F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07AGF (DGECON)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07AGF (DGECON) estimates the condition number of a real matrix $A$, where $A$ has been factorized by F07ADF (DGETRF).

## 2  Specification

 SUBROUTINE F07AGF ( NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
 INTEGER N, LDA, IWORK(N), INFO REAL (KIND=nag_wp) A(LDA,*), ANORM, RCOND, WORK(4*N) CHARACTER(1) NORM
The routine may be called by its LAPACK name dgecon.

## 3  Description

F07AGF (DGECON) estimates the condition number of a real matrix $A$, in either the $1$-norm or the $\infty$-norm:
 $κ1 A = A1 A-11 or κ∞ A = A∞ A-1∞ .$
Note that ${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{T}}\right)$.
Because the condition number is infinite if $A$ is singular, the routine actually returns an estimate of the reciprocal of the condition number.
The routine should be preceded by a call to F06RAF to compute ${‖A‖}_{1}$ or ${‖A‖}_{\infty }$, and a call to F07ADF (DGETRF) to compute the $LU$ factorization of $A$. The routine then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate ${‖{A}^{-1}‖}_{1}$ or ${‖{A}^{-1}‖}_{\infty }$.

## 4  References

Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

## 5  Parameters

1:     $\mathrm{NORM}$ – CHARACTER(1)Input
On entry: indicates whether ${\kappa }_{1}\left(A\right)$ or ${\kappa }_{\infty }\left(A\right)$ is estimated.
${\mathbf{NORM}}=\text{'1'}$ or $\text{'O'}$
${\kappa }_{1}\left(A\right)$ is estimated.
${\mathbf{NORM}}=\text{'I'}$
${\kappa }_{\infty }\left(A\right)$ is estimated.
Constraint: ${\mathbf{NORM}}=\text{'1'}$, $\text{'O'}$ or $\text{'I'}$.
2:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $LU$ factorization of $A$, as returned by F07ADF (DGETRF).
4:     $\mathrm{LDA}$ – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07AGF (DGECON) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
5:     $\mathrm{ANORM}$ – REAL (KIND=nag_wp)Input
On entry: if ${\mathbf{NORM}}=\text{'1'}$ or $\text{'O'}$, the $1$-norm of the original matrix $A$.
If ${\mathbf{NORM}}=\text{'I'}$, the $\infty$-norm of the original matrix $A$.
ANORM may be computed by calling F06RAF with the same value for the parameter NORM.
ANORM must be computed either before calling F07ADF (DGETRF) or else from a copy of the original matrix $A$ (see Section 10).
Constraint: ${\mathbf{ANORM}}\ge 0.0$.
6:     $\mathrm{RCOND}$ – REAL (KIND=nag_wp)Output
On exit: an estimate of the reciprocal of the condition number of $A$. RCOND is set to zero if exact singularity is detected or the estimate underflows. If RCOND is less than machine precision, $A$ is singular to working precision.
7:     $\mathrm{WORK}\left(4×{\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayWorkspace
8:     $\mathrm{IWORK}\left({\mathbf{N}}\right)$ – INTEGER arrayWorkspace
9:     $\mathrm{INFO}$ – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

The computed estimate RCOND is never less than the true value $\rho$, and in practice is nearly always less than $10\rho$, although examples can be constructed where RCOND is much larger.

## 8  Parallelism and Performance

F07AGF (DGECON) is not threaded by NAG in any implementation.
F07AGF (DGECON) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

A call to F07AGF (DGECON) involves solving a number of systems of linear equations of the form $Ax=b$ or ${A}^{\mathrm{T}}x=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2{n}^{2}$ floating-point operations but takes considerably longer than a call to F07AEF (DGETRS) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The complex analogue of this routine is F07AUF (ZGECON).

## 10  Example

This example estimates the condition number in the $1$-norm of the matrix $A$, where
 $A= 1.80 2.88 2.05 -0.89 5.25 -2.95 -0.95 -3.80 1.58 -2.69 -2.90 -1.04 -1.11 -0.66 -0.59 0.80 .$
Here $A$ is nonsymmetric and must first be factorized by F07ADF (DGETRF). The true condition number in the $1$-norm is $152.16$.

### 10.1  Program Text

Program Text (f07agfe.f90)

### 10.2  Program Data

Program Data (f07agfe.d)

### 10.3  Program Results

Program Results (f07agfe.r)