LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dgbsvx()

 subroutine dgbsvx ( character FACT, character TRANS, integer N, integer KL, integer KU, integer NRHS, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldafb, * ) AFB, integer LDAFB, integer, dimension( * ) IPIV, character EQUED, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

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

Purpose:
``` DGBSVX uses the LU factorization to compute the solution to a real
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 (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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DGBTRF. 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 DGBTRF; 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (3*N) On exit, WORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If WORK(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 366 of file dgbsvx.f.

369*
370* -- LAPACK driver routine --
371* -- LAPACK is a software package provided by Univ. of Tennessee, --
372* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
373*
374* .. Scalar Arguments ..
375 CHARACTER EQUED, FACT, TRANS
376 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
377 DOUBLE PRECISION RCOND
378* ..
379* .. Array Arguments ..
380 INTEGER IPIV( * ), IWORK( * )
381 DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
382 \$ BERR( * ), C( * ), FERR( * ), R( * ),
383 \$ WORK( * ), X( LDX, * )
384* ..
385*
386* =====================================================================
387*
388* .. Parameters ..
389 DOUBLE PRECISION ZERO, ONE
390 parameter( zero = 0.0d+0, one = 1.0d+0 )
391* ..
392* .. Local Scalars ..
393 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
394 CHARACTER NORM
395 INTEGER I, INFEQU, J, J1, J2
396 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
397 \$ ROWCND, RPVGRW, SMLNUM
398* ..
399* .. External Functions ..
400 LOGICAL LSAME
401 DOUBLE PRECISION DLAMCH, DLANGB, DLANTB
402 EXTERNAL lsame, dlamch, dlangb, dlantb
403* ..
404* .. External Subroutines ..
405 EXTERNAL dcopy, dgbcon, dgbequ, dgbrfs, dgbtrf, dgbtrs,
407* ..
408* .. Intrinsic Functions ..
409 INTRINSIC abs, max, min
410* ..
411* .. Executable Statements ..
412*
413 info = 0
414 nofact = lsame( fact, 'N' )
415 equil = lsame( fact, 'E' )
416 notran = lsame( trans, 'N' )
417 IF( nofact .OR. equil ) THEN
418 equed = 'N'
419 rowequ = .false.
420 colequ = .false.
421 ELSE
422 rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
423 colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
424 smlnum = dlamch( 'Safe minimum' )
425 bignum = one / smlnum
426 END IF
427*
428* Test the input parameters.
429*
430 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
431 \$ THEN
432 info = -1
433 ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
434 \$ lsame( trans, 'C' ) ) THEN
435 info = -2
436 ELSE IF( n.LT.0 ) THEN
437 info = -3
438 ELSE IF( kl.LT.0 ) THEN
439 info = -4
440 ELSE IF( ku.LT.0 ) THEN
441 info = -5
442 ELSE IF( nrhs.LT.0 ) THEN
443 info = -6
444 ELSE IF( ldab.LT.kl+ku+1 ) THEN
445 info = -8
446 ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
447 info = -10
448 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
449 \$ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
450 info = -12
451 ELSE
452 IF( rowequ ) THEN
453 rcmin = bignum
454 rcmax = zero
455 DO 10 j = 1, n
456 rcmin = min( rcmin, r( j ) )
457 rcmax = max( rcmax, r( j ) )
458 10 CONTINUE
459 IF( rcmin.LE.zero ) THEN
460 info = -13
461 ELSE IF( n.GT.0 ) THEN
462 rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
463 ELSE
464 rowcnd = one
465 END IF
466 END IF
467 IF( colequ .AND. info.EQ.0 ) THEN
468 rcmin = bignum
469 rcmax = zero
470 DO 20 j = 1, n
471 rcmin = min( rcmin, c( j ) )
472 rcmax = max( rcmax, c( j ) )
473 20 CONTINUE
474 IF( rcmin.LE.zero ) THEN
475 info = -14
476 ELSE IF( n.GT.0 ) THEN
477 colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
478 ELSE
479 colcnd = one
480 END IF
481 END IF
482 IF( info.EQ.0 ) THEN
483 IF( ldb.LT.max( 1, n ) ) THEN
484 info = -16
485 ELSE IF( ldx.LT.max( 1, n ) ) THEN
486 info = -18
487 END IF
488 END IF
489 END IF
490*
491 IF( info.NE.0 ) THEN
492 CALL xerbla( 'DGBSVX', -info )
493 RETURN
494 END IF
495*
496 IF( equil ) THEN
497*
498* Compute row and column scalings to equilibrate the matrix A.
499*
500 CALL dgbequ( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
501 \$ amax, infequ )
502 IF( infequ.EQ.0 ) THEN
503*
504* Equilibrate the matrix.
505*
506 CALL dlaqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
507 \$ amax, equed )
508 rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
509 colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
510 END IF
511 END IF
512*
513* Scale the right hand side.
514*
515 IF( notran ) THEN
516 IF( rowequ ) THEN
517 DO 40 j = 1, nrhs
518 DO 30 i = 1, n
519 b( i, j ) = r( i )*b( i, j )
520 30 CONTINUE
521 40 CONTINUE
522 END IF
523 ELSE IF( colequ ) THEN
524 DO 60 j = 1, nrhs
525 DO 50 i = 1, n
526 b( i, j ) = c( i )*b( i, j )
527 50 CONTINUE
528 60 CONTINUE
529 END IF
530*
531 IF( nofact .OR. equil ) THEN
532*
533* Compute the LU factorization of the band matrix A.
534*
535 DO 70 j = 1, n
536 j1 = max( j-ku, 1 )
537 j2 = min( j+kl, n )
538 CALL dcopy( j2-j1+1, ab( ku+1-j+j1, j ), 1,
539 \$ afb( kl+ku+1-j+j1, j ), 1 )
540 70 CONTINUE
541*
542 CALL dgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
543*
544* Return if INFO is non-zero.
545*
546 IF( info.GT.0 ) THEN
547*
548* Compute the reciprocal pivot growth factor of the
549* leading rank-deficient INFO columns of A.
550*
551 anorm = zero
552 DO 90 j = 1, info
553 DO 80 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
554 anorm = max( anorm, abs( ab( i, j ) ) )
555 80 CONTINUE
556 90 CONTINUE
557 rpvgrw = dlantb( 'M', 'U', 'N', info, min( info-1, kl+ku ),
558 \$ afb( max( 1, kl+ku+2-info ), 1 ), ldafb,
559 \$ work )
560 IF( rpvgrw.EQ.zero ) THEN
561 rpvgrw = one
562 ELSE
563 rpvgrw = anorm / rpvgrw
564 END IF
565 work( 1 ) = rpvgrw
566 rcond = zero
567 RETURN
568 END IF
569 END IF
570*
571* Compute the norm of the matrix A and the
572* reciprocal pivot growth factor RPVGRW.
573*
574 IF( notran ) THEN
575 norm = '1'
576 ELSE
577 norm = 'I'
578 END IF
579 anorm = dlangb( norm, n, kl, ku, ab, ldab, work )
580 rpvgrw = dlantb( 'M', 'U', 'N', n, kl+ku, afb, ldafb, work )
581 IF( rpvgrw.EQ.zero ) THEN
582 rpvgrw = one
583 ELSE
584 rpvgrw = dlangb( 'M', n, kl, ku, ab, ldab, work ) / rpvgrw
585 END IF
586*
587* Compute the reciprocal of the condition number of A.
588*
589 CALL dgbcon( norm, n, kl, ku, afb, ldafb, ipiv, anorm, rcond,
590 \$ work, iwork, info )
591*
592* Compute the solution matrix X.
593*
594 CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
595 CALL dgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
596 \$ info )
597*
598* Use iterative refinement to improve the computed solution and
599* compute error bounds and backward error estimates for it.
600*
601 CALL dgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
602 \$ b, ldb, x, ldx, ferr, berr, work, iwork, info )
603*
604* Transform the solution matrix X to a solution of the original
605* system.
606*
607 IF( notran ) THEN
608 IF( colequ ) THEN
609 DO 110 j = 1, nrhs
610 DO 100 i = 1, n
611 x( i, j ) = c( i )*x( i, j )
612 100 CONTINUE
613 110 CONTINUE
614 DO 120 j = 1, nrhs
615 ferr( j ) = ferr( j ) / colcnd
616 120 CONTINUE
617 END IF
618 ELSE IF( rowequ ) THEN
619 DO 140 j = 1, nrhs
620 DO 130 i = 1, n
621 x( i, j ) = r( i )*x( i, j )
622 130 CONTINUE
623 140 CONTINUE
624 DO 150 j = 1, nrhs
625 ferr( j ) = ferr( j ) / rowcnd
626 150 CONTINUE
627 END IF
628*
629* Set INFO = N+1 if the matrix is singular to working precision.
630*
631 IF( rcond.LT.dlamch( 'Epsilon' ) )
632 \$ info = n + 1
633*
634 work( 1 ) = rpvgrw
635 RETURN
636*
637* End of DGBSVX
638*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dlaqgb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
DLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.
Definition: dlaqgb.f:159
double precision function dlangb(NORM, N, KL, KU, AB, LDAB, WORK)
DLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlangb.f:124
subroutine dgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
DGBTRS
Definition: dgbtrs.f:138
subroutine dgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
DGBTRF
Definition: dgbtrf.f:144
subroutine dgbequ(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
DGBEQU
Definition: dgbequ.f:153
subroutine dgbcon(NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DGBCON
Definition: dgbcon.f:146
subroutine dgbrfs(TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DGBRFS
Definition: dgbrfs.f:205
double precision function dlantb(NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
DLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlantb.f:140
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