# ?pbcon

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

## Syntax

FORTRAN 77:

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 )

Fortran 95:

call pbcon( a, anorm, rcond [,uplo] [,info] )

## Description

This 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)`).

Before calling this routine:

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

- call ?pbtrf to compute the Cholesky factorization of A.

## Input Parameters

uplo

  CHARACTER*1. Must be 'U' or 'L'.

  Indicates how the input matrix A has been factored:

  If `uplo = 'U'`, the upper triangular factor is stored in ab.

  If `uplo = 'L'`, the lower triangular factor is stored in ab.

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 first dimension of the array ab. (ldab ≥ kd +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, DIMENSION at least `max(1, n)`.

rwork

  REAL for cpbcon

  DOUBLE PRECISION for zpbcon.

  Workspace array, DIMENSION 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.

## 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 pbcon interface are as follows:

a

  Stands for argument ab in Fortan 77 interface. 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.

Parent topic: Routines for Estimating the Condition Number