# NAG FL Interfacef03bff (real_​sym)

## 1Purpose

f03bff computes the determinant of a real $n$ by $n$ symmetric positive definite matrix $A$. f07fdf must be called first to supply the symmetric matrix $A$ in Cholesky factorized form. The storage (upper or lower triangular) used by f07fdf 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 <nag.h>
 void f03bff_ (const Integer *n, const double a[], const Integer *lda, double *d, Integer *id, Integer *ifail)
The routine may be called by the names f03bff or nagf_det_real_sym.

## 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.
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 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: $\mathbf{lda}$Integer Input
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: $\mathbf{id}$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.
6: $\mathbf{ifail}$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: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=3$
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 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 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)