LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ 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

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

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<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
                       value of RCOND would suggest.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

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,
406  $ dlacpy, dlaqgb, xerbla
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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