NAG FL Interface
f03bff (real_​sym)

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1 Purpose

f03bff computes the determinant of a real n×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.

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×n symmetric positive definite matrix A that has been factorized as A=UTU, where U is upper triangular, or A=LLT, 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.

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×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: ldan.
4: d Real (Kind=nag_wp) Output
5: id Integer Output
On exit: the determinant of A is given by d×2.0id. 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 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, n=value.
Constraint: n>0.
On entry, lda=value and n=value.
Constraint: ldan.
The matrix A is not positive definite.
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.
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.
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

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
( 6 7 6 5 7 11 8 7 6 8 11 9 5 7 9 11 ) .  

10.1 Program Text

Program Text (f03bffe.f90)

10.2 Program Data

Program Data (f03bffe.d)

10.3 Program Results

Program Results (f03bffe.r)