f03 Chapter Contents
f03 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_det_real_gen (f03bac)

## 1  Purpose

nag_det_real_gen (f03bac) computes the determinant of a real $n$ by $n$ matrix $A$. nag_dgetrf (f07adc) must be called first to supply the matrix $A$ in factorized form.

## 2  Specification

 #include #include
 void nag_det_real_gen (Nag_OrderType order, Integer n, const double a[], Integer pda, const Integer ipiv[], double *d, Integer *id, NagError *fail)

## 3  Description

nag_det_real_gen (f03bac) computes the determinant of a real $n$ by $n$ matrix $A$ that has been factorized by a call to nag_dgetrf (f07adc). The determinant of $A$ is the product of the diagonal elements of $U$ with the correct sign determined by the row interchanges.

## 4  References

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

## 5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}>0$.
3:     a[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the factorized form of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $n$ matrix $A$ in factorized form as returned by nag_dgetrf (f07adc).
4:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
5:     ipiv[n]const IntegerInput
On entry: the row interchanges used to factorize matrix $A$ as returned by nag_dgetrf (f07adc).
6:     ddouble *Output
7:     idInteger *Output
On exit: the determinant of $A$ is given by ${\mathbf{d}}×{2.0}^{{\mathbf{id}}}$. It is given in this form to avoid overflow or underflow.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_SINGULAR
The matrix $A$ is approximately singular.

## 7  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 (f03bac) is approximately proportional to $n$.

## 9  Example

This example computes the $LU$ factorization with partial pivoting, and calculates the determinant, of the real matrix
 $33 16 72 -24 -10 -57 -8 -4 -17 .$

### 9.1  Program Text

Program Text (f03bace.c)

### 9.2  Program Data

Program Data (f03bace.d)

### 9.3  Program Results

Program Results (f03bace.r)