?sycon

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?sycon

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

Syntax

call ssycon( uplo, n, a, lda, ipiv, anorm, rcond, work, iwork, info )

call dsycon( uplo, n, a, lda, ipiv, anorm, rcond, work, iwork, info )

call csycon( uplo, n, a, lda, ipiv, anorm, rcond, work, info )

call zsycon( uplo, n, a, lda, ipiv, anorm, rcond, work, info )

call sycon( a, ipiv, anorm, rcond [,uplo] [,info] )

Include Files

mkl.fi, lapack.f90

Description

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

κ₁(A) = ||A||₁ ||A^-1||₁ (since A is symmetric, 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 ?sytrf 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 `a` stores the upper triangular factor `U` of the factorization `A` = `UDU^T`. If `uplo = 'L'`, the array `a` stores the lower triangular factor `L` of the factorization `A` = `LD*L^T`.
n	INTEGER. The order of matrix `A`; `n`≥ 0.
a, work	REAL for ssycon DOUBLE PRECISION for dsycon COMPLEX for csycon DOUBLE COMPLEX for zsycon. Arrays: `a(lda,)`, `work()`. The array `a` contains the factored matrix `A`, as returned by ?sytrf. The second dimension of `a` must be at least `max(1,n)`. The array `work` is a workspace for the routine. The dimension of `work` must be at least `max(1, 2*n)`.
lda	INTEGER. The leading dimension of `a`; `lda≥ max(1, n)`.
ipiv	INTEGER. Array, size at least `max(1, n)`. The array `ipiv`, as returned by ?sytrf.
anorm	REAL for single precision flavors. DOUBLE PRECISION for double precision flavors. The norm of the original matrix `A` (see Description).
iwork	INTEGER. Workspace array, size at least `max(1, n)`.

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 sycon interface are as follows:

a	Holds the matrix `A` of size (`n`, `n`).
ipiv	Holds the vector of length `n`.
uplo	Must be 'U' or 'L'. The default value is 'U'.

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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?sycon

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