ZGBRFS(3f)

NAME
ZGBRFS - 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 ZGBRFS(
    TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )

CHARACTER
    TRANS

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

INTEGER
    IPIV( * )

DOUBLE
    PRECISION BERR( * ), FERR( * ), RWORK( * )

COMPLEX*16
    AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 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

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