# NAG Library Routine Document

## 1Purpose

f03bff computes the determinant of a real $n$ by $n$ symmetric positive definite matrix $A$. f07fdf (dpotrf) must be called first to supply the symmetric matrix $A$ in Cholesky factorized form. The storage (upper or lower triangular) used by f07fdf (dpotrf) is not relevant to f03bff since only the diagonal elements of the factorized $A$ are referenced.

## 2Specification

Fortran Interface
 Subroutine f03bff ( n, a, lda, d, id,
 Integer, Intent (In) :: n, lda 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 f03bff_ (const Integer *n, const double a[], const Integer *lda, double *d, Integer *id, Integer *ifail)

## 3Description

f03bff computes the determinant of a real $n$ by $n$ symmetric positive definite matrix $A$ that has been factorized as $A={U}^{\mathrm{T}}U$, where $U$ is upper triangular, or $A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular. The determinant is the product of the squares of the diagonal elements of $U$ or $L$. The Cholesky factorized form of the matrix must be supplied; this is returned by a call to f07fdf (dpotrf).
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 lower or upper triangle of the Cholesky factorized form of the $n$ by $n$ positive definite symmetric matrix $A$. Only the diagonal elements are referenced.
3:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f03bff is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
4:     $\mathbf{d}$ – Real (Kind=nag_wp)Output
5:     $\mathbf{id}$ – 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.
6:     $\mathbf{ifail}$ – 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: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=3$
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 not positive definite.
${\mathbf{ifail}}=-99$
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 25 of Wilkinson and Reinsch (1971).

## 8Parallelism and Performance

f03bff is not threaded in any implementation.

The time taken by f03bff is approximately proportional to $n$.

## 10Example

This example computes a Cholesky factorization and calculates the determinant of the real symmetric positive definite matrix
 $6 7 6 5 7 11 8 7 6 8 11 9 5 7 9 11 .$

### 10.1Program Text

Program Text (f03bffe.f90)

### 10.2Program Data

Program Data (f03bffe.d)

### 10.3Program Results

Program Results (f03bffe.r)

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