# ?gtcon

Estimates the reciprocal of the condition number of a tridiagonal matrix using the factorization computed by ?gttrf.

## Syntax

FORTRAN 77:

call sgtcon( norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, iwork, info )

call dgtcon( norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, iwork, info )

call cgtcon( norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, info )

call zgtcon( norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, info )

Fortran 95:

call gtcon( dl, d, du, du2, ipiv, anorm, rcond [,norm] [,info] )

## Description

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

κ1(A) = ||A||1||A-1||1

κ∞(A) = ||A||∞||A-1||∞

An estimate is obtained for `||A-1||`, and the reciprocal of the condition number is computed as `rcond = 1 / (||A|| ||A-1||)`.

Before calling this routine:

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

- call ?gttrf 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.

dl,d,du,du2

  REAL for sgtcon

  DOUBLE PRECISION for dgtcon

  COMPLEX for cgtcon

  DOUBLE COMPLEX for zgtcon.

  Arrays: `dl(n -1)`, `d(n)`, `du(n -1)`, `du2(n -2)`.

  The array dl contains the `(n - 1)` multipliers that define the matrix L from the LU factorization of A as computed by ?gttrf.

  The array d contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.

  The array du contains the (n - 1) elements of the first superdiagonal of U.

  The array du2 contains the (n - 2) elements of the second superdiagonal of U.

ipiv

  INTEGER.

  Array, DIMENSION (n). The array of pivot indices, as returned by ?gttrf.

anorm

  REAL for single precision flavors.

  DOUBLE PRECISION for double precision flavors.

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

work

  REAL for sgtcon

  DOUBLE PRECISION for dgtcon

  COMPLEX for cgtcon

  DOUBLE COMPLEX for zgtcon.

  Workspace array, DIMENSION (2*n).

iwork

  INTEGER. Workspace array, DIMENSION (n). Used for real flavors only.

## 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 gtcon interface are as follows:

dl

  Holds the vector of length (n-1).

d

  Holds the vector of length (n).

du

  Holds the vector of length (n-1).

du2

  Holds the vector of length (n-2).

ipiv

  Holds the vector of length (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`; 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