?gbcon

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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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_∞(A^T) = κ_∞(A^H)

κ_∞(A) = ||A||_∞||A^-1||_∞ = κ₁(A^T) = κ₁(A^H).

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 ?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, 3n)` for real flavors and `max(1, 2n)` 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 A^H*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.

Parent topic: Estimating the Condition Number: LAPACK Computational Routines

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