NAG Library Function Document

1Purpose

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.

2Specification

 #include #include
 void nag_det_real_sym (Nag_OrderType order, Integer n, const double a[], Integer pda, double *d, Integer *id, NagError *fail)

3Description

nag_det_real_sym (f03bfc) computes the determinant of a real $n$ by $n$ symmetric positive definite matrix $A$ that has been factorized as $A={U}^{\mathrm{T}}U$, where $U$ is upper triangular, or $A=L{L}^{\mathrm{T}}$, 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).
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

5Arguments

1:    $\mathbf{order}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. 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: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}>0$.
3:    $\mathbf{a}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the Cholesky factorization of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
5:    $\mathbf{d}$double *Output
6:    $\mathbf{id}$Integer *Output
On exit: the determinant of $A$ is given by ${\mathbf{d}}×{2.0}^{{\mathbf{id}}}$. It is given in this form to avoid overflow or underflow.
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
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.
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.

7Accuracy

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).

8Parallelism and Performance

nag_det_real_sym (f03bfc) is not threaded in any implementation.

The time taken by nag_det_real_sym (f03bfc) is approximately proportional to $n$.

10Example

This example computes a Cholesky factorization and calculates the determinant of the real symmetric positive definite matrix
 $6 7 6 5 7 11 8 7 6 8 11 9 5 7 9 11 .$

10.1Program Text

Program Text (f03bfce.c)

10.2Program Data

Program Data (f03bfce.d)

10.3Program Results

Program Results (f03bfce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017