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

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

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

Purpose:
``` SGBSVX 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 REAL 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 REAL 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 SGBTRF. 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 SGBTRF; 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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```

Definition at line 365 of file sgbsvx.f.

368*
369* -- LAPACK driver routine --
370* -- LAPACK is a software package provided by Univ. of Tennessee, --
371* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
372*
373* .. Scalar Arguments ..
374 CHARACTER EQUED, FACT, TRANS
375 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
376 REAL RCOND
377* ..
378* .. Array Arguments ..
379 INTEGER IPIV( * ), IWORK( * )
380 REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
381 \$ BERR( * ), C( * ), FERR( * ), R( * ),
382 \$ WORK( * ), X( LDX, * )
383* ..
384*
385* =====================================================================
386* Moved setting of INFO = N+1 so INFO does not subsequently get
387* overwritten. Sven, 17 Mar 05.
388* =====================================================================
389*
390* .. Parameters ..
391 REAL ZERO, ONE
392 parameter( zero = 0.0e+0, one = 1.0e+0 )
393* ..
394* .. Local Scalars ..
395 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
396 CHARACTER NORM
397 INTEGER I, INFEQU, J, J1, J2
398 REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
399 \$ ROWCND, RPVGRW, SMLNUM
400* ..
401* .. External Functions ..
402 LOGICAL LSAME
403 REAL SLAMCH, SLANGB, SLANTB
404 EXTERNAL lsame, slamch, slangb, slantb
405* ..
406* .. External Subroutines ..
407 EXTERNAL scopy, sgbcon, sgbequ, sgbrfs, sgbtrf, sgbtrs,
409* ..
410* .. Intrinsic Functions ..
411 INTRINSIC abs, max, min
412* ..
413* .. Executable Statements ..
414*
415 info = 0
416 nofact = lsame( fact, 'N' )
417 equil = lsame( fact, 'E' )
418 notran = lsame( trans, 'N' )
419 IF( nofact .OR. equil ) THEN
420 equed = 'N'
421 rowequ = .false.
422 colequ = .false.
423 ELSE
424 rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
425 colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
426 smlnum = slamch( 'Safe minimum' )
427 bignum = one / smlnum
428 END IF
429*
430* Test the input parameters.
431*
432 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
433 \$ THEN
434 info = -1
435 ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
436 \$ lsame( trans, 'C' ) ) THEN
437 info = -2
438 ELSE IF( n.LT.0 ) THEN
439 info = -3
440 ELSE IF( kl.LT.0 ) THEN
441 info = -4
442 ELSE IF( ku.LT.0 ) THEN
443 info = -5
444 ELSE IF( nrhs.LT.0 ) THEN
445 info = -6
446 ELSE IF( ldab.LT.kl+ku+1 ) THEN
447 info = -8
448 ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
449 info = -10
450 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
451 \$ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
452 info = -12
453 ELSE
454 IF( rowequ ) THEN
455 rcmin = bignum
456 rcmax = zero
457 DO 10 j = 1, n
458 rcmin = min( rcmin, r( j ) )
459 rcmax = max( rcmax, r( j ) )
460 10 CONTINUE
461 IF( rcmin.LE.zero ) THEN
462 info = -13
463 ELSE IF( n.GT.0 ) THEN
464 rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
465 ELSE
466 rowcnd = one
467 END IF
468 END IF
469 IF( colequ .AND. info.EQ.0 ) THEN
470 rcmin = bignum
471 rcmax = zero
472 DO 20 j = 1, n
473 rcmin = min( rcmin, c( j ) )
474 rcmax = max( rcmax, c( j ) )
475 20 CONTINUE
476 IF( rcmin.LE.zero ) THEN
477 info = -14
478 ELSE IF( n.GT.0 ) THEN
479 colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
480 ELSE
481 colcnd = one
482 END IF
483 END IF
484 IF( info.EQ.0 ) THEN
485 IF( ldb.LT.max( 1, n ) ) THEN
486 info = -16
487 ELSE IF( ldx.LT.max( 1, n ) ) THEN
488 info = -18
489 END IF
490 END IF
491 END IF
492*
493 IF( info.NE.0 ) THEN
494 CALL xerbla( 'SGBSVX', -info )
495 RETURN
496 END IF
497*
498 IF( equil ) THEN
499*
500* Compute row and column scalings to equilibrate the matrix A.
501*
502 CALL sgbequ( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
503 \$ amax, infequ )
504 IF( infequ.EQ.0 ) THEN
505*
506* Equilibrate the matrix.
507*
508 CALL slaqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
509 \$ amax, equed )
510 rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
511 colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
512 END IF
513 END IF
514*
515* Scale the right hand side.
516*
517 IF( notran ) THEN
518 IF( rowequ ) THEN
519 DO 40 j = 1, nrhs
520 DO 30 i = 1, n
521 b( i, j ) = r( i )*b( i, j )
522 30 CONTINUE
523 40 CONTINUE
524 END IF
525 ELSE IF( colequ ) THEN
526 DO 60 j = 1, nrhs
527 DO 50 i = 1, n
528 b( i, j ) = c( i )*b( i, j )
529 50 CONTINUE
530 60 CONTINUE
531 END IF
532*
533 IF( nofact .OR. equil ) THEN
534*
535* Compute the LU factorization of the band matrix A.
536*
537 DO 70 j = 1, n
538 j1 = max( j-ku, 1 )
539 j2 = min( j+kl, n )
540 CALL scopy( j2-j1+1, ab( ku+1-j+j1, j ), 1,
541 \$ afb( kl+ku+1-j+j1, j ), 1 )
542 70 CONTINUE
543*
544 CALL sgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
545*
546* Return if INFO is non-zero.
547*
548 IF( info.GT.0 ) THEN
549*
550* Compute the reciprocal pivot growth factor of the
551* leading rank-deficient INFO columns of A.
552*
553 anorm = zero
554 DO 90 j = 1, info
555 DO 80 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
556 anorm = max( anorm, abs( ab( i, j ) ) )
557 80 CONTINUE
558 90 CONTINUE
559 rpvgrw = slantb( 'M', 'U', 'N', info, min( info-1, kl+ku ),
560 \$ afb( max( 1, kl+ku+2-info ), 1 ), ldafb,
561 \$ work )
562 IF( rpvgrw.EQ.zero ) THEN
563 rpvgrw = one
564 ELSE
565 rpvgrw = anorm / rpvgrw
566 END IF
567 work( 1 ) = rpvgrw
568 rcond = zero
569 RETURN
570 END IF
571 END IF
572*
573* Compute the norm of the matrix A and the
574* reciprocal pivot growth factor RPVGRW.
575*
576 IF( notran ) THEN
577 norm = '1'
578 ELSE
579 norm = 'I'
580 END IF
581 anorm = slangb( norm, n, kl, ku, ab, ldab, work )
582 rpvgrw = slantb( 'M', 'U', 'N', n, kl+ku, afb, ldafb, work )
583 IF( rpvgrw.EQ.zero ) THEN
584 rpvgrw = one
585 ELSE
586 rpvgrw = slangb( 'M', n, kl, ku, ab, ldab, work ) / rpvgrw
587 END IF
588*
589* Compute the reciprocal of the condition number of A.
590*
591 CALL sgbcon( norm, n, kl, ku, afb, ldafb, ipiv, anorm, rcond,
592 \$ work, iwork, info )
593*
594* Compute the solution matrix X.
595*
596 CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
597 CALL sgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
598 \$ info )
599*
600* Use iterative refinement to improve the computed solution and
601* compute error bounds and backward error estimates for it.
602*
603 CALL sgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
604 \$ b, ldb, x, ldx, ferr, berr, work, iwork, info )
605*
606* Transform the solution matrix X to a solution of the original
607* system.
608*
609 IF( notran ) THEN
610 IF( colequ ) THEN
611 DO 110 j = 1, nrhs
612 DO 100 i = 1, n
613 x( i, j ) = c( i )*x( i, j )
614 100 CONTINUE
615 110 CONTINUE
616 DO 120 j = 1, nrhs
617 ferr( j ) = ferr( j ) / colcnd
618 120 CONTINUE
619 END IF
620 ELSE IF( rowequ ) THEN
621 DO 140 j = 1, nrhs
622 DO 130 i = 1, n
623 x( i, j ) = r( i )*x( i, j )
624 130 CONTINUE
625 140 CONTINUE
626 DO 150 j = 1, nrhs
627 ferr( j ) = ferr( j ) / rowcnd
628 150 CONTINUE
629 END IF
630*
631* Set INFO = N+1 if the matrix is singular to working precision.
632*
633 IF( rcond.LT.slamch( 'Epsilon' ) )
634 \$ info = n + 1
635*
636 work( 1 ) = rpvgrw
637 RETURN
638*
639* End of SGBSVX
640*
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine slaqgb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
SLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.
Definition: slaqgb.f:159
real function slangb(NORM, N, KL, KU, AB, LDAB, WORK)
SLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slangb.f:124
subroutine sgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
SGBTRS
Definition: sgbtrs.f:138
subroutine sgbrfs(TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SGBRFS
Definition: sgbrfs.f:205
subroutine sgbcon(NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
SGBCON
Definition: sgbcon.f:146
subroutine sgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
SGBTRF
Definition: sgbtrf.f:144
subroutine sgbequ(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
SGBEQU
Definition: sgbequ.f:153
real function slantb(NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
SLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slantb.f:140
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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