NAG CL Interface
f03bfc (real_​sym)

1 Purpose

f03bfc computes the determinant of a real n by n symmetric positive definite matrix A. f07fdc must be called first to supply the symmetric matrix A in Cholesky factorized form. The storage (upper or lower triangular) used by f07fdc is not relevant to f03bfc since only the diagonal elements of the factorized A are referenced.

2 Specification

#include <nag.h>
void  f03bfc (Nag_OrderType order, Integer n, const double a[], Integer pda, double *d, Integer *id, NagError *fail)
The function may be called by the names: f03bfc or nag_det_real_sym.

3 Description

f03bfc computes the determinant of a real n by 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 f07fdc.

4 References

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

5 Arguments

1: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n>0.
3: a[dim] const double Input
Note: the dimension, dim, of the array a must be at least pda×n.
the i,jth element of the Cholesky factorization of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
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.
4: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdan.
5: d double * Output
6: 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.
7: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdan.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MAT_NOT_POS_DEF
The matrix A is not positive definite.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL 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

f03bfc is not threaded in any implementation.

9 Further Comments

The time taken by f03bfc 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 (f03bfce.c)

10.2 Program Data

Program Data (f03bfce.d)

10.3 Program Results

Program Results (f03bfce.r)