Developer Reference for Intel® oneAPI Math Kernel Library for C
?pbrfs
Refines the solution of a system of linear equations with a band symmetric (Hermitian) positive-definite coefficient matrix and estimates its error.
Syntax
lapack_int LAPACKE_spbrfs( int matrix_layout, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, const float* ab, lapack_int ldab, const float* afb, lapack_int ldafb, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );
lapack_int LAPACKE_dpbrfs( int matrix_layout, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, const double* ab, lapack_int ldab, const double* afb, lapack_int ldafb, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );
lapack_int LAPACKE_cpbrfs( int matrix_layout, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, const lapack_complex_float* ab, lapack_int ldab, const lapack_complex_float* afb, lapack_int ldafb, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );
lapack_int LAPACKE_zpbrfs( int matrix_layout, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, const lapack_complex_double* ab, lapack_int ldab, const lapack_complex_double* afb, lapack_int ldafb, 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 with a symmetric (Hermitian) positive definite 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:
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 how the input matrix A has been factored: If uplo = 'U', the upper triangle of A is stored. If uplo = 'L', the lower triangle of A is stored. |
n |
The order of the matrix A; n≥ 0. |
kd |
The number of superdiagonals or subdiagonals in the matrix A; kd≥ 0. |
nrhs |
The number of right-hand sides; nrhs≥ 0. |
ab |
Array ab (size max(ldab*n)) contains the original band matrix A, as supplied to ?pbtrf. |
afb |
Array afb (size max(ldafb*n)) contains the factored band matrix A, as returned by ?pbtrf. |
b |
Array b of 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. |
x |
Array x of 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; ldab≥kd + 1. |
ldafb |
The leading dimension of afb; ldafb≥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; ldx≥ max(1, n) for column major layout and ldx≥nrhs for row major layout. |
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 8n*kd floating-point operations (for real flavors) or 32n*kd operations (for complex flavors). In addition, each step of iterative refinement involves 12n*kd operations (for real flavors) or 48n*kd 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 4n*kd floating-point operations for real flavors or 16n*kd for complex flavors.