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speed-> speed_main speed_utility speed_double speed_adolc speed_cppad speed_fadbad speed_sacado

speed_utility-> uniform_01 det_of_minor det_by_minor det_by_lu det_33 det_grad_33 mat_sum_sq ode_evaluate sparse_evaluate

det_of_minor-> det_of_minor.cpp det_of_minor.hpp

Headings-> Syntax Inclusion Purpose Determinant of A a m n r c d Scalar Example Source Code

Determinant of a Minor
Syntax
# include <cppad/speed/det_of_minor.hpp>
d = det_of_minor(a, m, n, r, c)

Inclusion
The template function det_of_minor is defined in the CppAD namespace by including the file cppad/speed/det_of_minor.hpp (relative to the CppAD distribution directory). It is only intended for example and testing purposes, so it is not automatically included by cppad.hpp .

Purpose
This template function returns the determinant of a minor of the matrix $\mathit{A}$ using expansion by minors. The elements of the $\mathit{n}\times \mathit{n}$ minor $\mathit{M}$ of the matrix $\mathit{A}$ are defined, for $\mathit{i}=0,\dots ,\mathit{n}-1$ and $\mathit{j}=0,\dots ,\mathit{n}-1$ , by $${\mathit{M}}_{\mathit{i},\mathit{j}}={\mathit{A}}_{\mathit{R}(\mathit{i}),\mathit{C}(\mathit{j})}$$ where the functions $\mathit{R}(\mathit{i})$ is defined by the argument r and $\mathit{C}(\mathit{j})$ is defined by the argument c .

This template function is for example and testing purposes only. Expansion by minors is chosen as an example because it uses a lot of floating point operations yet does not require much source code (on the order of m factorial floating point operations and about 70 lines of source code including comments). This is not an efficient method for computing a determinant; for example, using an LU factorization would be better.

Determinant of A
If the following conditions hold, the minor is the entire matrix $\mathit{A}$ and hence det_of_minor will return the determinant of $\mathit{A}$ :

1. $\mathit{n}=\mathit{m}$ .
2. for $\mathit{i}=0,\dots ,\mathit{m}-1$ , $\mathit{r}[\mathit{i}]=\mathit{i}+1$ , and $\mathit{r}[\mathit{m}]=0$ .
3. for $\mathit{j}=0,\dots ,\mathit{m}-1$ , $\mathit{c}[\mathit{j}]=\mathit{j}+1$ , and $\mathit{c}[\mathit{m}]=0$ .

a
The argument a has prototype
     const std::vector<Scalar>& a
and is a vector with size $\mathit{m}*\mathit{m}$ (see description of Scalar below). The elements of the $\mathit{m}\times \mathit{m}$ matrix $\mathit{A}$ are defined, for $\mathit{i}=0,\dots ,\mathit{m}-1$ and $\mathit{j}=0,\dots ,\mathit{m}-1$ , by $${\mathit{A}}_{\mathit{i},\mathit{j}}=\mathit{a}[\mathit{i}*\mathit{m}+\mathit{j}]$$
m
The argument m has prototype
     size_t m
and is the size of the square matrix $\mathit{A}$ .

n
The argument n has prototype
     size_t n
and is the size of the square minor $\mathit{M}$ .

r
The argument r has prototype
     std::vector<size_t>& r
and is a vector with $\mathit{m}+1$ elements. This vector defines the function $\mathit{R}(\mathit{i})$ which specifies the rows of the minor $\mathit{M}$ . To be specific, the function $\mathit{R}(\mathit{i})$ for $\mathit{i}=0,\dots ,\mathit{n}-1$ is defined by $$\begin{array}{ccc}\hfill \mathit{R}(0)& \hfill =\hfill & \mathit{r}[\mathit{m}]\hfill \\ \hfill \mathit{R}(\mathit{i}+1)& \hfill =\hfill & \mathit{r}[\mathit{R}(\mathit{i})]\hfill \end{array}$$ All the elements of r must have value less than or equal m . The elements of vector r are modified during the computation, and restored to their original value before the return from det_of_minor.

c
The argument c has prototype
     std::vector<size_t>& c
and is a vector with $\mathit{m}+1$ elements This vector defines the function $\mathit{C}(\mathit{i})$ which specifies the rows of the minor $\mathit{M}$ . To be specific, the function $\mathit{C}(\mathit{i})$ for $\mathit{j}=0,\dots ,\mathit{n}-1$ is defined by $$\begin{array}{ccc}\hfill \mathit{C}(0)& \hfill =\hfill & \mathit{c}[\mathit{m}]\hfill \\ \hfill \mathit{C}(\mathit{j}+1)& \hfill =\hfill & \mathit{c}[\mathit{C}(\mathit{j})]\hfill \end{array}$$ All the elements of c must have value less than or equal m . The elements of vector c are modified during the computation, and restored to their original value before the return from det_of_minor.

d
The result d has prototype
     Scalar d
and is equal to the determinant of the minor $\mathit{M}$ .

Scalar
If x and y are objects of type Scalar and i is an object of type int, the Scalar must support the following operations:

Syntax	Description	Result Type
Scalar x	default constructor for Scalar object.
x = i	set value of x to current value of i
x = y	set value of x to current value of y
x + y	value of x plus y	Scalar
x - y	value of x minus y	Scalar
x * y	value of x times value of y	Scalar

Example
The file det_of_minor.cpp contains an example and test of det_of_minor.hpp. It returns true if it succeeds and false otherwise.

Source Code
The file det_of_minor.hpp contains the source for this template function.

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Input File: cppad/speed/det_of_minor.hpp