NAG C Library Function Document

nag_det_real_sym (f03bfc)

1
Purpose

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

2
Specification

#include <nag.h>
#include <nagf03.h>
void  nag_det_real_sym (Nag_OrderType order, Integer n, const double a[], Integer pda, double *d, Integer *id, NagError *fail)

3
Description

nag_det_real_sym (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 nag_dpotrf (f07fdc).

4
References

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

5
Arguments

1:     order Nag_OrderTypeInput
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n>0.
3:     a[dim] const doubleInput
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 IntegerInput
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 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_det_real_sym (f03bfc) is not threaded in any implementation.

9
Further Comments

The time taken by nag_det_real_sym (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)