F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07AUF (ZGECON)

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

F07AUF (ZGECON) estimates the condition number of a complex matrix $A$, where $A$ has been factorized by F07ARF (ZGETRF).

## 2  Specification

 SUBROUTINE F07AUF ( NORM, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO)
 INTEGER N, LDA, INFO REAL (KIND=nag_wp) ANORM, RCOND, RWORK(2*N) COMPLEX (KIND=nag_wp) A(LDA,*), WORK(2*N) CHARACTER(1) NORM
The routine may be called by its LAPACK name zgecon.

## 3  Description

F07AUF (ZGECON) estimates the condition number of a complex matrix $A$, in either the $1$-norm or the $\infty$-norm:
 $κ1 A = A1 A-11 or κ∞ A = A∞ A-1∞ .$
Note that ${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{H}}\right)$.
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 F06UAF to compute ${‖A‖}_{1}$ or ${‖A‖}_{\infty }$, and a call to F07ARF (ZGETRF) 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:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     A(LDA,$*$) – COMPLEX (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 F07ARF (ZGETRF).
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07AUF (ZGECON) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
5:     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 F06UAF with the same value for the parameter NORM.
ANORM must be computed either before calling F07ARF (ZGETRF) or else from a copy of the original matrix $A$ (see Section 9).
Constraint: ${\mathbf{ANORM}}\ge 0.0$.
6:     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:     WORK($2×{\mathbf{N}}$) – COMPLEX (KIND=nag_wp) arrayWorkspace
8:     RWORK($2×{\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
9:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th parameter 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.

A call to F07AUF (ZGECON) involves solving a number of systems of linear equations of the form $Ax=b$ or ${A}^{\mathrm{H}}x=b$; the number is usually $5$ and never more than $11$. Each solution involves approximately $8{n}^{2}$ real floating point operations but takes considerably longer than a call to F07ASF (ZGETRS) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The real analogue of this routine is F07AGF (DGECON).

## 9  Example

This example estimates the condition number in the $1$-norm of the matrix $A$, where
 $A= -1.34+2.55i 0.28+3.17i -6.39-2.20i 0.72-0.92i -0.17-1.41i 3.31-0.15i -0.15+1.34i 1.29+1.38i -3.29-2.39i -1.91+4.42i -0.14-1.35i 1.72+1.35i 2.41+0.39i -0.56+1.47i -0.83-0.69i -1.96+0.67i .$
Here $A$ is nonsymmetric and must first be factorized by F07ARF (ZGETRF). The true condition number in the $1$-norm is $231.86$.

### 9.1  Program Text

Program Text (f07aufe.f90)

### 9.2  Program Data

Program Data (f07aufe.d)

### 9.3  Program Results

Program Results (f07aufe.r)