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

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 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.```

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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