 LAPACK  3.10.1 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

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.```

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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