f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dgecon (f07agc)

## 1  Purpose

nag_dgecon (f07agc) estimates the condition number of a real matrix $A$, where $A$ has been factorized by nag_dgetrf (f07adc).

## 2  Specification

 #include #include
 void nag_dgecon (Nag_OrderType order, Nag_NormType norm, Integer n, const double a[], Integer pda, double anorm, double *rcond, NagError *fail)

## 3  Description

nag_dgecon (f07agc) 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 function actually returns an estimate of the reciprocal of the condition number.
The function should be preceded by a call to nag_dge_norm (f16rac) to compute ${‖A‖}_{1}$ or ${‖A‖}_{\infty }$, and a call to nag_dgetrf (f07adc) to compute the $LU$ factorization of $A$. The function 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  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     normNag_NormTypeInput
On entry: indicates whether ${\kappa }_{1}\left(A\right)$ or ${\kappa }_{\infty }\left(A\right)$ is estimated.
${\mathbf{norm}}=\mathrm{Nag_OneNorm}$
${\kappa }_{1}\left(A\right)$ is estimated.
${\mathbf{norm}}=\mathrm{Nag_InfNorm}$
${\kappa }_{\infty }\left(A\right)$ is estimated.
Constraint: ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$ or $\mathrm{Nag_InfNorm}$.
3:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     a[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $LU$ factorization of $A$, as returned by nag_dgetrf (f07adc).
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6:     anormdoubleInput
On entry: if ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$, the $1$-norm of the original matrix $A$.
If ${\mathbf{norm}}=\mathrm{Nag_InfNorm}$, the $\infty$-norm of the original matrix $A$.
anorm may be computed by calling nag_dge_norm (f16rac) with the same value for the argument norm.
anorm must be computed either before calling nag_dgetrf (f07adc) or else from a copy of the original matrix $A$ (see Section 9).
Constraint: ${\mathbf{anorm}}\ge 0.0$.
7:     rconddouble *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.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, ${\mathbf{anorm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.

## 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 nag_dgecon (f07agc) 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 nag_dgetrs (f07aec) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The complex analogue of this function is nag_zgecon (f07auc).

## 9  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 nag_dgetrf (f07adc). The true condition number in the $1$-norm is $152.16$.

### 9.1  Program Text

Program Text (f07agce.c)

### 9.2  Program Data

Program Data (f07agce.d)

### 9.3  Program Results

Program Results (f07agce.r)