?tbrfs

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?tbrfs

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

Syntax

lapack_int LAPACKE_stbrfs( int matrix_layout, 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_layout, 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_layout, 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_layout, 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

mkl.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 errorβ. 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

matrix_layout	Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
uplo	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	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	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	The order of the matrix `A`; `n`≥ 0.
kd	The number of super-diagonals or sub-diagonals in the matrix `A`; `kd`≥ 0.
nrhs	The number of right-hand sides; `nrhs`≥ 0.
ab, b, x	Arrays: `ab`(size max(1, ldabn)) contains the upper or lower triangular matrix `A`, as specified by `uplo`, in band storage format. `b`of size max(1, ldbnrhs) for column major layout and max(1, ldbn) for row major layout contains the right-hand side matrix `B`. `x`of size max(1, ldxnrhs) for column major layout and max(1, ldx*n) for row major layout contains the solution matrix `X`.
ldab	The leading dimension of the array `ab`; `ldab`≥`kd` +1.
ldb	The leading dimension of `b`; `ldb`≥ max(1, n) for column major layout and `ldb`≥nrhs for row major layout.
ldx	The leading dimension of `x`; `ldb`≥ max(1, n) for column major layout and `ldb`≥nrhs for row major layout.

Output Parameters

ferr, berr

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

Return Values

This function returns a value info.

If info = 0, the execution is successful.

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

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.

Parent topic: Refining the Solution and Estimating Its Error: LAPACK Computational Routines

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