NAG Library Routine Document

F03BFF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F03BFF computes the determinant of a real

n

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.

2 Specification

SUBROUTINE F03BFF ( N, A, LDA, D, ID, IFAIL)

INTEGER N, LDA, ID, IFAIL

REAL (KIND=nag_wp) A(LDA,*), D

3 Description

F03BFF computes the determinant of a real

n

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 (DPOTRF).

4 References

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

5 Parameters

1: N – INTEGERInput: On entry: $n$ , the order of the matrix $A$ .
Constraint: $N > 0$ .
2: A(LDA, $*$ ) – REAL (KIND=nag_wp) arrayInput: 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 $n$ by $n$ positive definite symmetric matrix $A$ . Only the diagonal elements are referenced.
3: LDA – INTEGERInput: 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 – INTEGEROutput: 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 – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . 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, $N < 1$ .

$IFAIL = 3$: On entry, $LDA < N$ .

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

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 Further Comments

The time taken by F03BFF is approximately proportional to

n

9 Example

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

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

NAG Library Routine DocumentF03BFF