# ?gecon

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

## Syntax

FORTRAN 77:

call sgecon( norm, n, a, lda, anorm, rcond, work, iwork, info )

call dgecon( norm, n, a, lda, anorm, rcond, work, iwork, info )

call cgecon( norm, n, a, lda, anorm, rcond, work, rwork, info )

call zgecon( norm, n, a, lda, anorm, rcond, work, rwork, info )

Fortran 95:

call gecon( a, anorm, rcond [,norm] [,info] )

## Description

This routine estimates the reciprocal of the condition number of a general matrix A in the 1-norm or infinity-norm:

κ 1(A) =||A||1||A-1||1 = κ ∞(AT)= κ ∞(AH)

κ ∞(A) =||A||∞||A-1||∞ = κ 1(AT)= κ 1(AH).

Before calling this routine:

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

- call ?getrf 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.

a, work

  REAL for sgecon

  DOUBLE PRECISION for dgecon

  COMPLEX for cgecon

  DOUBLE COMPLEX for zgecon. Arrays: `a(lda,*)`, `work(*)`.

  The array a contains the LU-factored matrix A, as returned by ?getrf. 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, 4*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).

lda

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

iwork

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

rwork

  REAL for cgecon

  DOUBLE PRECISION for zgecon.

  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 gecon interface are as follows:

a

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

norm

  Must be '1', 'O', or 'I'. The default value is '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 `2*n2` floating-point operations for real flavors and `8*n2` for complex flavors.

Parent topic: Routines for Estimating the Condition Number