Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 3/22/2024
Public

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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 AT*X = B or AH*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:

|δaij| β|aij|, |δbi| β|bi| such that (A + δA)x = (b + δb).

Finally, the routine estimates the component-wise forward error in the computed solution ||x - xe||/||x|| (here xe 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 A*X = B.

If trans = 'T', the system has the form AT*X = B.

If trans = 'C', the system has the form AH*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, ldab*n)) 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, ldafb*n)) contains the factored band matrix A, as returned by ?gbtrf.

bof size max(1, ldb*nrhs) for column major layout and max(1, ldb*n) for row major layout contains the right-hand side matrix B.

xof size max(1, ldx*nrhs) for column major layout and max(1, ldx*n) for row major layout contains the solution matrix X.

ldab

The leading dimension of ab, ldabkl + 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 ldbnrhs 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.