.TH DTBRFS 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME DTBRFS - error bounds and backward error estimates for the solution to a system of linear equations with a triangular band coefficient matrix .SH SYNOPSIS .TP 19 SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) .TP 19 .ti +4 CHARACTER DIAG, TRANS, UPLO .TP 19 .ti +4 INTEGER INFO, KD, LDAB, LDB, LDX, N, NRHS .TP 19 .ti +4 INTEGER IWORK( * ) .TP 19 .ti +4 DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) .SH PURPOSE DTBRFS 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 DTBTRS or some other means before entering this routine. DTBRFS does not do iterative refinement because doing so cannot improve the backward error. .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 = \(aqU\(aq: A is upper triangular; .br = \(aqL\(aq: A is lower triangular. .TP 8 TRANS (input) CHARACTER*1 .br Specifies the form of the system of equations: .br = \(aqN\(aq: A * X = B (No transpose) .br = \(aqT\(aq: A**T * X = B (Transpose) .br = \(aqC\(aq: A**H * X = B (Conjugate transpose = Transpose) .TP 8 DIAG (input) CHARACTER*1 .br = \(aqN\(aq: A is non-unit triangular; .br = \(aqU\(aq: A is unit triangular. .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 KD (input) INTEGER The number of superdiagonals or subdiagonals of the triangular band matrix A. KD >= 0. .TP 8 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. .TP 8 AB (input) DOUBLE PRECISION 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 = \(aqU\(aq, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = \(aqL\(aq, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). If DIAG = \(aqU\(aq, the diagonal elements of A are not referenced and are assumed to be 1. .TP 8 LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= KD+1. .TP 8 B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) The right hand side matrix B. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) The solution matrix X. .TP 8 LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). .TP 8 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. .TP 8 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). .TP 8 WORK (workspace) DOUBLE PRECISION array, dimension (3*N) .TP 8 IWORK (workspace) INTEGER array, dimension (N) .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value