?sycon

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 10/15/2026

Version

Public

?sycon

Estimates the reciprocal of the condition number of a symmetric matrix.

Syntax

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

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

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

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

Include Files

mkl.h

Description

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

κ_{1}(A) = ||A||_{1} ||A^{-1}||_{1} (since A is symmetric, \(\kappa_{\infty}(A) = \kappa_{1}(A)\) ).

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 _j Σ _i | a_ij | or \(||A||_{\infty} = max_{i}\sum_{j} |a_{ij}|\) )
call ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric matrix.) 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 a stores the upper triangular factor U of the factorization A = U*D*U^{T} .

If uplo = 'L' , the array a stores the lower triangular factor L of the factorization A = L*D*L^{T} .

n: Specifies the order of the matrix A . The value of n must be at least zero.

a , work

The array a of size max(1, lda * n ) contains the factored matrix A , as returned by ?sytrf .

lda

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

ipiv

Array, size at least max(1, n) .

The array ipiv , as returned by ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric matrix.) .

anorm

The norm of the original matrix A (see Description ) .

iwork

Workspace array, size at least 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.

LAPACK 95 Interface Notes

There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.

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 10 r . 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 \(2n^{2}\) floating-point operations for real flavors and \(8n^{2}\) for complex flavors.

NOTE:

Matrix Storage Schemes

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?sycon

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

LAPACK 95 Interface Notes

Application Notes