?hpcon

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?hpcon

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

Syntax

lapack_int LAPACKE_chpcon( int matrix_layout, char uplo, lapack_int n, const lapack_complex_float* ap, const lapack_int* ipiv, float anorm, float* rcond );

lapack_int LAPACKE_zhpcon( int matrix_layout, char uplo, lapack_int n, const lapack_complex_double* ap, const lapack_int* ipiv, double anorm, double* rcond );

Include Files

mkl.h

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

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 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` = `LDL^T`.
n	The order of matrix `A`; `n`≥ 0.
ap	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).
ipiv	Array, size at least `max(1, n)`. The array `ipiv`, as returned by ?hptrf.
anorm	The norm of the original matrix `A` (see Description).

Output Parameters

rcond

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.

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 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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Developer Reference for Intel® oneAPI Math Kernel Library for C

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