Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpocon (f07fu)

## Purpose

nag_lapack_zpocon (f07fu) estimates the condition number of a complex Hermitian positive definite matrix $A$, where $A$ has been factorized by nag_lapack_zpotrf (f07fr).

## Syntax

[rcond, info] = f07fu(uplo, a, anorm, 'n', n)
[rcond, info] = nag_lapack_zpocon(uplo, a, anorm, 'n', n)

## Description

nag_lapack_zpocon (f07fu) estimates the condition number (in the $1$-norm) of a complex Hermitian positive definite matrix $A$:
 $κ1A=A1A-11 .$
Since $A$ is Hermitian, ${\kappa }_{1}\left(A\right)={\kappa }_{\infty }\left(A\right)={‖A‖}_{\infty }{‖{A}^{-1}‖}_{\infty }$.
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_zpotrf (f07fr) 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.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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_zpotrf (f07fr).
3:     $\mathrm{anorm}$ – double scalar
The $1$-norm of the original matrix $A$. anorm must be computed either before calling nag_lapack_zpotrf (f07fr) 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_zpocon (f07fu) involves solving a number of systems of linear equations of the form $Ax=b$; 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 nag_lapack_zpotrs (f07fs) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The real analogue of this function is nag_lapack_dpocon (f07fg).

## Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .$
Here $A$ is Hermitian positive definite and must first be factorized by nag_lapack_zpotrf (f07fr). The true condition number in the $1$-norm is $201.92$.
```function f07fu_example

fprintf('f07fu example results\n\n');

% Hermitian matrix A
uplo = 'L';
a = [ 3.23,          1.51 - 1.92i,  1.90 + 0.84i,  0.42 + 2.50i;
1.51 + 1.92i,  3.58 + 0i,    -0.23 + 1.11i, -1.18 + 1.37i;
1.90 - 0.84i, -0.23 - 1.11i,  4.09 + 0i,     2.33 - 0.14i;
0.42 - 2.50i, -1.18 - 1.37i,  2.33 + 0.14i,  4.29 + 0i];
anorm = norm(a, 1);

% Factorize A
[L, info] = f07fr( ...
uplo, a);

% Condition number estimator
[rcond, info] = f07fu( ...
uplo, L, anorm);

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

```
```f07fu example results

Estimate of condition number =  1.51e+02
```