Second Order Exponential Approximation

@(@\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}} }@)@Second Order Exponential Approximation
Syntax
# include "exp_2.hpp"

y = exp_2(x)

Purpose
This is a simple example algorithm that is used to demonstrate Algorithmic Differentiation (see exp_eps for a more complex example).

Mathematical Form
The exponential function can be defined by @[@ \exp (x) = 1 + x^1 / 1 ! + x^2 / 2 ! + \cdots @]@ The second order approximation for the exponential function is @[@ {\rm exp\_2} (x) = 1 + x + x^2 / 2 @]@

include
The include command in the syntax is relative to



     cppad-yyyymmdd/introduction/exp_apx

where cppad-yyyymmdd is the distribution directory created during the beginning steps of the installation of CppAD.

x
The argument x has prototype



     const Type &x

(see Type below). It specifies the point at which to evaluate the approximation for the second order exponential approximation.

y
The result y has prototype



     Type y

It is the value of the exponential function approximation defined above.

Type
If u and v are Type objects and i is an int:

Operation	Result Type	Description
`Type(i)`	`Type`	construct object with value equal to `i`
`Type u = v`	`Type`	construct `u` with value equal to `v`
`u * v`	`Type`	result is value of @(@ u * v @)@
`u / v`	`Type`	result is value of @(@ u / v @)@
`u + v`	`Type`	result is value of @(@ u + v @)@

Contents

exp_2.hpp	exp_2: Implementation
exp_2.cpp	exp_2: Test
exp_2_for0	exp_2: Operation Sequence and Zero Order Forward Mode
exp_2_for1	exp_2: First Order Forward Mode
exp_2_rev1	exp_2: First Order Reverse Mode
exp_2_for2	exp_2: Second Order Forward Mode
exp_2_rev2	exp_2: Second Order Reverse Mode
exp_2_cppad	exp_2: CppAD Forward and Reverse Sweeps

Implementation
The file exp_2.hpp contains a C++ implementation of this function.

Test
The file exp_2.cpp contains a test of this implementation. It returns true for success and false for failure.

Exercises

Suppose that we make the call double x = .1; double y = exp_2(x); What is the value assigned to v1, v2, ... ,v5 in exp_2.hpp ?
Extend the routine exp_2.hpp to a routine exp_3.hpp that computes @[@ 1 + x^2 / 2 ! + x^3 / 3 ! @]@ Do this in a way that only assigns one value to each variable (as exp_2 does).
Suppose that we make the call double x = .5; double y = exp_3(x); using exp_3 created in the previous problem. What is the value assigned to the new variables in exp_3 (variables that are in exp_3 and not in exp_2) ?

Input File: introduction/exp_2.hpp