?pocon

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/22/2024

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?pocon

Estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite matrix.

Syntax

lapack_int LAPACKE_spocon( int matrix_layout, char uplo, lapack_int n, const float* a, lapack_int lda, float anorm, float* rcond );

lapack_int LAPACKE_dpocon( int matrix_layout, char uplo, lapack_int n, const double* a, lapack_int lda, double anorm, double* rcond );

lapack_int LAPACKE_cpocon( int matrix_layout, char uplo, lapack_int n, const lapack_complex_float* a, lapack_int lda, float anorm, float* rcond );

lapack_int LAPACKE_zpocon( int matrix_layout, char uplo, lapack_int n, const lapack_complex_double* a, lapack_int lda, double anorm, double* rcond );

Include Files

mkl.h

Description

The routine estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite matrix A:

κ₁(A) = ||A||₁ ||A^-1||₁ (since A is symmetric or Hermitian, κ_∞(A) = κ₁(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 ?potrf to compute the Cholesky 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'`, `A` is factored as `A = U^TU` for real flavors or `A = U^HU` for complex flavors, and `U` is stored. If `uplo = 'L'`, `A` is factored as `A = LL^T` for real flavors or `A = LL^H` for complex flavors, and `L` is stored.
n	The order of the matrix `A`; `n`≥ 0.
a	The array `a` of size max(1, lda*n) contains the factored matrix `A`, as returned by ?potrf.
lda	The leading dimension of `a`; `lda≥ max(1, n)`.
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 4 or 5 and never more than 11. Each solution requires approximately 2n² floating-point operations for real flavors and 8n² for complex flavors.

Parent topic: Estimating the Condition Number: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?pocon

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

Application Notes

See Also