?gecon

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?gecon

Estimates the reciprocal of the condition number of a general matrix in the 1-norm or the infinity-norm.

Syntax

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

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

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

lapack_int LAPACKE_zgecon( int matrix_layout, char norm, 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 general matrix A in the 1-norm or infinity-norm:

κ₁(A) =||A||₁||A^-1||₁ = κ_∞(A^T) = κ_∞(A^H)

κ_∞(A) =||A||_∞||A^-1||_∞ = κ₁(A^T) = κ₁(A^H).

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 ?getrf to compute the LU factorization of A.

Input Parameters

matrix_layout	Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
norm	Must be '1' or 'O' or 'I'. If `norm` = '1' or 'O', then the routine estimates the condition number of matrix `A` in 1-norm. If `norm = 'I'`, then the routine estimates the condition number of matrix `A` in infinity-norm.
n	The order of the matrix `A`; `n`≥ 0.
a	The array `a` contains the `LU`-factored matrix `A`, as returned by ?getrf.
anorm	The norm of the original matrix `A` (see Description).
lda	The leading dimension of `a; lda≥ max(1, n)`.

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 or A^H*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2*n² floating-point operations for real flavors and 8*n² for complex flavors.

Parent topic: Estimating the Condition Number: LAPACK Computational Routines

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?gecon

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