nag_det_real_sym (f03bfc) (PDF version)
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NAG Library Manual

# NAG Library Function Documentnag_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 #include
 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 $A={U}^{\mathrm{T}}U$, 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

## 5  Arguments

1:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}>0$.
3:     a[$\mathit{dim}$]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:     pdaIntegerInput
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:     ddouble *Output
6:     idInteger *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:     failNagError *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 $⟨\mathit{\text{value}}⟩$ 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.
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).

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
 $6 7 6 5 7 11 8 7 6 8 11 9 5 7 9 11 .$

### 10.1  Program Text

Program Text (f03bfce.c)

### 10.2  Program Data

Program Data (f03bfce.d)

### 10.3  Program Results

Program Results (f03bfce.r)

nag_det_real_sym (f03bfc) (PDF version)
f03 Chapter Contents
f03 Chapter Introduction
NAG Library Manual