Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 7/13/2023
Public

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?pbcon

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

Syntax

call spbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info )

call dpbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info )

call cpbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, rwork, info )

call zpbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, rwork, info )

call pbcon( ab, 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 band 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 ?pbtrf to compute the Cholesky factorization of A.

Input Parameters

n

INTEGER. The order of the matrix A; n 0.

kd

INTEGER. The number of superdiagonals or subdiagonals in the matrix A; kd 0.

ldab

INTEGER. The leading dimension of the array ab. (ldabkd +1).

ab, work

REAL for spbcon

DOUBLE PRECISION for dpbcon

COMPLEX for cpbcon

DOUBLE COMPLEX for zpbcon.

Arrays: ab(ldab,*), work(*).

The array ab contains the factored matrix A in band form, as returned by ?pbtrf. The second dimension of ab 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.

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 cpbcon

DOUBLE PRECISION for zpbcon.

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 pbcon interface are as follows:

ab

Holds the array A of size (kd+1,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 4*n(kd + 1) floating-point operations for real flavors and 16*n(kd + 1) for complex flavors.