LAPACK  3.8.0
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

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

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<INFO<=N, then
          WORK(1) contains the reciprocal pivot growth factor for the
          leading INFO columns of A.
[out]IWORK
          IWORK is INTEGER array, dimension (N)
[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
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
April 2012

Definition at line 370 of file sgbsvx.f.

370 *
371 * -- LAPACK driver routine (version 3.7.0) --
372 * -- LAPACK is a software package provided by Univ. of Tennessee, --
373 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
374 * April 2012
375 *
376 * .. Scalar Arguments ..
377  CHARACTER equed, fact, trans
378  INTEGER info, kl, ku, ldab, ldafb, ldb, ldx, n, nrhs
379  REAL rcond
380 * ..
381 * .. Array Arguments ..
382  INTEGER ipiv( * ), iwork( * )
383  REAL ab( ldab, * ), afb( ldafb, * ), b( ldb, * ),
384  $ berr( * ), c( * ), ferr( * ), r( * ),
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  REAL zero, one
395  parameter( zero = 0.0e+0, one = 1.0e+0 )
396 * ..
397 * .. Local Scalars ..
398  LOGICAL colequ, equil, nofact, notran, rowequ
399  CHARACTER norm
400  INTEGER i, infequ, j, j1, j2
401  REAL amax, anorm, bignum, colcnd, rcmax, rcmin,
402  $ rowcnd, rpvgrw, smlnum
403 * ..
404 * .. External Functions ..
405  LOGICAL lsame
406  REAL slamch, slangb, slantb
407  EXTERNAL lsame, slamch, slangb, slantb
408 * ..
409 * .. External Subroutines ..
410  EXTERNAL scopy, sgbcon, sgbequ, sgbrfs, sgbtrf, sgbtrs,
411  $ slacpy, slaqgb, xerbla
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 = slamch( '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( 'SGBSVX', -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 sgbequ( 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 slaqgb( 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 scopy( 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 sgbtrf( 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 = slantb( 'M', 'U', 'N', info, min( info-1, kl+ku ),
563  $ afb( max( 1, kl+ku+2-info ), 1 ), ldafb,
564  $ work )
565  IF( rpvgrw.EQ.zero ) THEN
566  rpvgrw = one
567  ELSE
568  rpvgrw = anorm / rpvgrw
569  END IF
570  work( 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 = slangb( norm, n, kl, ku, ab, ldab, work )
585  rpvgrw = slantb( 'M', 'U', 'N', n, kl+ku, afb, ldafb, work )
586  IF( rpvgrw.EQ.zero ) THEN
587  rpvgrw = one
588  ELSE
589  rpvgrw = slangb( 'M', n, kl, ku, ab, ldab, work ) / rpvgrw
590  END IF
591 *
592 * Compute the reciprocal of the condition number of A.
593 *
594  CALL sgbcon( norm, n, kl, ku, afb, ldafb, ipiv, anorm, rcond,
595  $ work, iwork, info )
596 *
597 * Compute the solution matrix X.
598 *
599  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
600  CALL sgbtrs( 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 sgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
607  $ b, ldb, x, ldx, ferr, berr, work, iwork, 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.slamch( 'Epsilon' ) )
637  $ info = n + 1
638 *
639  work( 1 ) = rpvgrw
640  RETURN
641 *
642 * End of SGBSVX
643 *
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:84
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine sgbcon(NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
SGBCON
Definition: sgbcon.f:148
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, or the element of largest absolute value of a triangular band matrix.
Definition: slantb.f:142
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine sgbequ(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
SGBEQU
Definition: sgbequ.f:155
subroutine sgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
SGBTRF
Definition: sgbtrf.f:146
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine sgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
SGBTRS
Definition: sgbtrs.f:140
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:207
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:126
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:161
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
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