NAG Library Function Document

nag_det_real_sym (f03bfc)

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

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 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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

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 \times n

The

(i, j)

th element of the Cholesky factorization of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

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.

4: pda – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq n

5: d – double *Output

6: 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.

7: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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: $pda \geq n$ .
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.
NE_MAT_NOT_POS_DEF: 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 Parallelism and Performance

Not applicable.

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

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

NAG Library Function Documentnag_det_real_sym (f03bfc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_det_real_sym (f03bfc)