Refines the solution of a system of linear equations with a packed complex Hermitian coefficient matrix and estimates the solution error.

Syntax

lapack_int LAPACKE_chprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_float* ap, const lapack_complex_float* afp, 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_zhprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_double* ap, const lapack_complex_double* afp, 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 with a packed complex Hermitian 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 ?hptrf

  • call the solver routine ?hptrs.

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'.

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,afp,b,x

Arrays:

apmax(1, n(n + 1)/2) contains the original packed matrix A, as supplied to ?hptrf.

afpmax(1, n(n + 1)/2) contains the factored packed matrix A, as returned by ?hptrf.

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.

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) for column major layout and ldxnrhs for row major layout.

ipiv

Array, size at least max(1, n). The ipiv array, as returned by ?hptrf.

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 16n2 operations. In addition, each step of iterative refinement involves 24n2 operations; 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 8n2 floating-point operations.

The real counterpart of this routine is ?ssprfs/?dsprfs.

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