?tbrfs

Estimates the error in the solution of a system of linear equations with a triangular band matrix.

Syntax

FORTRAN 77:

call stbrfs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call dtbrfs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call ctbrfs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call ztbrfs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, rwork, info )

FORTRAN 95:

call tbrfs( ab, b, x [,uplo] [,trans] [,diag] [,ferr] [,berr] [,info] )

lapack_int LAPACKE_stbrfs( int matrix_order, char uplo, char trans, char diag, lapack_int n, lapack_int kd, lapack_int nrhs, const float* ab, lapack_int ldab, const float* b, lapack_int ldb, const float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_dtbrfs( int matrix_order, char uplo, char trans, char diag, lapack_int n, lapack_int kd, lapack_int nrhs, const double* ab, lapack_int ldab, const double* b, lapack_int ldb, const double* x, lapack_int ldx, double* ferr, double* berr );

lapack_int LAPACKE_ctbrfs( int matrix_order, char uplo, char trans, char diag, lapack_int n, lapack_int kd, lapack_int nrhs, const lapack_complex_float* ab, lapack_int ldab, const lapack_complex_float* b, lapack_int ldb, const lapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_ztbrfs( int matrix_order, char uplo, char trans, char diag, lapack_int n, lapack_int kd, lapack_int nrhs, const lapack_complex_double* ab, lapack_int ldab, const lapack_complex_double* b, lapack_int ldb, const lapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );

Include Files

Fortran: mkl_lapack.fi and mkl_lapack.h
Fortran 95: lapack.f90
C: mkl_lapacke.h

Description

The routine estimates the errors in the solution to a system of linear equations A*X = B or A^T*X = B or A^H*X = B with a triangular band matrix A, with multiple right-hand sides. For each computed solution vector x, the routine computes the component-wise backward error β. This error is the smallest relative perturbation in elements of A and b such that x is the exact solution of the perturbed system:

|δa_ij| ≤ β|a_ij|, |δb_i| ≤ β|b_i| such that (A + δA)x = (b + δb).

The routine also estimates the component-wise forward error in the computed solution ||x - x_e||_∞/||x||_∞ (here x_e is the exact solution).

Before calling this routine, call the solver routine ?tbtrs.

Input Parameters

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

uplo	CHARACTER*1. Must be 'U' or 'L'. Indicates whether `A` is upper or lower triangular: If `uplo = 'U'`, then `A` is upper triangular. If `uplo = 'L'`, then `A` is lower triangular.
trans	CHARACTER1. Must be 'N' or 'T' or 'C'. Indicates the form of the equations: If `trans = 'N'`, the system has the form `AX = B`. If `trans = 'T'`, the system has the form `A`^T`X` = `B`. If `trans = 'C'`, the system has the form `A`^H`X` = `B`.
diag	CHARACTER*1. Must be 'N' or 'U'. If `diag = 'N'`, `A` is not a unit triangular matrix. If `diag = 'U'`, `A` is unit triangular: diagonal elements of `A` are assumed to be 1 and not referenced in the array `ab`.
n	INTEGER. The order of the matrix `A`; `n` ≥ 0.
kd	INTEGER. The number of super-diagonals or sub-diagonals in the matrix `A`; `kd` ≥ 0.
nrhs	INTEGER. The number of right-hand sides; `nrhs` ≥ 0.
ab, b, x, work	REAL for stbrfs DOUBLE PRECISION for dtbrfs COMPLEX for ctbrfs DOUBLE COMPLEX for ztbrfs. Arrays: `ab(ldab,)` contains the upper or lower triangular matrix `A`, as specified by `uplo`, in band storage format. `b(ldb,)` contains the right-hand side matrix `B`. `x(ldx,)` contains the solution matrix `X`. `work()` is a workspace array. The second dimension of `a` must be at least `max(1,n)`; the second dimension of `b` and `x` must be at least `max(1,nrhs)`. The dimension of `work` must be at least `max(1,3n)` for real flavors and `max(1,2n)` for complex flavors.
ldab	INTEGER. The leading dimension of the array `ab`; `ldab` ≥ `kd` +1.
ldb	INTEGER. The leading dimension of `b`; `ldb ≥ max(1, n)`.
ldx	INTEGER. The leading dimension of `x`; `ldx ≥ max(1, n)`.
iwork	INTEGER. Workspace array, DIMENSION at least `max(1, n)`.
rwork	REAL for ctbrfs DOUBLE PRECISION for ztbrfs. Workspace array, DIMENSION at least `max(1, n)`.

Output Parameters

ferr, berr

REAL for single precision flavors.

DOUBLE PRECISION for double precision flavors.

Arrays, DIMENSION at least max(1, nrhs). Contain the component-wise forward and backward errors, respectively, for each solution vector.

info

INTEGER. If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.

Specific details for the routine tbrfs interface are as follows:

ab	Holds the array `A` of size (`kd+1,n`).
b	Holds the matrix `B` of size (`n,nrhs`).
x	Holds the matrix `X` of size (`n,nrhs`).
ferr	Holds the vector of length (`nrhs`).
berr	Holds the vector of length (`nrhs`).
uplo	Must be 'U' or 'L'. The default value is 'U'.
trans	Must be 'N', 'C', or 'T'. The default value is 'N'.
diag	Must be 'N' or 'U'. The default value is 'N'.

Application Notes

The bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual error.

A call to this routine involves, for each right-hand side, solving a number of systems of linear equations A*x = b; the number of systems is usually 4 or 5 and never more than 11. Each solution requires approximately 2n*kd floating-point operations for real flavors or 8n*kd operations for complex flavors.