in Cholesky factorized form. The storage (upper or lower triangular) used by nag_lapack_dpotrf (f07fd) is not relevant to nag_det_real_sym (f03bf) since only the diagonal elements of the factorized

A

are referenced.

Syntax

[d, id, ifail] = f03bf(a, 'n', n)

[d, id, ifail] = nag_det_real_sym(a, 'n', n)

Description

nag_det_real_sym (f03bf) 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_lapack_dpotrf (f07fd).

References

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

Parameters

Compulsory Input Parameters

1: $a (lda :)$ – double array: The first dimension of the array a must be at least $n$ .
The second dimension of the array a must be at least $n$ .

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.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a.
$n$ , the order of the matrix $A$ .

Constraint: $n > 0$ .

Output Parameters

1: $d$ – double scalar
2: $id$ – int64int32nag_int scalar: The determinant of $A$ is given by $d \times {2.0}^{id}$ . It is given in this form to avoid overflow or underflow.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $n > 0$ .

$ifail = 3$: Constraint: $lda \geq n$ .

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

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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).

Further Comments

The time taken by nag_det_real_sym (f03bf) is approximately proportional to

n

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}) .

Open in the MATLAB editor: f03bf_example

function f03bf_example


fprintf('f03bf example results\n\n');

a = [6,   7,   6,   5;
     7,  11,   8,   7;
     6,   8,  11,   9;
     5,   7,   9,  11];
% Factorize a
[a, info] = f07fd('l', a);

fprintf('\n');
[ifail] = x04ca('l', 'n', a, 'Array a after factorization');

[d, id, ifail] = f03bf(a);

fprintf('\nd = %13.5f id = %d\n', d, id);
fprintf('Value of determinant = %13.5e\n', d*2^id);

f03bf example results


 Array a after factorization
             1          2          3          4
 1      2.4495
 2      2.8577     1.6833
 3      2.4495     0.5941     2.1557
 4      2.0412     0.6931     1.6645     1.8927

d =       0.06909 id = 12
Value of determinant =   2.83000e+02

PDF version (NAG web site, 64-bit version, 64-bit version)

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