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# NAG Toolbox: nag_det_withdraw_real_gen (f03aa)

## Purpose

nag_det_withdraw_real_gen (f03aa) calculates the determinant of a real matrix using an LU$LU$ factorization with partial pivoting.
Note: this function is scheduled to be withdrawn, please see f03aa in Advice on Replacement Calls for Withdrawn/Superseded Routines..

## Syntax

[a, det, ifail] = f03aa(a, 'n', n)
[a, det, ifail] = nag_det_withdraw_real_gen(a, 'n', n)

## Description

nag_det_withdraw_real_gen (f03aa) calculates the determinant of A$A$ using the LU$LU$ factorization with partial pivoting, PA = LU$PA=LU$, where P$P$ is a permutation matrix, L$L$ is lower triangular and U$U$ is unit upper triangular. The determinant of A$A$ is the product of the diagonal elements of L$L$ with the correct sign determined by the row interchanges.

## References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## Parameters

### Compulsory Input Parameters

1:     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 n$n$ by n$n$ matrix A$A$.

### 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 wkspce

### Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
A$A$ stores the factors L$L$ and U$U$, except that the unit diagonal elements of U$U$ are not stored.
2:     det – double scalar
The determinant of A$A$.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
The matrix A$A$ is singular, possibly due to rounding errors. The factorization could not be completed. det is set to 0.0$0.0$.
ifail = 2${\mathbf{ifail}}=2$
Overflow. The value of the determinant is too large to be held in the computer.
ifail = 3${\mathbf{ifail}}=3$
Underflow. The value of the determinant is too small to be held in the computer.
ifail = 4${\mathbf{ifail}}=4$
 On entry, n < 0${\mathbf{n}}<0$, or lda < max (1,n)$\mathit{lda}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

## Accuracy

The accuracy of the determinant depends on the conditioning of the original matrix. For a detailed error analysis see page 107 of Wilkinson and Reinsch (1971).

The time taken by nag_det_withdraw_real_gen (f03aa) is approximately proportional to n3${n}^{3}$.

## Example

```function nag_det_withdraw_real_gen_example
a = [33, 16, 72;
-24, -10, -57;
-8, -4, -17];
[aOut, det, ifail] = nag_det_withdraw_real_gen(a)
```
```

aOut =

-8.0000    0.5000    2.1250
-24.0000    2.0000   -3.0000
33.0000   -0.5000    0.3750

det =

6

ifail =

0

```
```function f03aa_example
a = [33, 16, 72;
-24, -10, -57;
-8, -4, -17];
[aOut, det, ifail] = f03aa(a)
```
```

aOut =

-8.0000    0.5000    2.1250
-24.0000    2.0000   -3.0000
33.0000   -0.5000    0.3750

det =

6

ifail =

0

```

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Chapter Introduction
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