# ?hpcon

Estimates the reciprocal of the condition number of a packed Hermitian matrix.

## Syntax

FORTRAN 77:

call chpcon( uplo, n, ap, ipiv, anorm, rcond, work, info )

call zhpcon( uplo, n, ap, ipiv, anorm, rcond, work, info )

Fortran 95:

call hpcon( a, ipiv, anorm, rcond [,uplo] [,info] )

## Description

This routine estimates the reciprocal of the condition number of a Hermitian matrix A:

`κ1(A) = ||A||1 ||A-1||1` (since A is Hermitian, `κ∞(A) = k1(A)`).

Before calling this routine:

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

- call ?hptrf to compute the 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 array ap stores the packed upper triangular factor U of the factorization A = `P*U*D*UT*PT`.

  If `uplo = 'L'`, the array ap stores the packed lower triangular factor L of the factorization A = `P*L*D*LT*PT`.

n

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

ap, work

  COMPLEX for chpcon

  DOUBLE COMPLEX for zhpcon.

  Arrays: `ap(*)`, `work(*)`.

  The array ap contains the packed factored matrix A, as returned by ?hptrf. The dimension of ap must be at least max(1,n(n+1)/2).

  The array work is a workspace for the routine.

  The dimension of work must be at least `max(1, 2*n)`.

ipiv

  INTEGER.

  Array, DIMENSION at least `max(1, n)`. The array ipiv, as returned by ?hptrf.

anorm

  REAL for single precision flavors

  DOUBLE PRECISION for double precision flavors.

  The norm of the original matrix A (see Description).

## 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 hbcon interface are as follows:

a

  Stands for argument ap in Fortan 77 interface. Holds the array A of size `(n*(n+1)/2)`.

ipiv

  Holds the vector of length (n).

## 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 5 and never more than 11. Each solution requires approximately `8n2` floating-point operations.

Parent topic: Routines for Estimating the Condition Number