f03 Chapter Contents
f03 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_det_complex_gen (f03bnc)

## 1  Purpose

nag_det_complex_gen (f03bnc) computes the determinant of a complex $n$ by $n$ matrix $A$. nag_zgetrf (f07arc) must be called first to supply the matrix $A$ in factorized form.

## 2  Specification

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

## 3  Description

nag_det_complex_gen (f03bnc) computes the determinant of a complex $n$ by $n$ matrix $A$ that has been factorized by a call to nag_zgetrf (f07arc). 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 ComplexInput
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_zgetrf (f07arc).
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_zgetrf (f07arc).
6:     dComplex *Output
On exit: the mantissa of the real and imaginary parts of the determinant.
7:     id[$2$]IntegerOutput
On exit: the exponents for the real and imaginary parts of the determinant. The determinant, $d=\left({d}_{r},{d}_{i}\right)$, is returned as ${d}_{r}={D}_{r}×{2}^{j}$ and ${d}_{i}={D}_{i}×{2}^{k}$, where ${\mathbf{d}}=\left({D}_{r},{D}_{i}\right)$ and $j$ and $k$ are stored in the first and second elements respectively of the array id on successful exit.
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_complex_gen (f03bnc) is approximately proportional to $n$.

## 9  Example

This example calculates the determinant of the complex matrix
 $1 1+2i 2+10i 1+i 3i -5+14i 1+i 5i -8+20i .$

### 9.1  Program Text

Program Text (f03bnce.c)

### 9.2  Program Data

Program Data (f03bnce.d)

### 9.3  Program Results

Program Results (f03bnce.r)