 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

◆ 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,
408  \$ slacpy, slaqgb, xerbla
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
Here is the call graph for this function:
Here is the caller graph for this function: