AD Theory for Solving ODE's Using Taylor's Method

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$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$ This is cppad-20221105 documentation. Here is a link to its current documentation . AD Theory for Solving ODE's Using Taylor's Method
Problem
We are given an initial value problem for

$y : \B{R} \rightarrow \B{R}^n$ ; i.e., we know

$y(0) \in \B{R}^n$ and we know a function

$g : \B{R}^n \rightarrow \B{R}^n$ such that

$y^1 (t) = g[ y(t) ]$ where

$y^k (t)$ is the k-th derivative of

$y(t)$ .

z(t)
We define the function

$z : \B{R} \rightarrow \B{R}^n$ by

$z(t) = g[ y(t) ]$ . Given the Taylor coefficients

$y^{(k)} (t)$ for

$k = 0 , \ldots , p$ , we can compute

$z^{(p)} (t)$ using forward mode AD on the function

$g(y)$ ; see forward_order . It follows from

$y^1 (t) = z(t)$ that

$y^{p+1} (t) = z^p (t)$

$y^{(p+1)} (t) = z^{(p)} (t) / (k + 1)$ where

$y^{(k)} (t)$ is the k-th order Taylor coefficient for

$y(t)$ ; i.e.,

$y^k (t) / k !$ . Starting with the known value

$y^{(0)} (t)$ , this gives a prescription for computing

$y^{(k)} (t)$ for any

$k$ .

Taylor's Method
The p-th order Taylor method for approximates

$y( t + \Delta t ) \approx y^{(0)} (t) + y^{(1)} (t) \Delta t + \cdots + y^{(p)} (t) \Delta t^p$

Example
The file taylor_ode.cpp contains an example and test of this method.

Input File: omh/theory/taylor_ode.omh