# ?gbcon

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

## Syntax

FORTRAN 77:

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 )

Fortran 95:

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

## Description

This 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).

Before calling this routine:

- compute anorm (either ||A||1 =maxj Σi |ai j| or ||A||∞ = maxi Σj |ai j|)

- 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 first dimension of the array ab. (ldab ≥ 2*kl + ku +1).

ipiv

  INTEGER. Array, DIMENSION 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, DIMENSION at least `max(1, n)`.

rwork

  REAL for cgbcon

  DOUBLE PRECISION for zgbcon.

  Workspace array, DIMENSION 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.

## Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine gbcon interface are as follows:

a

  Stands for argument ab in Fortan 77 interface. 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.

Parent topic: Routines for Estimating the Condition Number