# NAG FL Interfacef03bnf (complex_​gen)

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## 1Purpose

f03bnf computes the determinant of a complex $n×n$ matrix $A$. f07arf 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 <nag.h>
 void f03bnf_ (const Integer *n, const Complex a[], const Integer *lda, const Integer ipiv[], Complex *d, Integer id[], Integer *ifail)
The routine may be called by the names f03bnf or nagf_det_complex_gen.

## 3Description

f03bnf computes the determinant of a complex $n×n$ matrix $A$ that has been factorized by a call to f07arf. 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: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}>0$.
2: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input
Note: the second dimension of the array a must be at least ${\mathbf{n}}$.
On entry: the $n×n$ matrix $A$ in factorized form as returned by f07arf.
3: $\mathbf{lda}$Integer Input
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 array Input
On entry: the row interchanges used to factorize matrix $A$ as returned by f07arf.
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 array Output
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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

## 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 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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)