LAPACK  3.6.0
LAPACK: Linear Algebra PACKage
Collaboration diagram for complex16:

Functions

subroutine zgbsv (N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
  ZGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver) More...
 
subroutine zgbsvx (FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
  ZGBSVX computes the solution to system of linear equations A * X = B for GB matrices More...
 
subroutine zgbsvxx (FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
  ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices More...
 

Detailed Description

This is the group of complex16 solve driver functions for GB matrices

Function Documentation

subroutine zgbsv ( integer  N,
integer  KL,
integer  KU,
integer  NRHS,
complex*16, dimension( ldab, * )  AB,
integer  LDAB,
integer, dimension( * )  IPIV,
complex*16, dimension( ldb, * )  B,
integer  LDB,
integer  INFO 
)

ZGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)

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

Purpose:
 ZGBSV computes the solution to a complex system of linear equations
 A * 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.

 The LU decomposition with partial pivoting and row interchanges is
 used to factor A 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.  The factored form of A
 is then used to solve the system of equations A * X = B.
Parameters
[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 matrix B.  NRHS >= 0.
[in,out]AB
          AB is COMPLEX*16 array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows KL+1 to
          2*KL+KU+1; rows 1 to KL of the array need not be set.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
          On exit, details of the factorization: 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.
          See below for further details.
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
[out]IPIV
          IPIV is INTEGER array, dimension (N)
          The pivot indices that define the permutation matrix P;
          row i of the matrix was interchanged with row IPIV(i).
[in,out]B
          B is COMPLEX*16 array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,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, U(i,i) is exactly zero.  The factorization
                has been completed, but the factor U is exactly
                singular, and the solution has not been computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
November 2011
Further Details:
  The band storage scheme is illustrated by the following example, when
  M = N = 6, KL = 2, KU = 1:

  On entry:                       On exit:

      *    *    *    +    +    +       *    *    *   u14  u25  u36
      *    *    +    +    +    +       *    *   u13  u24  u35  u46
      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

  Array elements marked * are not used by the routine; elements marked
  + need not be set on entry, but are required by the routine to store
  elements of U because of fill-in resulting from the row interchanges.

Definition at line 164 of file zgbsv.f.

164 *
165 * -- LAPACK driver routine (version 3.4.0) --
166 * -- LAPACK is a software package provided by Univ. of Tennessee, --
167 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
168 * November 2011
169 *
170 * .. Scalar Arguments ..
171  INTEGER info, kl, ku, ldab, ldb, n, nrhs
172 * ..
173 * .. Array Arguments ..
174  INTEGER ipiv( * )
175  COMPLEX*16 ab( ldab, * ), b( ldb, * )
176 * ..
177 *
178 * =====================================================================
179 *
180 * .. External Subroutines ..
181  EXTERNAL xerbla, zgbtrf, zgbtrs
182 * ..
183 * .. Intrinsic Functions ..
184  INTRINSIC max
185 * ..
186 * .. Executable Statements ..
187 *
188 * Test the input parameters.
189 *
190  info = 0
191  IF( n.LT.0 ) THEN
192  info = -1
193  ELSE IF( kl.LT.0 ) THEN
194  info = -2
195  ELSE IF( ku.LT.0 ) THEN
196  info = -3
197  ELSE IF( nrhs.LT.0 ) THEN
198  info = -4
199  ELSE IF( ldab.LT.2*kl+ku+1 ) THEN
200  info = -6
201  ELSE IF( ldb.LT.max( n, 1 ) ) THEN
202  info = -9
203  END IF
204  IF( info.NE.0 ) THEN
205  CALL xerbla( 'ZGBSV ', -info )
206  RETURN
207  END IF
208 *
209 * Compute the LU factorization of the band matrix A.
210 *
211  CALL zgbtrf( n, n, kl, ku, ab, ldab, ipiv, info )
212  IF( info.EQ.0 ) THEN
213 *
214 * Solve the system A*X = B, overwriting B with X.
215 *
216  CALL zgbtrs( 'No transpose', n, kl, ku, nrhs, ab, ldab, ipiv,
217  $ b, ldb, info )
218  END IF
219  RETURN
220 *
221 * End of ZGBSV
222 *
subroutine zgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
ZGBTRF
Definition: zgbtrf.f:146
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
ZGBTRS
Definition: zgbtrs.f:140

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subroutine zgbsvx ( character  FACT,
character  TRANS,
integer  N,
integer  KL,
integer  KU,
integer  NRHS,
complex*16, dimension( ldab, * )  AB,
integer  LDAB,
complex*16, dimension( ldafb, * )  AFB,
integer  LDAFB,
integer, dimension( * )  IPIV,
character  EQUED,
double precision, dimension( * )  R,
double precision, dimension( * )  C,
complex*16, dimension( ldb, * )  B,
integer  LDB,
complex*16, dimension( ldx, * )  X,
integer  LDX,
double precision  RCOND,
double precision, dimension( * )  FERR,
double precision, dimension( * )  BERR,
complex*16, dimension( * )  WORK,
double precision, dimension( * )  RWORK,
integer  INFO 
)

ZGBSVX computes the solution to system of linear equations A * X = B for GB matrices

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

Purpose:
 ZGBSVX uses the LU factorization to compute the solution to a complex
 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  (Conjugate 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 COMPLEX*16 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 COMPLEX*16 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 ZGBTRF.  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 ZGBTRF; 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 array, dimension (2*N)
[out]RWORK
          RWORK is DOUBLE PRECISION array, dimension (N)
          On exit, RWORK(1) contains the reciprocal pivot growth
          factor norm(A)/norm(U). The "max absolute element" norm is
          used. If RWORK(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
          RWORK(1) contains the reciprocal pivot growth factor for the
          leading INFO columns of A.
[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.
Date
April 2012

Definition at line 372 of file zgbsvx.f.

372 *
373 * -- LAPACK driver routine (version 3.4.1) --
374 * -- LAPACK is a software package provided by Univ. of Tennessee, --
375 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
376 * April 2012
377 *
378 * .. Scalar Arguments ..
379  CHARACTER equed, fact, trans
380  INTEGER info, kl, ku, ldab, ldafb, ldb, ldx, n, nrhs
381  DOUBLE PRECISION rcond
382 * ..
383 * .. Array Arguments ..
384  INTEGER ipiv( * )
385  DOUBLE PRECISION berr( * ), c( * ), ferr( * ), r( * ),
386  $ rwork( * )
387  COMPLEX*16 ab( ldab, * ), afb( ldafb, * ), b( ldb, * ),
388  $ work( * ), x( ldx, * )
389 * ..
390 *
391 * =====================================================================
392 * Moved setting of INFO = N+1 so INFO does not subsequently get
393 * overwritten. Sven, 17 Mar 05.
394 * =====================================================================
395 *
396 * .. Parameters ..
397  DOUBLE PRECISION zero, one
398  parameter( zero = 0.0d+0, one = 1.0d+0 )
399 * ..
400 * .. Local Scalars ..
401  LOGICAL colequ, equil, nofact, notran, rowequ
402  CHARACTER norm
403  INTEGER i, infequ, j, j1, j2
404  DOUBLE PRECISION amax, anorm, bignum, colcnd, rcmax, rcmin,
405  $ rowcnd, rpvgrw, smlnum
406 * ..
407 * .. External Functions ..
408  LOGICAL lsame
409  DOUBLE PRECISION dlamch, zlangb, zlantb
410  EXTERNAL lsame, dlamch, zlangb, zlantb
411 * ..
412 * .. External Subroutines ..
413  EXTERNAL xerbla, zcopy, zgbcon, zgbequ, zgbrfs, zgbtrf,
414  $ zgbtrs, zlacpy, zlaqgb
415 * ..
416 * .. Intrinsic Functions ..
417  INTRINSIC abs, max, min
418 * ..
419 * .. Executable Statements ..
420 *
421  info = 0
422  nofact = lsame( fact, 'N' )
423  equil = lsame( fact, 'E' )
424  notran = lsame( trans, 'N' )
425  IF( nofact .OR. equil ) THEN
426  equed = 'N'
427  rowequ = .false.
428  colequ = .false.
429  ELSE
430  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
431  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
432  smlnum = dlamch( 'Safe minimum' )
433  bignum = one / smlnum
434  END IF
435 *
436 * Test the input parameters.
437 *
438  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
439  $ THEN
440  info = -1
441  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
442  $ lsame( trans, 'C' ) ) THEN
443  info = -2
444  ELSE IF( n.LT.0 ) THEN
445  info = -3
446  ELSE IF( kl.LT.0 ) THEN
447  info = -4
448  ELSE IF( ku.LT.0 ) THEN
449  info = -5
450  ELSE IF( nrhs.LT.0 ) THEN
451  info = -6
452  ELSE IF( ldab.LT.kl+ku+1 ) THEN
453  info = -8
454  ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
455  info = -10
456  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
457  $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
458  info = -12
459  ELSE
460  IF( rowequ ) THEN
461  rcmin = bignum
462  rcmax = zero
463  DO 10 j = 1, n
464  rcmin = min( rcmin, r( j ) )
465  rcmax = max( rcmax, r( j ) )
466  10 CONTINUE
467  IF( rcmin.LE.zero ) THEN
468  info = -13
469  ELSE IF( n.GT.0 ) THEN
470  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
471  ELSE
472  rowcnd = one
473  END IF
474  END IF
475  IF( colequ .AND. info.EQ.0 ) THEN
476  rcmin = bignum
477  rcmax = zero
478  DO 20 j = 1, n
479  rcmin = min( rcmin, c( j ) )
480  rcmax = max( rcmax, c( j ) )
481  20 CONTINUE
482  IF( rcmin.LE.zero ) THEN
483  info = -14
484  ELSE IF( n.GT.0 ) THEN
485  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
486  ELSE
487  colcnd = one
488  END IF
489  END IF
490  IF( info.EQ.0 ) THEN
491  IF( ldb.LT.max( 1, n ) ) THEN
492  info = -16
493  ELSE IF( ldx.LT.max( 1, n ) ) THEN
494  info = -18
495  END IF
496  END IF
497  END IF
498 *
499  IF( info.NE.0 ) THEN
500  CALL xerbla( 'ZGBSVX', -info )
501  RETURN
502  END IF
503 *
504  IF( equil ) THEN
505 *
506 * Compute row and column scalings to equilibrate the matrix A.
507 *
508  CALL zgbequ( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
509  $ amax, infequ )
510  IF( infequ.EQ.0 ) THEN
511 *
512 * Equilibrate the matrix.
513 *
514  CALL zlaqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
515  $ amax, equed )
516  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
517  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
518  END IF
519  END IF
520 *
521 * Scale the right hand side.
522 *
523  IF( notran ) THEN
524  IF( rowequ ) THEN
525  DO 40 j = 1, nrhs
526  DO 30 i = 1, n
527  b( i, j ) = r( i )*b( i, j )
528  30 CONTINUE
529  40 CONTINUE
530  END IF
531  ELSE IF( colequ ) THEN
532  DO 60 j = 1, nrhs
533  DO 50 i = 1, n
534  b( i, j ) = c( i )*b( i, j )
535  50 CONTINUE
536  60 CONTINUE
537  END IF
538 *
539  IF( nofact .OR. equil ) THEN
540 *
541 * Compute the LU factorization of the band matrix A.
542 *
543  DO 70 j = 1, n
544  j1 = max( j-ku, 1 )
545  j2 = min( j+kl, n )
546  CALL zcopy( j2-j1+1, ab( ku+1-j+j1, j ), 1,
547  $ afb( kl+ku+1-j+j1, j ), 1 )
548  70 CONTINUE
549 *
550  CALL zgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
551 *
552 * Return if INFO is non-zero.
553 *
554  IF( info.GT.0 ) THEN
555 *
556 * Compute the reciprocal pivot growth factor of the
557 * leading rank-deficient INFO columns of A.
558 *
559  anorm = zero
560  DO 90 j = 1, info
561  DO 80 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
562  anorm = max( anorm, abs( ab( i, j ) ) )
563  80 CONTINUE
564  90 CONTINUE
565  rpvgrw = zlantb( 'M', 'U', 'N', info, min( info-1, kl+ku ),
566  $ afb( max( 1, kl+ku+2-info ), 1 ), ldafb,
567  $ rwork )
568  IF( rpvgrw.EQ.zero ) THEN
569  rpvgrw = one
570  ELSE
571  rpvgrw = anorm / rpvgrw
572  END IF
573  rwork( 1 ) = rpvgrw
574  rcond = zero
575  RETURN
576  END IF
577  END IF
578 *
579 * Compute the norm of the matrix A and the
580 * reciprocal pivot growth factor RPVGRW.
581 *
582  IF( notran ) THEN
583  norm = '1'
584  ELSE
585  norm = 'I'
586  END IF
587  anorm = zlangb( norm, n, kl, ku, ab, ldab, rwork )
588  rpvgrw = zlantb( 'M', 'U', 'N', n, kl+ku, afb, ldafb, rwork )
589  IF( rpvgrw.EQ.zero ) THEN
590  rpvgrw = one
591  ELSE
592  rpvgrw = zlangb( 'M', n, kl, ku, ab, ldab, rwork ) / rpvgrw
593  END IF
594 *
595 * Compute the reciprocal of the condition number of A.
596 *
597  CALL zgbcon( norm, n, kl, ku, afb, ldafb, ipiv, anorm, rcond,
598  $ work, rwork, info )
599 *
600 * Compute the solution matrix X.
601 *
602  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
603  CALL zgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
604  $ info )
605 *
606 * Use iterative refinement to improve the computed solution and
607 * compute error bounds and backward error estimates for it.
608 *
609  CALL zgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
610  $ b, ldb, x, ldx, ferr, berr, work, rwork, info )
611 *
612 * Transform the solution matrix X to a solution of the original
613 * system.
614 *
615  IF( notran ) THEN
616  IF( colequ ) THEN
617  DO 110 j = 1, nrhs
618  DO 100 i = 1, n
619  x( i, j ) = c( i )*x( i, j )
620  100 CONTINUE
621  110 CONTINUE
622  DO 120 j = 1, nrhs
623  ferr( j ) = ferr( j ) / colcnd
624  120 CONTINUE
625  END IF
626  ELSE IF( rowequ ) THEN
627  DO 140 j = 1, nrhs
628  DO 130 i = 1, n
629  x( i, j ) = r( i )*x( i, j )
630  130 CONTINUE
631  140 CONTINUE
632  DO 150 j = 1, nrhs
633  ferr( j ) = ferr( j ) / rowcnd
634  150 CONTINUE
635  END IF
636 *
637 * Set INFO = N+1 if the matrix is singular to working precision.
638 *
639  IF( rcond.LT.dlamch( 'Epsilon' ) )
640  $ info = n + 1
641 *
642  rwork( 1 ) = rpvgrw
643  RETURN
644 *
645 * End of ZGBSVX
646 *
subroutine zgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
ZGBTRF
Definition: zgbtrf.f:146
double precision function zlantb(NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
ZLANTB 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: zlantb.f:143
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:52
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
ZGBTRS
Definition: zgbtrs.f:140
subroutine zgbequ(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
ZGBEQU
Definition: zgbequ.f:156
subroutine zgbrfs(TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZGBRFS
Definition: zgbrfs.f:208
subroutine zgbcon(NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, RWORK, INFO)
ZGBCON
Definition: zgbcon.f:149
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
double precision function zlangb(NORM, N, KL, KU, AB, LDAB, WORK)
ZLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlangb.f:127
subroutine zlaqgb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
ZLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ...
Definition: zlaqgb.f:162

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subroutine zgbsvxx ( character  FACT,
character  TRANS,
integer  N,
integer  KL,
integer  KU,
integer  NRHS,
complex*16, dimension( ldab, * )  AB,
integer  LDAB,
complex*16, dimension( ldafb, * )  AFB,
integer  LDAFB,
integer, dimension( * )  IPIV,
character  EQUED,
double precision, dimension( * )  R,
double precision, dimension( * )  C,
complex*16, dimension( ldb, * )  B,
integer  LDB,
complex*16, dimension( ldx , * )  X,
integer  LDX,
double precision  RCOND,
double precision  RPVGRW,
double precision, dimension( * )  BERR,
integer  N_ERR_BNDS,
double precision, dimension( nrhs, * )  ERR_BNDS_NORM,
double precision, dimension( nrhs, * )  ERR_BNDS_COMP,
integer  NPARAMS,
double precision, dimension( * )  PARAMS,
complex*16, dimension( * )  WORK,
double precision, dimension( * )  RWORK,
integer  INFO 
)

ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

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

Purpose:
    ZGBSVXX uses the LU factorization to compute the solution to a
    complex*16 system of linear equations  A * X = B,  where A is an
    N-by-N matrix and X and B are N-by-NRHS matrices.

    If requested, both normwise and maximum componentwise error bounds
    are returned. ZGBSVXX will return a solution with a tiny
    guaranteed error (O(eps) where eps is the working machine
    precision) unless the matrix is very ill-conditioned, in which
    case a warning is returned. Relevant condition numbers also are
    calculated and returned.

    ZGBSVXX accepts user-provided factorizations and equilibration
    factors; see the definitions of the FACT and EQUED options.
    Solving with refinement and using a factorization from a previous
    ZGBSVXX call will also produce a solution with either O(eps)
    errors or warnings, but we cannot make that claim for general
    user-provided factorizations and equilibration factors if they
    differ from what ZGBSVXX would itself produce.
Description:
    The following steps are performed:

    1. If FACT = 'E', double precision 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 = P * L * U,

    where P is a permutation matrix, L is a unit lower triangular
    matrix, and U is upper triangular.

    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 (see
    argument RCOND). If the reciprocal of the condition number is less
    than machine precision, 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. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
    the routine will use iterative refinement to try to get a small
    error and error bounds.  Refinement calculates the residual to at
    least twice the working precision.

    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.
     Some optional parameters are bundled in the PARAMS array.  These
     settings determine how refinement is performed, but often the
     defaults are acceptable.  If the defaults are acceptable, users
     can pass NPARAMS = 0 which prevents the source code from accessing
     the PARAMS argument.
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, AF 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.
               A, AF, and IPIV are not modified.
       = 'N':  The matrix A will be copied to AF and factored.
       = 'E':  The matrix A will be equilibrated if necessary, then
               copied to AF 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  (Conjugate Transpose = 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 COMPLEX*16 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 AB 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 COMPLEX*16 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 ZGBTRF.  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 AF is an output argument and on exit
     returns the factors L and U from the factorization A = P*L*U
     of the original matrix A.

     If FACT = 'E', then AF is an output argument and on exit
     returns the factors L and U from the factorization A = P*L*U
     of the equilibrated matrix A (see the description of A 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 = P*L*U
     as computed by DGETRF; 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 = P*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 = P*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.
     If R is output, each element of R is a power of the radix.
     If R is input, each element of R should be a power of the radix
     to ensure a reliable solution and error estimates. Scaling by
     powers of the radix does not cause rounding errors unless the
     result underflows or overflows. Rounding errors during scaling
     lead to refining with a matrix that is not equivalent to the
     input matrix, producing error estimates that may not be
     reliable.
[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.
     If C is output, each element of C is a power of the radix.
     If C is input, each element of C should be a power of the radix
     to ensure a reliable solution and error estimates. Scaling by
     powers of the radix does not cause rounding errors unless the
     result underflows or overflows. Rounding errors during scaling
     lead to refining with a matrix that is not equivalent to the
     input matrix, producing error estimates that may not be
     reliable.
[in,out]B
          B is COMPLEX*16 array, dimension (LDB,NRHS)
     On entry, the N-by-NRHS 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 COMPLEX*16 array, dimension (LDX,NRHS)
     If INFO = 0, 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
     Reciprocal scaled condition number.  This is an estimate of the
     reciprocal Skeel condition number of the matrix A after
     equilibration (if done).  If this is less than the machine
     precision (in particular, if it is zero), the matrix is singular
     to working precision.  Note that the error may still be small even
     if this number is very small and the matrix appears ill-
     conditioned.
[out]RPVGRW
          RPVGRW is DOUBLE PRECISION
     Reciprocal pivot growth.  On exit, this contains the reciprocal
     pivot growth factor norm(A)/norm(U). The "max absolute element"
     norm is used.  If this 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, estimated condition numbers,
     and error bounds could be unreliable. If factorization fails with
     0<INFO<=N, then this contains the reciprocal pivot growth factor
     for the leading INFO columns of A.  In DGESVX, this quantity is
     returned in WORK(1).
[out]BERR
          BERR is DOUBLE PRECISION array, dimension (NRHS)
     Componentwise relative backward error.  This is 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).
[in]N_ERR_BNDS
          N_ERR_BNDS is INTEGER
     Number of error bounds to return for each right hand side
     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
     ERR_BNDS_COMP below.
[out]ERR_BNDS_NORM
          ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
     For each right-hand side, this array contains information about
     various error bounds and condition numbers corresponding to the
     normwise relative error, which is defined as follows:

     Normwise relative error in the ith solution vector:
             max_j (abs(XTRUE(j,i) - X(j,i)))
            ------------------------------
                  max_j abs(X(j,i))

     The array is indexed by the type of error information as described
     below. There currently are up to three pieces of information
     returned.

     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
     right-hand side.

     The second index in ERR_BNDS_NORM(:,err) contains the following
     three fields:
     err = 1 "Trust/don't trust" boolean. Trust the answer if the
              reciprocal condition number is less than the threshold
              sqrt(n) * dlamch('Epsilon').

     err = 2 "Guaranteed" error bound: The estimated forward error,
              almost certainly within a factor of 10 of the true error
              so long as the next entry is greater than the threshold
              sqrt(n) * dlamch('Epsilon'). This error bound should only
              be trusted if the previous boolean is true.

     err = 3  Reciprocal condition number: Estimated normwise
              reciprocal condition number.  Compared with the threshold
              sqrt(n) * dlamch('Epsilon') to determine if the error
              estimate is "guaranteed". These reciprocal condition
              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
              appropriately scaled matrix Z.
              Let Z = S*A, where S scales each row by a power of the
              radix so all absolute row sums of Z are approximately 1.

     See Lapack Working Note 165 for further details and extra
     cautions.
[out]ERR_BNDS_COMP
          ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
     For each right-hand side, this array contains information about
     various error bounds and condition numbers corresponding to the
     componentwise relative error, which is defined as follows:

     Componentwise relative error in the ith solution vector:
                    abs(XTRUE(j,i) - X(j,i))
             max_j ----------------------
                         abs(X(j,i))

     The array is indexed by the right-hand side i (on which the
     componentwise relative error depends), and the type of error
     information as described below. There currently are up to three
     pieces of information returned for each right-hand side. If
     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
     the first (:,N_ERR_BNDS) entries are returned.

     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
     right-hand side.

     The second index in ERR_BNDS_COMP(:,err) contains the following
     three fields:
     err = 1 "Trust/don't trust" boolean. Trust the answer if the
              reciprocal condition number is less than the threshold
              sqrt(n) * dlamch('Epsilon').

     err = 2 "Guaranteed" error bound: The estimated forward error,
              almost certainly within a factor of 10 of the true error
              so long as the next entry is greater than the threshold
              sqrt(n) * dlamch('Epsilon'). This error bound should only
              be trusted if the previous boolean is true.

     err = 3  Reciprocal condition number: Estimated componentwise
              reciprocal condition number.  Compared with the threshold
              sqrt(n) * dlamch('Epsilon') to determine if the error
              estimate is "guaranteed". These reciprocal condition
              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
              appropriately scaled matrix Z.
              Let Z = S*(A*diag(x)), where x is the solution for the
              current right-hand side and S scales each row of
              A*diag(x) by a power of the radix so all absolute row
              sums of Z are approximately 1.

     See Lapack Working Note 165 for further details and extra
     cautions.
[in]NPARAMS
          NPARAMS is INTEGER
     Specifies the number of parameters set in PARAMS.  If .LE. 0, the
     PARAMS array is never referenced and default values are used.
[in,out]PARAMS
          PARAMS is DOUBLE PRECISION array, dimension NPARAMS
     Specifies algorithm parameters.  If an entry is .LT. 0.0, then
     that entry will be filled with default value used for that
     parameter.  Only positions up to NPARAMS are accessed; defaults
     are used for higher-numbered parameters.

       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
            refinement or not.
         Default: 1.0D+0
            = 0.0 : No refinement is performed, and no error bounds are
                    computed.
            = 1.0 : Use the extra-precise refinement algorithm.
              (other values are reserved for future use)

       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
            computations allowed for refinement.
         Default: 10
         Aggressive: Set to 100 to permit convergence using approximate
                     factorizations or factorizations other than LU. If
                     the factorization uses a technique other than
                     Gaussian elimination, the guarantees in
                     err_bnds_norm and err_bnds_comp may no longer be
                     trustworthy.

       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
            will attempt to find a solution with small componentwise
            relative error in the double-precision algorithm.  Positive
            is true, 0.0 is false.
         Default: 1.0 (attempt componentwise convergence)
[out]WORK
          WORK is COMPLEX*16 array, dimension (2*N)
[out]RWORK
          RWORK is DOUBLE PRECISION array, dimension (2*N)
[out]INFO
          INFO is INTEGER
       = 0:  Successful exit. The solution to every right-hand side is
         guaranteed.
       < 0:  If INFO = -i, the i-th argument had an illegal value
       > 0 and <= N:  U(INFO,INFO) 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+J: The solution corresponding to the Jth right-hand side is
         not guaranteed. The solutions corresponding to other right-
         hand sides K with K > J may not be guaranteed as well, but
         only the first such right-hand side is reported. If a small
         componentwise error is not requested (PARAMS(3) = 0.0) then
         the Jth right-hand side is the first with a normwise error
         bound that is not guaranteed (the smallest J such
         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
         the Jth right-hand side is the first with either a normwise or
         componentwise error bound that is not guaranteed (the smallest
         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
         about all of the right-hand sides check ERR_BNDS_NORM or
         ERR_BNDS_COMP.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
April 2012

Definition at line 562 of file zgbsvxx.f.

562 *
563 * -- LAPACK driver routine (version 3.4.1) --
564 * -- LAPACK is a software package provided by Univ. of Tennessee, --
565 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
566 * April 2012
567 *
568 * .. Scalar Arguments ..
569  CHARACTER equed, fact, trans
570  INTEGER info, ldab, ldafb, ldb, ldx, n, nrhs, nparams,
571  $ n_err_bnds
572  DOUBLE PRECISION rcond, rpvgrw
573 * ..
574 * .. Array Arguments ..
575  INTEGER ipiv( * )
576  COMPLEX*16 ab( ldab, * ), afb( ldafb, * ), b( ldb, * ),
577  $ x( ldx , * ),work( * )
578  DOUBLE PRECISION r( * ), c( * ), params( * ), berr( * ),
579  $ err_bnds_norm( nrhs, * ),
580  $ err_bnds_comp( nrhs, * ), rwork( * )
581 * ..
582 *
583 * ==================================================================
584 *
585 * .. Parameters ..
586  DOUBLE PRECISION zero, one
587  parameter( zero = 0.0d+0, one = 1.0d+0 )
588  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
589  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
590  INTEGER cmp_err_i, piv_growth_i
591  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
592  $ berr_i = 3 )
593  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
594  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
595  $ piv_growth_i = 9 )
596 * ..
597 * .. Local Scalars ..
598  LOGICAL colequ, equil, nofact, notran, rowequ
599  INTEGER infequ, i, j, kl, ku
600  DOUBLE PRECISION amax, bignum, colcnd, rcmax, rcmin,
601  $ rowcnd, smlnum
602 * ..
603 * .. External Functions ..
604  EXTERNAL lsame, dlamch, zla_gbrpvgrw
605  LOGICAL lsame
606  DOUBLE PRECISION dlamch, zla_gbrpvgrw
607 * ..
608 * .. External Subroutines ..
609  EXTERNAL zgbequb, zgbtrf, zgbtrs, zlacpy, zlaqgb,
611 * ..
612 * .. Intrinsic Functions ..
613  INTRINSIC max, min
614 * ..
615 * .. Executable Statements ..
616 *
617  info = 0
618  nofact = lsame( fact, 'N' )
619  equil = lsame( fact, 'E' )
620  notran = lsame( trans, 'N' )
621  smlnum = dlamch( 'Safe minimum' )
622  bignum = one / smlnum
623  IF( nofact .OR. equil ) THEN
624  equed = 'N'
625  rowequ = .false.
626  colequ = .false.
627  ELSE
628  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
629  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
630  END IF
631 *
632 * Default is failure. If an input parameter is wrong or
633 * factorization fails, make everything look horrible. Only the
634 * pivot growth is set here, the rest is initialized in ZGBRFSX.
635 *
636  rpvgrw = zero
637 *
638 * Test the input parameters. PARAMS is not tested until DGERFSX.
639 *
640  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
641  $ lsame( fact, 'F' ) ) THEN
642  info = -1
643  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
644  $ lsame( trans, 'C' ) ) THEN
645  info = -2
646  ELSE IF( n.LT.0 ) THEN
647  info = -3
648  ELSE IF( kl.LT.0 ) THEN
649  info = -4
650  ELSE IF( ku.LT.0 ) THEN
651  info = -5
652  ELSE IF( nrhs.LT.0 ) THEN
653  info = -6
654  ELSE IF( ldab.LT.kl+ku+1 ) THEN
655  info = -8
656  ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
657  info = -10
658  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
659  $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
660  info = -12
661  ELSE
662  IF( rowequ ) THEN
663  rcmin = bignum
664  rcmax = zero
665  DO 10 j = 1, n
666  rcmin = min( rcmin, r( j ) )
667  rcmax = max( rcmax, r( j ) )
668  10 CONTINUE
669  IF( rcmin.LE.zero ) THEN
670  info = -13
671  ELSE IF( n.GT.0 ) THEN
672  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
673  ELSE
674  rowcnd = one
675  END IF
676  END IF
677  IF( colequ .AND. info.EQ.0 ) THEN
678  rcmin = bignum
679  rcmax = zero
680  DO 20 j = 1, n
681  rcmin = min( rcmin, c( j ) )
682  rcmax = max( rcmax, c( j ) )
683  20 CONTINUE
684  IF( rcmin.LE.zero ) THEN
685  info = -14
686  ELSE IF( n.GT.0 ) THEN
687  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
688  ELSE
689  colcnd = one
690  END IF
691  END IF
692  IF( info.EQ.0 ) THEN
693  IF( ldb.LT.max( 1, n ) ) THEN
694  info = -15
695  ELSE IF( ldx.LT.max( 1, n ) ) THEN
696  info = -16
697  END IF
698  END IF
699  END IF
700 *
701  IF( info.NE.0 ) THEN
702  CALL xerbla( 'ZGBSVXX', -info )
703  RETURN
704  END IF
705 *
706  IF( equil ) THEN
707 *
708 * Compute row and column scalings to equilibrate the matrix A.
709 *
710  CALL zgbequb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
711  $ amax, infequ )
712  IF( infequ.EQ.0 ) THEN
713 *
714 * Equilibrate the matrix.
715 *
716  CALL zlaqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
717  $ amax, equed )
718  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
719  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
720  END IF
721 *
722 * If the scaling factors are not applied, set them to 1.0.
723 *
724  IF ( .NOT.rowequ ) THEN
725  DO j = 1, n
726  r( j ) = 1.0d+0
727  END DO
728  END IF
729  IF ( .NOT.colequ ) THEN
730  DO j = 1, n
731  c( j ) = 1.0d+0
732  END DO
733  END IF
734  END IF
735 *
736 * Scale the right-hand side.
737 *
738  IF( notran ) THEN
739  IF( rowequ ) CALL zlascl2( n, nrhs, r, b, ldb )
740  ELSE
741  IF( colequ ) CALL zlascl2( n, nrhs, c, b, ldb )
742  END IF
743 *
744  IF( nofact .OR. equil ) THEN
745 *
746 * Compute the LU factorization of A.
747 *
748  DO 40, j = 1, n
749  DO 30, i = kl+1, 2*kl+ku+1
750  afb( i, j ) = ab( i-kl, j )
751  30 CONTINUE
752  40 CONTINUE
753  CALL zgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
754 *
755 * Return if INFO is non-zero.
756 *
757  IF( info.GT.0 ) THEN
758 *
759 * Pivot in column INFO is exactly 0
760 * Compute the reciprocal pivot growth factor of the
761 * leading rank-deficient INFO columns of A.
762 *
763  rpvgrw = zla_gbrpvgrw( n, kl, ku, info, ab, ldab, afb,
764  $ ldafb )
765  RETURN
766  END IF
767  END IF
768 *
769 * Compute the reciprocal pivot growth factor RPVGRW.
770 *
771  rpvgrw = zla_gbrpvgrw( n, kl, ku, n, ab, ldab, afb, ldafb )
772 *
773 * Compute the solution matrix X.
774 *
775  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
776  CALL zgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
777  $ info )
778 *
779 * Use iterative refinement to improve the computed solution and
780 * compute error bounds and backward error estimates for it.
781 *
782  CALL zgbrfsx( trans, equed, n, kl, ku, nrhs, ab, ldab, afb, ldafb,
783  $ ipiv, r, c, b, ldb, x, ldx, rcond, berr,
784  $ n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params,
785  $ work, rwork, info )
786 
787 *
788 * Scale solutions.
789 *
790  IF ( colequ .AND. notran ) THEN
791  CALL zlascl2( n, nrhs, c, x, ldx )
792  ELSE IF ( rowequ .AND. .NOT.notran ) THEN
793  CALL zlascl2( n, nrhs, r, x, ldx )
794  END IF
795 *
796  RETURN
797 *
798 * End of ZGBSVXX
799 *
subroutine zgbrfsx(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
ZGBRFSX
Definition: zgbrfsx.f:442
subroutine zgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
ZGBTRF
Definition: zgbtrf.f:146
subroutine zlascl2(M, N, D, X, LDX)
ZLASCL2 performs diagonal scaling on a vector.
Definition: zlascl2.f:93
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
ZGBTRS
Definition: zgbtrs.f:140
subroutine zgbequb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
ZGBEQUB
Definition: zgbequb.f:163
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
double precision function zla_gbrpvgrw(N, KL, KU, NCOLS, AB, LDAB, AFB, LDAFB)
ZLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix...
Definition: zla_gbrpvgrw.f:119
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zlaqgb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
ZLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ...
Definition: zlaqgb.f:162

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