?hpcon

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?hpcon

Estimates the reciprocal of the condition number of a packed Hermitian matrix.

Syntax

call chpcon( uplo, n, ap, ipiv, anorm, rcond, work, info )

call zhpcon( uplo, n, ap, ipiv, anorm, rcond, work, info )

call hpcon( ap, ipiv, anorm, rcond [,uplo] [,info] )

Include Files

mkl.fi, lapack.f90

Description

The routine estimates the reciprocal of the condition number of a Hermitian matrix A:

κ₁(A) = ||A||₁ ||A^-1||₁ (since A is Hermitian, κ_∞(A) = k₁(A)).

An estimate is obtained for ||A^-1||, and the reciprocal of the condition number is computed as rcond = 1 / (||A|| ||A^-1||).

Before calling this routine:

compute anorm (either ||A||₁ =max_jΣ_i |a_ij| or ||A||_∞ =max_iΣ_j |a_ij|)
call ?hptrf to compute the factorization of A.

Input Parameters

uplo	CHARACTER1. Must be 'U' or 'L'. Indicates how the input matrix `A` has been factored: If `uplo = 'U'`, the array `ap` stores the packed upper triangular factor `U` of the factorization `A` = `UDU^T`. If `uplo = 'L'`, the array `ap` stores the packed lower triangular factor `L` of the factorization `A` = `LD*L^T`.
n	INTEGER. The order of matrix `A`; `n`≥ 0.
ap, work	COMPLEX for chpcon DOUBLE COMPLEX for zhpcon. The array `ap()` contains the packed factored matrix `A`, as returned by ?hptrf. The dimension of `ap` must be at least max(1,`n`(`n`+1)/2). The array `work()` is a workspace for the routine. The dimension of `work` must be at least `max(1, 2*n)`.
ipiv	INTEGER. Array, size at least `max(1, n)`. The array `ipiv`, as returned by ?hptrf.
anorm	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. The norm of the original matrix `A` (see Description).

Output Parameters

rcond

REAL for single precision flavors.

DOUBLE PRECISION for double precision flavors.

An estimate of the reciprocal of the condition number. The routine sets rcond =0 if the estimate underflows; in this case the matrix is singular (to working precision). However, anytime rcond is small compared to 1.0, for the working precision, the matrix may be poorly conditioned or even singular.

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 hbcon interface are as follows:

ap	Holds the array `A` of size `(n*(n+1)/2)`.
ipiv	Holds the vector of length `n`.

Application Notes

The computed rcond is never less than r (the reciprocal of the true condition number) and in practice is nearly always less than 10r. A call to this routine involves solving a number of systems of linear equations A*x = b; the number is usually 5 and never more than 11. Each solution requires approximately 8n² floating-point operations.

Parent topic: Estimating the Condition Number: LAPACK Computational Routines

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?hpcon

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