F03 Chapter Contents
F03 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF03BNF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F03BNF computes the determinant of a complex $n$ by $n$ matrix $A$. F07ARF (ZGETRF) must be called first to supply the matrix $A$ in factorized form.

## 2  Specification

 SUBROUTINE F03BNF ( N, A, LDA, IPIV, D, ID, IFAIL)
 INTEGER N, LDA, IPIV(N), ID(2), IFAIL COMPLEX (KIND=nag_wp) A(LDA,*), D

## 3  Description

F03BNF computes the determinant of a complex $n$ by $n$ matrix $A$ that has been factorized by a call to F07ARF (ZGETRF). The determinant of $A$ is the product of the diagonal elements of $U$ with the correct sign determined by the row interchanges.
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## 5  Parameters

1:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}>0$.
2:     $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput
Note: the second dimension of the array A must be at least ${\mathbf{N}}$.
On entry: the $n$ by $n$ matrix $A$ in factorized form as returned by F07ARF (ZGETRF).
3:     $\mathrm{LDA}$ – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F03BNF is called.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
4:     $\mathrm{IPIV}\left({\mathbf{N}}\right)$ – INTEGER arrayInput
On entry: the row interchanges used to factorize matrix $A$ as returned by F07ARF (ZGETRF).
5:     $\mathrm{D}$ – COMPLEX (KIND=nag_wp)Output
On exit: the mantissa of the real and imaginary parts of the determinant.
6:     $\mathrm{ID}\left(2\right)$ – INTEGER arrayOutput
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.
7:     $\mathrm{IFAIL}$ – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, ${\mathbf{N}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{N}}\ge 1$.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{LDA}}=〈\mathit{\text{value}}〉$ and ${\mathbf{N}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
${\mathbf{IFAIL}}=4$
The matrix $A$ is approximately singular.
${\mathbf{IFAIL}}=-99$
See Section 3.8 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

## 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).

## 8  Parallelism and Performance

Not applicable.

The time taken by F03BNF is approximately proportional to $n$.

## 10  Example

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

### 10.1  Program Text

Program Text (f03bnfe.f90)

### 10.2  Program Data

Program Data (f03bnfe.d)

### 10.3  Program Results

Program Results (f03bnfe.r)