sgbrfs(3S)

NAME
     SGBRFS - improve 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

SYNOPSIS
     SUBROUTINE SGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B,
                        LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )

         CHARACTER      TRANS

         INTEGER        INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS

         INTEGER        IPIV( * ), IWORK( * )

         REAL           AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( *
                        ), FERR( * ), WORK( * ), X( LDX, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     SGBRFS 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 = 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) REAL 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) REAL array, dimension (LDAFB,N)
             Details of the LU factorization of the band matrix A, as computed
             by SGBTRF.  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 SGBTRF; for 1<=i<=N, row i of the matrix
             was interchanged with row IPIV(i).

     B       (input) REAL 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) REAL array, dimension (LDX,NRHS)
             On entry, the solution matrix X, as computed by SGBTRS.  On exit,
             the improved solution matrix X.

     LDX     (input) INTEGER
             The leading dimension of the array X.  LDX >= max(1,N).

     FERR    (output) REAL 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) REAL 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) REAL array, dimension (3*N)

     IWORK   (workspace) INTEGER array, dimension (N)

     INFO    (output) INTEGER
             = 0:  successful exit
             < 0:  if INFO = -i, the i-th argument had an illegal value

PARAMETERS
     ITMAX is the maximum number of steps of iterative refinement.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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