?gbrfs

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/22/2024

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?gbrfs

Refines the solution of a system of linear equations with a general band coefficient matrix and estimates its error.

Syntax

lapack_int LAPACKE_sgbrfs( int matrix_layout, char trans, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, const float* ab, lapack_int ldab, const float* afb, lapack_int ldafb, const lapack_int* ipiv, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_dgbrfs( int matrix_layout, char trans, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, const double* ab, lapack_int ldab, const double* afb, lapack_int ldafb, const lapack_int* ipiv, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );

lapack_int LAPACKE_cgbrfs( int matrix_layout, char trans, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, const lapack_complex_float* ab, lapack_int ldab, const lapack_complex_float* afb, lapack_int ldafb, const lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_zgbrfs( int matrix_layout, char trans, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, const lapack_complex_double* ab, lapack_int ldab, const lapack_complex_double* afb, lapack_int ldafb, const lapack_int* ipiv, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );

Include Files

mkl.h

Description

The routine performs an iterative refinement of the solution to a system of linear equations A*X = B or A^T*X = B or A^H*X = B with a 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).

Finally, the routine 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 factorization routine ?gbtrf
call the solver routine ?gbtrs.

Input Parameters

matrix_layout	Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
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`.
n	The order of the matrix `A`; `n`≥ 0.
kl	The number of sub-diagonals within the band of `A`; `kl`≥ 0.
ku	The number of super-diagonals within the band of `A`; `ku`≥ 0.
nrhs	The number of right-hand sides; `nrhs`≥ 0.
ab,afb,b,x	Arrays: `ab`(size max(1, ldabn)) contains the original band matrix `A`, as supplied to ?gbtrf, but stored in rows from 1 to `kl` + `ku` + 1 for column major layout, and columns from 1 to `kl` + `ku` + 1 for row major layout. `afb`(size max(1, ldafbn)) contains the factored band matrix `A`, as returned by ?gbtrf. `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, ldxn) for row major layout contains the solution matrix `X`.
ldab	The leading dimension of `ab`, ldab≥kl + ku + 1.
ldafb	The leading dimension of `afb`, ldafb≥ 2*kl + ku + 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`; `ldx≥ max(1, n)`.
ipiv	Array, size at least `max(1, n)`. The `ipiv` array, as returned by ?gbtrf.

Output Parameters

x	The refined solution matrix `X`.
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.

For each right-hand side, computation of the backward error involves a minimum of 4n(kl + ku) floating-point operations (for real flavors) or 16n(kl + ku) operations (for complex flavors). In addition, each step of iterative refinement involves 2n(4kl + 3ku) operations (for real flavors) or 8n(4kl + 3ku) operations (for complex flavors); the number of iterations may range from 1 to 5. Estimating the forward error involves solving a number of systems of linear equations A*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n² floating-point operations for real flavors or 8n² for complex flavors.

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

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?gbrfs

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

Application Notes

See Also