 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
402 $ ROWCND, RPVGRW, SMLNUM
403* ..
404* .. External Functions ..
405 LOGICAL LSAME
406 DOUBLE PRECISION DLAMCH, ZLANGB, ZLANTB
407 EXTERNAL lsame, dlamch, zlangb, zlantb
408* ..
409* .. External Subroutines ..
410 EXTERNAL xerbla, zcopy, zgbcon, zgbequ, zgbrfs, zgbtrf,
412* ..
413* .. Intrinsic Functions ..
414 INTRINSIC abs, max, min
415* ..
416* .. Executable Statements ..
417*
418 info = 0
419 nofact = lsame( fact, 'N' )
420 equil = lsame( fact, 'E' )
421 notran = lsame( trans, 'N' )
422 IF( nofact .OR. equil ) THEN
423 equed = 'N'
424 rowequ = .false.
425 colequ = .false.
426 ELSE
427 rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
428 colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
429 smlnum = dlamch( 'Safe minimum' )
430 bignum = one / smlnum
431 END IF
432*
433* Test the input parameters.
434*
435 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
436 $ THEN
437 info = -1
438 ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
439 $ lsame( trans, 'C' ) ) THEN
440 info = -2
441 ELSE IF( n.LT.0 ) THEN
442 info = -3
443 ELSE IF( kl.LT.0 ) THEN
444 info = -4
445 ELSE IF( ku.LT.0 ) THEN
446 info = -5
447 ELSE IF( nrhs.LT.0 ) THEN
448 info = -6
449 ELSE IF( ldab.LT.kl+ku+1 ) THEN
450 info = -8
451 ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
452 info = -10
453 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
454 $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
455 info = -12
456 ELSE
457 IF( rowequ ) THEN
458 rcmin = bignum
459 rcmax = zero
460 DO 10 j = 1, n
461 rcmin = min( rcmin, r( j ) )
462 rcmax = max( rcmax, r( j ) )
463 10 CONTINUE
464 IF( rcmin.LE.zero ) THEN
465 info = -13
466 ELSE IF( n.GT.0 ) THEN
467 rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
468 ELSE
469 rowcnd = one
470 END IF
471 END IF
472 IF( colequ .AND. info.EQ.0 ) THEN
473 rcmin = bignum
474 rcmax = zero
475 DO 20 j = 1, n
476 rcmin = min( rcmin, c( j ) )
477 rcmax = max( rcmax, c( j ) )
478 20 CONTINUE
479 IF( rcmin.LE.zero ) THEN
480 info = -14
481 ELSE IF( n.GT.0 ) THEN
482 colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
483 ELSE
484 colcnd = one
485 END IF
486 END IF
487 IF( info.EQ.0 ) THEN
488 IF( ldb.LT.max( 1, n ) ) THEN
489 info = -16
490 ELSE IF( ldx.LT.max( 1, n ) ) THEN
491 info = -18
492 END IF
493 END IF
494 END IF
495*
496 IF( info.NE.0 ) THEN
497 CALL xerbla( 'ZGBSVX', -info )
498 RETURN
499 END IF
500*
501 IF( equil ) THEN
502*
503* Compute row and column scalings to equilibrate the matrix A.
504*
505 CALL zgbequ( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
506 $ amax, infequ )
507 IF( infequ.EQ.0 ) THEN
508*
509* Equilibrate the matrix.
510*
511 CALL zlaqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
512 $ amax, equed )
513 rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
514 colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
515 END IF
516 END IF
517*
518* Scale the right hand side.
519*
520 IF( notran ) THEN
521 IF( rowequ ) THEN
522 DO 40 j = 1, nrhs
523 DO 30 i = 1, n
524 b( i, j ) = r( i )*b( i, j )
525 30 CONTINUE
526 40 CONTINUE
527 END IF
528 ELSE IF( colequ ) THEN
529 DO 60 j = 1, nrhs
530 DO 50 i = 1, n
531 b( i, j ) = c( i )*b( i, j )
532 50 CONTINUE
533 60 CONTINUE
534 END IF
535*
536 IF( nofact .OR. equil ) THEN
537*
538* Compute the LU factorization of the band matrix A.
539*
540 DO 70 j = 1, n
541 j1 = max( j-ku, 1 )
542 j2 = min( j+kl, n )
543 CALL zcopy( j2-j1+1, ab( ku+1-j+j1, j ), 1,
544 $ afb( kl+ku+1-j+j1, j ), 1 )
545 70 CONTINUE
546*
547 CALL zgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
548*
549* Return if INFO is non-zero.
550*
551 IF( info.GT.0 ) THEN
552*
553* Compute the reciprocal pivot growth factor of the
554* leading rank-deficient INFO columns of A.
555*
556 anorm = zero
557 DO 90 j = 1, info
558 DO 80 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
559 anorm = max( anorm, abs( ab( i, j ) ) )
560 80 CONTINUE
561 90 CONTINUE
562 rpvgrw = zlantb( 'M', 'U', 'N', info, min( info-1, kl+ku ),
563 $ afb( max( 1, kl+ku+2-info ), 1 ), ldafb,
564 $ rwork )
565 IF( rpvgrw.EQ.zero ) THEN
566 rpvgrw = one
567 ELSE
568 rpvgrw = anorm / rpvgrw
569 END IF
570 rwork( 1 ) = rpvgrw
571 rcond = zero
572 RETURN
573 END IF
574 END IF
575*
576* Compute the norm of the matrix A and the
577* reciprocal pivot growth factor RPVGRW.
578*
579 IF( notran ) THEN
580 norm = '1'
581 ELSE
582 norm = 'I'
583 END IF
584 anorm = zlangb( norm, n, kl, ku, ab, ldab, rwork )
585 rpvgrw = zlantb( 'M', 'U', 'N', n, kl+ku, afb, ldafb, rwork )
586 IF( rpvgrw.EQ.zero ) THEN
587 rpvgrw = one
588 ELSE
589 rpvgrw = zlangb( 'M', n, kl, ku, ab, ldab, rwork ) / rpvgrw
590 END IF
591*
592* Compute the reciprocal of the condition number of A.
593*
594 CALL zgbcon( norm, n, kl, ku, afb, ldafb, ipiv, anorm, rcond,
595 $ work, rwork, info )
596*
597* Compute the solution matrix X.
598*
599 CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
600 CALL zgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
601 $ info )
602*
603* Use iterative refinement to improve the computed solution and
604* compute error bounds and backward error estimates for it.
605*
606 CALL zgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
607 $ b, ldb, x, ldx, ferr, berr, work, rwork, info )
608*
609* Transform the solution matrix X to a solution of the original
610* system.
611*
612 IF( notran ) THEN
613 IF( colequ ) THEN
614 DO 110 j = 1, nrhs
615 DO 100 i = 1, n
616 x( i, j ) = c( i )*x( i, j )
617 100 CONTINUE
618 110 CONTINUE
619 DO 120 j = 1, nrhs
620 ferr( j ) = ferr( j ) / colcnd
621 120 CONTINUE
622 END IF
623 ELSE IF( rowequ ) THEN
624 DO 140 j = 1, nrhs
625 DO 130 i = 1, n
626 x( i, j ) = r( i )*x( i, j )
627 130 CONTINUE
628 140 CONTINUE
629 DO 150 j = 1, nrhs
630 ferr( j ) = ferr( j ) / rowcnd
631 150 CONTINUE
632 END IF
633*
634* Set INFO = N+1 if the matrix is singular to working precision.
635*
636 IF( rcond.LT.dlamch( 'Epsilon' ) )
637 $ info = n + 1
638*
639 rwork( 1 ) = rpvgrw
640 RETURN
641*
642* End of ZGBSVX
643*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zcopy(n, zx, incx, zy, incy)
ZCOPY
Definition zcopy.f:81
subroutine zgbcon(norm, n, kl, ku, ab, ldab, ipiv, anorm, rcond, work, rwork, info)
ZGBCON
Definition zgbcon.f:147
subroutine zgbequ(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info)
ZGBEQU
Definition zgbequ.f:154
subroutine zgbrfs(trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZGBRFS
Definition zgbrfs.f:206
subroutine zgbtrf(m, n, kl, ku, ab, ldab, ipiv, info)
ZGBTRF
Definition zgbtrf.f:144
subroutine zgbtrs(trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
ZGBTRS
Definition zgbtrs.f:138
subroutine zlacpy(uplo, m, n, a, lda, b, ldb)
ZLACPY copies all or part of one two-dimensional array to another.
Definition zlacpy.f:103
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
double precision function zlangb(norm, n, kl, ku, ab, ldab, work)
ZLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition zlangb.f:125
double precision function zlantb(norm, uplo, diag, n, k, ab, ldab, work)
ZLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition zlantb.f:141
subroutine zlaqgb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, equed)
ZLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.
Definition zlaqgb.f:160
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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