# ?sycon

Estimates the reciprocal of the condition number of a symmetric matrix.

## Syntax

FORTRAN 77:

call ssycon( uplo, n, a, lda, ipiv, anorm, rcond, work, iwork, info )

call dsycon( uplo, n, a, lda, ipiv, anorm, rcond, work, iwork, info )

call csycon( uplo, n, a, lda, ipiv, anorm, rcond, work, info )

call zsycon( uplo, n, a, lda, ipiv, anorm, rcond, work, info )

Fortran 95:

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

## Description

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

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

Before calling this routine:

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

- call ?sytrf 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 a stores the upper triangular factor U of the factorization A = `P*U*D*UT*PT`.

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

n

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

a, work

  REAL for ssycon

  DOUBLE PRECISION for dsycon

  COMPLEX for csycon

  DOUBLE COMPLEX for zsycon.

  Arrays: `a(lda,*)`, `work(*)`.

  The array a contains the factored matrix A, as returned by ?sytrf. The second dimension of a 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, 2*n)`.

lda

  INTEGER. The first dimension of a; `lda ≥ max(1, n)`.

ipiv

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

  The array ipiv, as returned by ?sytrf.

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)`.

## 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 sycon interface are as follows:

a

  Holds the matrix A of size (n, n).

ipiv

  Holds the vector of length (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 `2n2` floating-point operations for real flavors and `8n2` for complex flavors.

Parent topic: Routines for Estimating the Condition Number