LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 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.```
Date
April 2012

Definition at line 371 of file dgbsvx.f.

371 *
372 * -- LAPACK driver routine (version 3.4.1) --
373 * -- LAPACK is a software package provided by Univ. of Tennessee, --
374 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
375 * April 2012
376 *
377 * .. Scalar Arguments ..
378  CHARACTER equed, fact, trans
379  INTEGER info, kl, ku, ldab, ldafb, ldb, ldx, n, nrhs
380  DOUBLE PRECISION rcond
381 * ..
382 * .. Array Arguments ..
383  INTEGER ipiv( * ), iwork( * )
384  DOUBLE PRECISION ab( ldab, * ), afb( ldafb, * ), b( ldb, * ),
385  \$ berr( * ), c( * ), ferr( * ), r( * ),
386  \$ work( * ), x( ldx, * )
387 * ..
388 *
389 * =====================================================================
390 *
391 * .. Parameters ..
392  DOUBLE PRECISION zero, one
393  parameter ( zero = 0.0d+0, one = 1.0d+0 )
394 * ..
395 * .. Local Scalars ..
396  LOGICAL colequ, equil, nofact, notran, rowequ
397  CHARACTER norm
398  INTEGER i, infequ, j, j1, j2
399  DOUBLE PRECISION amax, anorm, bignum, colcnd, rcmax, rcmin,
400  \$ rowcnd, rpvgrw, smlnum
401 * ..
402 * .. External Functions ..
403  LOGICAL lsame
404  DOUBLE PRECISION dlamch, dlangb, dlantb
405  EXTERNAL lsame, dlamch, dlangb, dlantb
406 * ..
407 * .. External Subroutines ..
408  EXTERNAL dcopy, dgbcon, dgbequ, dgbrfs, dgbtrf, dgbtrs,
409  \$ dlacpy, dlaqgb, xerbla
410 * ..
411 * .. Intrinsic Functions ..
412  INTRINSIC abs, max, min
413 * ..
414 * .. Executable Statements ..
415 *
416  info = 0
417  nofact = lsame( fact, 'N' )
418  equil = lsame( fact, 'E' )
419  notran = lsame( trans, 'N' )
420  IF( nofact .OR. equil ) THEN
421  equed = 'N'
422  rowequ = .false.
423  colequ = .false.
424  ELSE
425  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
426  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
427  smlnum = dlamch( 'Safe minimum' )
428  bignum = one / smlnum
429  END IF
430 *
431 * Test the input parameters.
432 *
433  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
434  \$ THEN
435  info = -1
436  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
437  \$ lsame( trans, 'C' ) ) THEN
438  info = -2
439  ELSE IF( n.LT.0 ) THEN
440  info = -3
441  ELSE IF( kl.LT.0 ) THEN
442  info = -4
443  ELSE IF( ku.LT.0 ) THEN
444  info = -5
445  ELSE IF( nrhs.LT.0 ) THEN
446  info = -6
447  ELSE IF( ldab.LT.kl+ku+1 ) THEN
448  info = -8
449  ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
450  info = -10
451  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
452  \$ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
453  info = -12
454  ELSE
455  IF( rowequ ) THEN
456  rcmin = bignum
457  rcmax = zero
458  DO 10 j = 1, n
459  rcmin = min( rcmin, r( j ) )
460  rcmax = max( rcmax, r( j ) )
461  10 CONTINUE
462  IF( rcmin.LE.zero ) THEN
463  info = -13
464  ELSE IF( n.GT.0 ) THEN
465  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
466  ELSE
467  rowcnd = one
468  END IF
469  END IF
470  IF( colequ .AND. info.EQ.0 ) THEN
471  rcmin = bignum
472  rcmax = zero
473  DO 20 j = 1, n
474  rcmin = min( rcmin, c( j ) )
475  rcmax = max( rcmax, c( j ) )
476  20 CONTINUE
477  IF( rcmin.LE.zero ) THEN
478  info = -14
479  ELSE IF( n.GT.0 ) THEN
480  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
481  ELSE
482  colcnd = one
483  END IF
484  END IF
485  IF( info.EQ.0 ) THEN
486  IF( ldb.LT.max( 1, n ) ) THEN
487  info = -16
488  ELSE IF( ldx.LT.max( 1, n ) ) THEN
489  info = -18
490  END IF
491  END IF
492  END IF
493 *
494  IF( info.NE.0 ) THEN
495  CALL xerbla( 'DGBSVX', -info )
496  RETURN
497  END IF
498 *
499  IF( equil ) THEN
500 *
501 * Compute row and column scalings to equilibrate the matrix A.
502 *
503  CALL dgbequ( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
504  \$ amax, infequ )
505  IF( infequ.EQ.0 ) THEN
506 *
507 * Equilibrate the matrix.
508 *
509  CALL dlaqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
510  \$ amax, equed )
511  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
512  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
513  END IF
514  END IF
515 *
516 * Scale the right hand side.
517 *
518  IF( notran ) THEN
519  IF( rowequ ) THEN
520  DO 40 j = 1, nrhs
521  DO 30 i = 1, n
522  b( i, j ) = r( i )*b( i, j )
523  30 CONTINUE
524  40 CONTINUE
525  END IF
526  ELSE IF( colequ ) THEN
527  DO 60 j = 1, nrhs
528  DO 50 i = 1, n
529  b( i, j ) = c( i )*b( i, j )
530  50 CONTINUE
531  60 CONTINUE
532  END IF
533 *
534  IF( nofact .OR. equil ) THEN
535 *
536 * Compute the LU factorization of the band matrix A.
537 *
538  DO 70 j = 1, n
539  j1 = max( j-ku, 1 )
540  j2 = min( j+kl, n )
541  CALL dcopy( j2-j1+1, ab( ku+1-j+j1, j ), 1,
542  \$ afb( kl+ku+1-j+j1, j ), 1 )
543  70 CONTINUE
544 *
545  CALL dgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
546 *
547 * Return if INFO is non-zero.
548 *
549  IF( info.GT.0 ) THEN
550 *
551 * Compute the reciprocal pivot growth factor of the
552 * leading rank-deficient INFO columns of A.
553 *
554  anorm = zero
555  DO 90 j = 1, info
556  DO 80 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
557  anorm = max( anorm, abs( ab( i, j ) ) )
558  80 CONTINUE
559  90 CONTINUE
560  rpvgrw = dlantb( 'M', 'U', 'N', info, min( info-1, kl+ku ),
561  \$ afb( max( 1, kl+ku+2-info ), 1 ), ldafb,
562  \$ work )
563  IF( rpvgrw.EQ.zero ) THEN
564  rpvgrw = one
565  ELSE
566  rpvgrw = anorm / rpvgrw
567  END IF
568  work( 1 ) = rpvgrw
569  rcond = zero
570  RETURN
571  END IF
572  END IF
573 *
574 * Compute the norm of the matrix A and the
575 * reciprocal pivot growth factor RPVGRW.
576 *
577  IF( notran ) THEN
578  norm = '1'
579  ELSE
580  norm = 'I'
581  END IF
582  anorm = dlangb( norm, n, kl, ku, ab, ldab, work )
583  rpvgrw = dlantb( 'M', 'U', 'N', n, kl+ku, afb, ldafb, work )
584  IF( rpvgrw.EQ.zero ) THEN
585  rpvgrw = one
586  ELSE
587  rpvgrw = dlangb( 'M', n, kl, ku, ab, ldab, work ) / rpvgrw
588  END IF
589 *
590 * Compute the reciprocal of the condition number of A.
591 *
592  CALL dgbcon( norm, n, kl, ku, afb, ldafb, ipiv, anorm, rcond,
593  \$ work, iwork, info )
594 *
595 * Compute the solution matrix X.
596 *
597  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
598  CALL dgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
599  \$ info )
600 *
601 * Use iterative refinement to improve the computed solution and
602 * compute error bounds and backward error estimates for it.
603 *
604  CALL dgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
605  \$ b, ldb, x, ldx, ferr, berr, work, iwork, info )
606 *
607 * Transform the solution matrix X to a solution of the original
608 * system.
609 *
610  IF( notran ) THEN
611  IF( colequ ) THEN
612  DO 110 j = 1, nrhs
613  DO 100 i = 1, n
614  x( i, j ) = c( i )*x( i, j )
615  100 CONTINUE
616  110 CONTINUE
617  DO 120 j = 1, nrhs
618  ferr( j ) = ferr( j ) / colcnd
619  120 CONTINUE
620  END IF
621  ELSE IF( rowequ ) THEN
622  DO 140 j = 1, nrhs
623  DO 130 i = 1, n
624  x( i, j ) = r( i )*x( i, j )
625  130 CONTINUE
626  140 CONTINUE
627  DO 150 j = 1, nrhs
628  ferr( j ) = ferr( j ) / rowcnd
629  150 CONTINUE
630  END IF
631 *
632 * Set INFO = N+1 if the matrix is singular to working precision.
633 *
634  IF( rcond.LT.dlamch( 'Epsilon' ) )
635  \$ info = n + 1
636 *
637  work( 1 ) = rpvgrw
638  RETURN
639 *
640 * End of DGBSVX
641 *
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:207
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:53
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
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:161
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dgbequ(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
DGBEQU
Definition: dgbequ.f:155
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:126
subroutine dgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
DGBTRF
Definition: dgbtrf.f:146
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, or the element of largest absolute value of a triangular band matrix.
Definition: dlantb.f:142
subroutine dgbcon(NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DGBCON
Definition: dgbcon.f:148
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
DGBTRS
Definition: dgbtrs.f:140

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