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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dpocon (f07fg)

## Purpose

nag_lapack_dpocon (f07fg) estimates the condition number of a real symmetric positive definite matrix $A$, where $A$ has been factorized by nag_lapack_dpotrf (f07fd).

## Syntax

[rcond, info] = f07fg(uplo, a, anorm, 'n', n)
[rcond, info] = nag_lapack_dpocon(uplo, a, anorm, 'n', n)

## Description

nag_lapack_dpocon (f07fg) estimates the condition number (in the $1$-norm) of a real symmetric positive definite matrix $A$:
 $κ1A=A1A-11 .$
Since $A$ is symmetric, ${\kappa }_{1}\left(A\right)={\kappa }_{\infty }\left(A\right)={‖A‖}_{\infty }{‖{A}^{-1}‖}_{\infty }$.
Because ${\kappa }_{1}\left(A\right)$ is infinite if $A$ is singular, the function actually returns an estimate of the reciprocal of ${\kappa }_{1}\left(A\right)$.
The function should be preceded by a computation of ${‖A‖}_{1}$ and a call to nag_lapack_dpotrf (f07fd) to compute the Cholesky factorization of $A$. The function then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate ${‖{A}^{-1}‖}_{1}$.

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The Cholesky factor of $A$, as returned by nag_lapack_dpotrf (f07fd).
3:     $\mathrm{anorm}$ – double scalar
The $1$-norm of the original matrix $A$. anorm must be computed either before calling nag_lapack_dpotrf (f07fd) or else from a copy of the original matrix $A$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{rcond}$ – double scalar
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.
2:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## 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.

## 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_lapack_dpocon (f07fg) involves solving a number of systems of linear equations of the form $Ax=b$; 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_lapack_dpotrs (f07fe) 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_lapack_zpocon (f07fu).

## Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 .$
Here $A$ is symmetric positive definite and must first be factorized by nag_lapack_dpotrf (f07fd). The true condition number in the $1$-norm is $97.32$.
```function f07fg_example

fprintf('f07fg example results\n\n');

a = [ 4.16, -3.12,  0.56, -0.10;
-3.12,  5.03, -0.83,  1.18;
0.56, -0.83,  0.76,  0.34;
-0.10,  1.18,  0.34,  1.18];

% Factorize
uplo = 'L';
[af, info] = f07fd(uplo, a);

% Estimate condition number
anorm = norm(a, 1);
[rcond, info] = f07fg( ...
uplo, af, anorm);

fprintf('Estimate of condition number = %9.2e\n', 1/rcond);

```
```f07fg example results

Estimate of condition number =  9.73e+01
```