# NAG FL Interfacef03baf (real_​gen)

## 1Purpose

f03baf computes the determinant of a real $n$ by $n$ matrix $A$. f07adf 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
C Header Interface
#include <nag.h>
 void f03baf_ (const Integer *n, const double a[], const Integer *lda, const Integer ipiv[], double *d, Integer *id, Integer *ifail)
The routine may be called by the names f03baf or nagf_det_real_gen.

## 3Description

f03baf computes the determinant of a real $n$ by $n$ matrix $A$ that has been factorized by a call to f07adf. 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

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}>0$.
2: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input
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.
3: $\mathbf{lda}$Integer Input
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 array Input
On entry: the row interchanges used to factorize matrix $A$ as returned by f07adf.
5: $\mathbf{d}$Real (Kind=nag_wp) Output
6: $\mathbf{id}$Integer Output
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: $\mathbf{ifail}$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.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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

f03baf is not threaded in any implementation.

## 9Further Comments

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)