\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

ode_evaluate

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Evaluate a Function Defined in Terms of an ODE

Syntax

# include <cppad/speed/ode_evaluate.hpp>

ode_evaluate ( x , p , fp )

Purpose

This routine evaluates a function \(f : \B{R}^n \rightarrow \B{R}^n\) defined by

\[f(x) = y(x, 1)\]

where \(y(x, t)\) solves the ordinary differential equation

\begin{eqnarray} y(x, 0) & = & x \\ \partial_t y (x, t ) & = & g[ y(x,t) , t ] \end{eqnarray}

where \(g : \B{R}^n \times \B{R} \rightarrow \B{R}^n\) is an unspecified function.

Inclusion

The template function ode_evaluate is defined in the CppAD namespace by including the file cppad/speed/ode_evaluate.hpp (relative to the CppAD distribution directory).

Float

Operation Sequence

The type Float must be a NumericType . The Float operation sequence for this routine does not depend on the value of the argument x , hence it does not need to be retaped for each value of \(x\).

fabs

If y and z are Float objects, the syntax

y = fabs ( z )

must be supported. Note that it does not matter if the operation sequence for fabs depends on z because the corresponding results are not actually used by ode_evaluate ; see fabs in Runge45 .

x

The argument x has prototype

const CppAD::vector< Float >& x

It contains he argument value for which the function, or its derivative, is being evaluated. The value \(n\) is determined by the size of the vector x .

p

The argument p has prototype

size_t p

p == 0

In this case a numerical method is used to solve the ode and obtain an accurate approximation for \(y(x, 1)\). This numerical method has a fixed that does not depend on x .

p = 1

In this case an analytic solution for the partial derivative \(\partial_x y(x, 1)\) is returned.

fp

The argument fp has prototype

CppAD::vector< Float >& fp

The input value of the elements of fp does not matter.

Function

If p is zero, fp has size equal to \(n\) and contains the value of \(y(x, 1)\).

Gradient

If p is one, fp has size equal to n^2 and for \(i = 0 , \ldots 1\), \(j = 0 , \ldots , n-1\)

\[\D{y[i]}{x[j]} (x, 1) = fp [ i \cdot n + j ]\]

Example

The file ode_evaluate.cpp contains an example and test of ode_evaluate.hpp .

Source Code

The file ode_evaluate.hpp contains the source code for this template function.