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NAG Toolbox

# NAG Toolbox: nag_lapack_dgecon (f07ag)

## Purpose

nag_lapack_dgecon (f07ag) estimates the condition number of a real matrix A$A$, where A$A$ has been factorized by nag_lapack_dgetrf (f07ad).

## Syntax

[rcond, info] = f07ag(norm_p, a, anorm, 'n', n)
[rcond, info] = nag_lapack_dgecon(norm_p, a, anorm, 'n', n)

## Description

nag_lapack_dgecon (f07ag) estimates the condition number of a real matrix A$A$, in either the 1$1$-norm or the $\infty$-norm:
 κ1 (A) = ‖A‖1 ‖A − 1‖1   or   κ∞ (A) = ‖A‖∞ ‖A − 1‖∞ . $κ1 (A) = ‖A‖1 ‖A-1‖1 or κ∞ (A) = ‖A‖∞ ‖A-1‖∞ .$
Note that κ(A) = κ1(AT)${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{T}}\right)$.
Because the condition number is infinite if A$A$ is singular, the function actually returns an estimate of the reciprocal of the condition number.

## 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:     norm_p – string (length ≥ 1)
Indicates whether κ1(A)${\kappa }_{1}\left(A\right)$ or κ(A)${\kappa }_{\infty }\left(A\right)$ is estimated.
norm = '1'${\mathbf{norm}}=\text{'1'}$ or 'O'$\text{'O'}$
κ1(A)${\kappa }_{1}\left(A\right)$ is estimated.
norm = 'I'${\mathbf{norm}}=\text{'I'}$
κ(A)${\kappa }_{\infty }\left(A\right)$ is estimated.
Constraint: norm = '1'${\mathbf{norm}}=\text{'1'}$, 'O'$\text{'O'}$ or 'I'$\text{'I'}$.
2:     a(lda, : $:$) – double array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The LU$LU$ factorization of A$A$, as returned by nag_lapack_dgetrf (f07ad).
3:     anorm – double scalar
If norm = '1'${\mathbf{norm}}=\text{'1'}$ or 'O'$\text{'O'}$, the 1$1$-norm of the original matrix A$A$.
If norm = 'I'${\mathbf{norm}}=\text{'I'}$, the $\infty$-norm of the original matrix A$A$.
anorm may be computed by calling nag_blas_dlange (f06ra) with the same value for the parameter norm_p.
anorm must be computed either before calling nag_lapack_dgetrf (f07ad) or else from a copy of the original matrix A$A$ (see Section [Example]).
Constraint: anorm0.0${\mathbf{anorm}}\ge 0.0$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the array a.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda work iwork

### Output Parameters

1:     rcond – double scalar
An estimate of the reciprocal of the condition number of A$A$. rcond is set to zero if exact singularity is detected or the estimate underflows. If rcond is less than machine precision, A$A$ is singular to working precision.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: norm_p, 2: n, 3: a, 4: lda, 5: anorm, 6: rcond, 7: work, 8: iwork, 9: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

## Accuracy

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

A call to nag_lapack_dgecon (f07ag) involves solving a number of systems of linear equations of the form Ax = b$Ax=b$ or ATx = b${A}^{\mathrm{T}}x=b$; the number is usually 4$4$ or 5$5$ and never more than 11$11$. Each solution involves approximately 2n2$2{n}^{2}$ floating point operations but takes considerably longer than a call to nag_lapack_dgetrs (f07ae) with one right-hand side, because extra care is taken to avoid overflow when A$A$ is approximately singular.
The complex analogue of this function is nag_lapack_zgecon (f07au).

## Example

```function nag_lapack_dgecon_example
norm_p = '1';
a = [1.8, 2.88, 2.05, -0.89;
5.25, -2.95, -0.95, -3.8;
1.58, -2.69, -2.9, -1.04;
-1.11, -0.66, -0.59, 0.8];

anorm = norm(a, 1);

% Factorize A
[a, ipiv, info] = nag_lapack_dgetrf(a);

% Estimate condition number
[rcond, info] = nag_lapack_dgecon(norm_p, a, anorm);

if rcond > nag_machine_precision
fprintf('\nEstimate of condition number = %10.2e\n', 1/rcond);
else
fprintf('\nA is singular to working precision\n');
end
```
```

Estimate of condition number =   1.52e+02

```
```function f07ag_example
norm_p = '1';
a = [1.8, 2.88, 2.05, -0.89;
5.25, -2.95, -0.95, -3.8;
1.58, -2.69, -2.9, -1.04;
-1.11, -0.66, -0.59, 0.8];

anorm = norm(a, 1);

% Factorize A

% Estimate condition number
[rcond, info] = f07ag(norm_p, a, anorm);

if rcond > x02aj
fprintf('\nEstimate of condition number = %10.2e\n', 1/rcond);
else
fprintf('\nA is singular to working precision\n');
end
```
```

Estimate of condition number =   1.52e+02

```

Chapter Contents
Chapter Introduction
NAG Toolbox

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