# NAG Library Routine Document

## 1Purpose

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.

## 2Specification

Fortran Interface
 Subroutine f03bnf ( n, a, lda, ipiv, d, id,
 Integer, Intent (In) :: n, lda, ipiv(n) Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: id(2) Complex (Kind=nag_wp), Intent (In) :: a(lda,*) Complex (Kind=nag_wp), Intent (Out) :: d
#include <nagmk26.h>
 void f03bnf_ (const Integer *n, const Complex a[], const Integer *lda, const Integer ipiv[], Complex *d, Integer id[], Integer *ifail)

## 3Description

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.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}>0$.
2:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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:     $\mathbf{lda}$ – 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:     $\mathbf{ipiv}\left({\mathbf{n}}\right)$ – Integer arrayInput
On entry: the row interchanges used to factorize matrix $A$ as returned by f07arf (zgetrf).
5:     $\mathbf{d}$ – Complex (Kind=nag_wp)Output
On exit: the mantissa of the real and imaginary parts of the determinant.
6:     $\mathbf{id}\left(2\right)$ – 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:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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).

## 8Parallelism and Performance

f03bnf is not threaded in any implementation.

The time taken by f03bnf is approximately proportional to $n$.

## 10Example

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

### 10.1Program Text

Program Text (f03bnfe.f90)

### 10.2Program Data

Program Data (f03bnfe.d)

### 10.3Program Results

Program Results (f03bnfe.r)