Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 3/31/2023
Public

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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 = maxjΣi |aij| or ||A|| = maxiΣj |aij|)

  • 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, 3*n) for real flavors and max(1, 2*n) 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.