ZTBRFS(3f)

NAME
ZTBRFS - error bounds and backward error estimates for the solution to a system of linear equations with a triangular band coefficient matrix

SYNOPSIS

SUBROUTINE ZTBRFS(
    UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )

CHARACTER
    DIAG, TRANS, UPLO

INTEGER
    INFO, KD, LDAB, LDB, LDX, N, NRHS

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

COMPLEX*16
    AB( LDAB, * ), B( LDB, * ), WORK( * ), X( LDX, * )

PURPOSE
ZTBRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular band coefficient matrix. The solution matrix X must be computed by ZTBTRS or some other means before entering this routine. ZTBRFS does not do iterative refinement because doing so cannot improve the backward error.

ARGUMENTS

UPLO    (input) CHARACTER*1
    = 'U': A is upper triangular;
    = 'L': A is lower triangular.

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)

DIAG    (input) CHARACTER*1
    = 'N': A is non-unit triangular;
    = 'U': A is unit triangular.

N       (input) INTEGER
    The order of the matrix A. N >= 0.

KD      (input) INTEGER
    The number of superdiagonals or subdiagonals of the triangular band matrix A. KD >= 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 upper or lower triangular band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). If DIAG = 'U', the diagonal elements of A are not referenced and are assumed to be 1.

LDAB    (input) INTEGER
    The leading dimension of the array AB. LDAB >= KD+1.

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) COMPLEX*16 array, dimension (LDX,NRHS)
    The 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