?pprfs

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?pprfs

Refines the solution of a system of linear equations with a symmetric (Hermitian) positive-definite coefficient matrix stored in a packed format and estimates its error.

Syntax

lapack_int LAPACKE_spprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs, const float* ap, const float* afp, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_dpprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs, const double* ap, const double* afp, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );

lapack_int LAPACKE_cpprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_float* ap, const lapack_complex_float* afp, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_zpprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_double* ap, const lapack_complex_double* afp, 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 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||_∞

where x_e is the exact solution.

Before calling this routine:

call the factorization routine ?pptrf
call the solver routine ?pptrs.

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.
nrhs	The number of right-hand sides; `nrhs`≥ 0.
ap	`ap` contains the original matrix `A` in a packed format, as supplied to ?pptrf. The dimension of `ap` must be at least `max(1,n(n+1)/2)`.
afp	`afp` contains the factored matrix `A` in a packed format, as returned by ?pptrf. The dimension of `afp` must be at least `max(1,n(n+1)/2)`.
b	Array `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	Array `x` of size max(1, ldxnrhs) for column major layout and max(1, ldxn) for row major layout contains the solution matrix `X`.
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 4n² floating-point operations (for real flavors) or 16n² operations (for complex flavors). In addition, each step of iterative refinement involves 6n² operations (for real flavors) or 24n² 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 of systems 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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