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version 1.7, 2010/12/21 13:53:42 version 1.8, 2011/11/21 20:43:08
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   *> \brief \b ZGBRFS
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZGBRFS + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbrfs.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgbrfs.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbrfs.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
   *                          IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
   *                          INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          TRANS
   *       INTEGER            INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IPIV( * )
   *       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
   *       COMPLEX*16         AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
   *      $                   WORK( * ), X( LDX, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGBRFS improves the computed solution to a system of linear
   *> equations when the coefficient matrix is banded, and provides
   *> error bounds and backward error estimates for the solution.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] TRANS
   *> \verbatim
   *>          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)
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] KL
   *> \verbatim
   *>          KL is INTEGER
   *>          The number of subdiagonals within the band of A.  KL >= 0.
   *> \endverbatim
   *>
   *> \param[in] KU
   *> \verbatim
   *>          KU is INTEGER
   *>          The number of superdiagonals within the band of A.  KU >= 0.
   *> \endverbatim
   *>
   *> \param[in] NRHS
   *> \verbatim
   *>          NRHS is INTEGER
   *>          The number of right hand sides, i.e., the number of columns
   *>          of the matrices B and X.  NRHS >= 0.
   *> \endverbatim
   *>
   *> \param[in] AB
   *> \verbatim
   *>          AB is COMPLEX*16 array, dimension (LDAB,N)
   *>          The original band matrix A, stored 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).
   *> \endverbatim
   *>
   *> \param[in] LDAB
   *> \verbatim
   *>          LDAB is INTEGER
   *>          The leading dimension of the array AB.  LDAB >= KL+KU+1.
   *> \endverbatim
   *>
   *> \param[in] AFB
   *> \verbatim
   *>          AFB is COMPLEX*16 array, dimension (LDAFB,N)
   *>          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.
   *> \endverbatim
   *>
   *> \param[in] LDAFB
   *> \verbatim
   *>          LDAFB is INTEGER
   *>          The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.
   *> \endverbatim
   *>
   *> \param[in] IPIV
   *> \verbatim
   *>          IPIV is INTEGER array, dimension (N)
   *>          The pivot indices from ZGBTRF; for 1<=i<=N, row i of the
   *>          matrix was interchanged with row IPIV(i).
   *> \endverbatim
   *>
   *> \param[in] B
   *> \verbatim
   *>          B is COMPLEX*16 array, dimension (LDB,NRHS)
   *>          The right hand side matrix B.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B.  LDB >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] X
   *> \verbatim
   *>          X is COMPLEX*16 array, dimension (LDX,NRHS)
   *>          On entry, the solution matrix X, as computed by ZGBTRS.
   *>          On exit, the improved solution matrix X.
   *> \endverbatim
   *>
   *> \param[in] LDX
   *> \verbatim
   *>          LDX is INTEGER
   *>          The leading dimension of the array X.  LDX >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] FERR
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] BERR
   *> \verbatim
   *>          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).
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension (2*N)
   *> \endverbatim
   *>
   *> \param[out] RWORK
   *> \verbatim
   *>          RWORK is DOUBLE PRECISION array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit
   *>          < 0:  if INFO = -i, the i-th argument had an illegal value
   *> \endverbatim
   *
   *> \par Internal Parameters:
   *  =========================
   *>
   *> \verbatim
   *>  ITMAX is the maximum number of steps of iterative refinement.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup complex16GBcomputational
   *
   *  =====================================================================
       SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,        SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
      $                   IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,       $                   IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
      $                   INFO )       $                   INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  *     November 2011
 *  
 *     Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          TRANS        CHARACTER          TRANS
Line 20 Line 222
      $                   WORK( * ), X( LDX, * )       $                   WORK( * ), X( LDX, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGBRFS improves the computed solution to a system of linear  
 *  equations when the coefficient matrix is banded, and provides  
 *  error bounds and backward error estimates for the solution.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  TRANS   (input) 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)  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix A.  N >= 0.  
 *  
 *  KL      (input) INTEGER  
 *          The number of subdiagonals within the band of A.  KL >= 0.  
 *  
 *  KU      (input) INTEGER  
 *          The number of superdiagonals within the band of A.  KU >= 0.  
 *  
 *  NRHS    (input) INTEGER  
 *          The number of right hand sides, i.e., the number of columns  
 *          of the matrices B and X.  NRHS >= 0.  
 *  
 *  AB      (input) COMPLEX*16 array, dimension (LDAB,N)  
 *          The original band matrix A, stored 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).  
 *  
 *  LDAB    (input) INTEGER  
 *          The leading dimension of the array AB.  LDAB >= KL+KU+1.  
 *  
 *  AFB     (input) COMPLEX*16 array, dimension (LDAFB,N)  
 *          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.  
 *  
 *  LDAFB   (input) INTEGER  
 *          The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.  
 *  
 *  IPIV    (input) INTEGER array, dimension (N)  
 *          The pivot indices from ZGBTRF; for 1<=i<=N, row i of the  
 *          matrix was interchanged with row IPIV(i).  
 *  
 *  B       (input) COMPLEX*16 array, dimension (LDB,NRHS)  
 *          The right hand side matrix B.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B.  LDB >= max(1,N).  
 *  
 *  X       (input/output) COMPLEX*16 array, dimension (LDX,NRHS)  
 *          On entry, the solution matrix X, as computed by ZGBTRS.  
 *          On exit, the improved solution matrix X.  
 *  
 *  LDX     (input) INTEGER  
 *          The leading dimension of the array X.  LDX >= max(1,N).  
 *  
 *  FERR    (output) 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.  
 *  
 *  BERR    (output) 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).  
 *  
 *  WORK    (workspace) COMPLEX*16 array, dimension (2*N)  
 *  
 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value  
 *  
 *  Internal Parameters  
 *  ===================  
 *  
 *  ITMAX is the maximum number of steps of iterative refinement.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.7  
changed lines
  Added in v.1.8

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