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 );

• 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 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. 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.