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# NAG Toolbox: nag_det_real_gen (f03ba)

## Purpose

nag_det_real_gen (f03ba) computes the determinant of a real n$n$ by n$n$ matrix A$A$. nag_lapack_dgetrf (f07ad) must be called first to supply the matrix A$A$ in factorized form.

## Syntax

[d, id, ifail] = f03ba(a, ipiv, 'n', n)
[d, id, ifail] = nag_det_real_gen(a, ipiv, 'n', n)

## Description

nag_det_real_gen (f03ba) computes the determinant of a real n$n$ by n$n$ matrix A$A$ that has been factorized by a call to nag_lapack_dgetrf (f07ad). The determinant of A$A$ is the product of the diagonal elements of U$U$ 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 n${\mathbf{n}}$
The second dimension of the array must be at least n${\mathbf{n}}$
The n$n$ by n$n$ matrix A$A$ in factorized form as returned by nag_lapack_dgetrf (f07ad).
2:     ipiv(n) – int64int32nag_int array
n, the dimension of the array, must satisfy the constraint n > 0${\mathbf{n}}>0$.
The row interchanges used to factorize matrix A$A$ as returned by nag_lapack_dgetrf (f07ad).

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a and the dimension of the array ipiv. (An error is raised if these dimensions are not equal.)
n$n$, the order of the matrix A$A$.
Constraint: n > 0${\mathbf{n}}>0$.

lda

### Output Parameters

1:     d – double scalar
2:     id – int64int32nag_int scalar
The determinant of A$A$ is given by d × 2.0id${\mathbf{d}}×{2.0}^{{\mathbf{id}}}$. It is given in this form to avoid overflow or underflow.
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${\mathbf{ifail}}=1$
Constraint: n1${\mathbf{n}}\ge 1$.
ifail = 3${\mathbf{ifail}}=3$
Constraint: ldan$\mathit{lda}\ge {\mathbf{n}}$.
ifail = 4${\mathbf{ifail}}=4$
The matrix A$A$ is approximately singular.

## 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_real_gen (f03ba) is approximately proportional to n$n$.

## Example

```function nag_det_real_gen_example
a = [ 33,  16,  72;
-24, -10, -57;
-8,  -4, -17];
% Compute LU factorisation of a
[a, ipiv, info] = nag_lapack_dgetrf(a);

fprintf('\n');
[ifail] = nag_file_print_matrix_real_gen('g', 'n', a, 'Array a after factorization');

fprintf('\nPivots:\n');
fprintf(' %d', ipiv);
fprintf('\n\n');

[d, id, ifail] = nag_det_real_gen(a, ipiv);

fprintf('d = %13.5f id = %d\n', d, id);
fprintf('Value of determinant = %13.5e\n', d*2^id);
```
```

Array a after factorization
1          2          3
1     33.0000    16.0000    72.0000
2     -0.7273     1.6364    -4.6364
3     -0.2424    -0.0741     0.1111

Pivots:
1 2 3

d =       0.37500 id = 4
Value of determinant =   6.00000e+00

```
```function f03ba_example
a = [ 33,  16,  72;
-24, -10, -57;
-8,  -4, -17];
% Compute LU factorisation of a
[a, ipiv, info] = f07ad(a);

fprintf('\n');
[ifail] = x04ca('g', 'n', a, 'Array a after factorization');

fprintf('\nPivots:\n');
fprintf(' %d', ipiv);
fprintf('\n\n');

[d, id, ifail] = f03ba(a, ipiv);

fprintf('d = %13.5f id = %d\n', d, id);
fprintf('Value of determinant = %13.5e\n', d*2^id);
```
```

Array a after factorization
1          2          3
1     33.0000    16.0000    72.0000
2     -0.7273     1.6364    -4.6364
3     -0.2424    -0.0741     0.1111

Pivots:
1 2 3

d =       0.37500 id = 4
Value of determinant =   6.00000e+00

```

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