Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 11/07/2023
Public

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?gbcon

Estimates the reciprocal of the condition number of a band matrix in the 1-norm or the infinity-norm.

Syntax

call sgbcon( norm, n, kl, ku, ab, ldab, ipiv, anorm, rcond, work, iwork, info )

call dgbcon( norm, n, kl, ku, ab, ldab, ipiv, anorm, rcond, work, iwork, info )

call cgbcon( norm, n, kl, ku, ab, ldab, ipiv, anorm, rcond, work, rwork, info )

call zgbcon( norm, n, kl, ku, ab, ldab, ipiv, anorm, rcond, work, rwork, info )

call gbcon( ab, ipiv, anorm, rcond [,kl] [,norm] [,info] )

Include Files

  • mkl.fi, lapack.f90

Description

The routine estimates the reciprocal of the condition number of a general band matrix A in the 1-norm or infinity-norm:

κ1(A) = ||A||1||A-1||1 = κ(AT) = κ(AH)

κ(A) = ||A||||A-1|| = κ1(AT) = κ1(AH).

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 ?gbtrf to compute the LU factorization of A.

Input Parameters

norm

CHARACTER*1. Must be '1' or 'O' or 'I'.

If norm = '1' or 'O', then the routine estimates the condition number of matrix A in 1-norm.

If norm = 'I', then the routine estimates the condition number of matrix A in infinity-norm.

n

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

kl

INTEGER. The number of subdiagonals within the band of A; kl 0.

ku

INTEGER. The number of superdiagonals within the band of A; ku 0.

ldab

INTEGER. The leading dimension of the array ab. (ldab 2*kl + ku +1).

ipiv

INTEGER. Array, size at least max(1, n). The ipiv array, as returned by ?gbtrf.

ab, work

REAL for sgbcon

DOUBLE PRECISION for dgbcon

COMPLEX for cgbcon

DOUBLE COMPLEX for zgbcon.

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

The array ab contains the factored band matrix A, as returned by ?gbtrf.

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 cgbcon

DOUBLE PRECISION for zgbcon.

Workspace array, size at least max(1, 2*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 gbcon interface are as follows:

ab

Holds the array A of size (2*kl+ku+1,n).

ipiv

Holds the vector of length n.

norm

Must be '1', 'O', or 'I'. The default value is '1'.

kl

If omitted, assumed kl = ku.

ku

Restored as ku = lda-2*kl-1.

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 or AH*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n(ku + 2kl) floating-point operations for real flavors and 8n(ku + 2kl) for complex flavors.