Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 3/22/2024
Public

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?herfs

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

Syntax

call cherfs( uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call zherfs( uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call herfs( a, af, ipiv, b, x [,uplo] [,ferr] [,berr] [,info] )

Include Files

  • mkl.fi, lapack.f90

Description

The routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a complex Hermitian full-storage 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 ?hetrf

  • call the solver routine ?hetrs.

Input Parameters

uplo

CHARACTER*1. 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

INTEGER. The order of the matrix A; n 0.

nrhs

INTEGER. The number of right-hand sides; nrhs 0.

a,af,b,x,work

COMPLEX for cherfs

DOUBLE COMPLEX for zherfs.

Arrays:

a(size lda by *) contains the original matrix A, as supplied to ?hetrf.

af(size ldaf by *) contains the factored matrix A, as returned by ?hetrf.

b(size ldb by *) contains the right-hand side matrix B.

x(size ldx by *) contains the solution matrix X.

work(*) is a workspace array.

The second dimension of a and af must be at least max(1, n); the second dimension of b and x must be at least max(1, nrhs); the dimension of work must be at least max(1, 2*n).

lda

INTEGER. The leading dimension of a; lda max(1, n).

ldaf

INTEGER. The leading dimension of af; ldaf max(1, n).

ldb

INTEGER. The leading dimension of b; ldb max(1, n).

ldx

INTEGER. The leading dimension of x; ldx max(1, n).

ipiv

INTEGER.

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

rwork

REAL for cherfs

DOUBLE PRECISION for zherfs.

Workspace array, size at least max(1, n).

Output Parameters

x

The refined solution matrix X.

ferr, berr

REAL for cherfs

DOUBLE PRECISION for zherfs.

Arrays, size at least max(1, nrhs). Contain the component-wise forward and backward errors, respectively, for each solution vector.

info

INTEGER. If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine herfs interface are as follows:

a

Holds the matrix A of size (n,n).

af

Holds the matrix AF of size (n,n).

ipiv

Holds the vector of length n.

b

Holds the matrix B of size (n,nrhs).

x

Holds the matrix X of size (n,nrhs).

ferr

Holds the vector of length (nrhs).

berr

Holds the vector of length (nrhs).

uplo

Must be 'U' or 'L'. The default value is 'U'.

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 ?ssyrfs/?dsyrfs