# NAG Library Routine Document

## 1Purpose

f03baf computes the determinant of a real $n$ by $n$ matrix $A$. f07adf (dgetrf) must be called first to supply the matrix $A$ in factorized form.

## 2Specification

Fortran Interface
 Subroutine f03baf ( n, a, lda, ipiv, d, id,
 Integer, Intent (In) :: n, lda, ipiv(n) Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: id Real (Kind=nag_wp), Intent (In) :: a(lda,*) Real (Kind=nag_wp), Intent (Out) :: d
#include nagmk26.h
 void f03baf_ (const Integer *n, const double a[], const Integer *lda, const Integer ipiv[], double *d, Integer *id, Integer *ifail)

## 3Description

f03baf computes the determinant of a real $n$ by $n$ matrix $A$ that has been factorized by a call to f07adf (dgetrf). 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:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (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 f07adf (dgetrf).
3:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f03baf 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 f07adf (dgetrf).
5:     $\mathbf{d}$ – Real (Kind=nag_wp)Output
6:     $\mathbf{id}$ – IntegerOutput
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.
7:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, 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).

## 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

f03baf is not threaded in any implementation.

The time taken by f03baf is approximately proportional to $n$.

## 10Example

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

### 10.1Program Text

Program Text (f03bafe.f90)

### 10.2Program Data

Program Data (f03bafe.d)

### 10.3Program Results

Program Results (f03bafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017