?pocon

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?pocon

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

Syntax

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

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

call cpocon( uplo, n, a, lda, anorm, rcond, work, rwork, info )

call zpocon( uplo, n, a, lda, anorm, rcond, work, rwork, info )

call pocon( a, anorm, rcond [,uplo] [,info] )

Include Files

mkl.fi, lapack.f90

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

n	INTEGER. The order of the matrix `A`; `n`≥ 0.
a, work	REAL for spocon DOUBLE PRECISION for dpocon COMPLEX for cpocon DOUBLE COMPLEX for zpocon. Arrays: `a(lda,)`, `work()`. The array `a` contains the factored matrix `A`, as returned by ?potrf. 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, 3n)` for real flavors and `max(1, 2n)` for complex flavors.
lda	INTEGER. The leading dimension of `a`; `lda≥ max(1, n)`.
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)`.
rwork	REAL for cpocon DOUBLE PRECISION for zpocon. 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 pocon interface are as follows:

a	Holds the matrix `A` of size (`n`, `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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