NAG FL Interface
f03bff (real_sym)

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.

2 Specification

Fortran Interface

Subroutine f03bff (

n, a, lda, d, id, ifail)

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

C Header Interface

#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.

3 Description

f03bff computes the determinant of a real

n \times n

symmetric positive definite matrix

A

that has been factorized as

A = U^{T} U

, where

U

is upper triangular, or

A = L L^{T}

, where

L

is lower triangular. The determinant is the product of the squares of the diagonal elements of

U

L

. The Cholesky factorized form of the matrix must be supplied; this is returned by a call to f07fdf.

4 References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n > 0$ .
2: $a (lda, *)$ – Real (Kind=nag_wp) array Input: Note: the second dimension of the array a must be at least $n$ .

On entry: the lower or upper triangle of the Cholesky factorized form of the $n \times n$ positive definite symmetric matrix $A$ . Only the diagonal elements are referenced.
3: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f03bff is called.

Constraint: $lda \geq n$ .
4: $d$ – Real (Kind=nag_wp) Output
5: $id$ – Integer Output: On exit: the determinant of $A$ is given by $d \times {2.0}^{id}$ . It is given in this form to avoid overflow or underflow.
6: $ifail$ – 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 $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .

$ifail = 3$: On entry, $lda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $lda \geq n$ .

$ifail = 4$: The matrix $A$ is not positive definite.

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

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).

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f03bff is not threaded in any implementation.

9 Further Comments

The time taken by f03bff is approximately proportional to

n

10 Example

This example computes a Cholesky factorization and calculates the determinant of the real symmetric positive definite matrix

(\begin{array}{r} 6 & 7 & 6 & 5 \\ 7 & 11 & 8 & 7 \\ 6 & 8 & 11 & 9 \\ 5 & 7 & 9 & 11 \end{array}) .

NAG FL Interfacef03bff (real_​sym)

▸▿ Contents

1 Purpose