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

Introduction

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An Introduction by Example to Algorithmic Differentiation

Purpose

This is an introduction by example to Algorithmic Differentiation. Its purpose is to aid in understand what AD calculates, how the calculations are preformed, and the amount of computation and memory required for a forward or reverse sweep.

Preface

Algorithmic Differentiation

Algorithmic Differentiation (often referred to as Automatic Differentiation or just AD) uses the software representation of a function to obtain an efficient method for calculating its derivatives. These derivatives can be of arbitrary order and are analytic in nature (do not have any truncation error).

Forward Mode

A forward mode sweep computes the partial derivative of all the dependent variables with respect to one independent variable (or independent variable direction).

Reverse Mode

A reverse mode sweep computes the derivative of one dependent variable (or one dependent variable direction) with respect to all the independent variables.

Operation Count

The number of floating point operations for either a forward or reverse mode sweep is a small multiple of the number required to evaluate the original function. Thus, using reverse mode, you can evaluate the derivative of a scalar valued function with respect to thousands of variables in a small multiple of the work to evaluate the original function.

Efficiency

AD automatically takes advantage of the speed of your algorithmic representation of a function. For example, if you calculate a determinant using LU factorization, AD will use the LU representation for the derivative of the determinant (which is faster than using the definition of the determinant).

Outline

A. get_started

See get_started.cpp for an example that uses CppAD to calculate derivatives of a polynomial.

B. Example Algorithms

Present two algorithms that approximate the exponential function. The first algorithm exp_2.hpp is simpler and does not include any logical variables or loops. The second algorithm exp_eps.hpp includes logical operations and a while loop.

C. Example Steps

For each of the algorithms, exp_2 and exp_eps, do the following:

1. Define the mathematical function corresponding to the algorithm (exp_2 and exp_eps ).

2. Write out the floating point operation sequence, and corresponding values, that correspond to executing the algorithm for a specific input (exp_2_for0 and exp_eps_for0 ).

3. Compute a forward sweep derivative of the operation sequence (exp_2_for1 and exp_eps_for1 ).

4. Compute a reverse sweep derivative of the operation sequence (exp_2_rev1 and exp_eps_rev1 ).

5. Use CppAD to compute both a forward and reverse sweep of the operation sequence (exp_2_cppad and exp_eps_cppad ).

D. Testing Examples

The program exp_apx.cpp runs all of the test routines that validate the calculations in the exp_2 and exp_eps presentation.

Reference

An in-depth review of AD theory and methods can be found in the book Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation , Andreas Griewank, SIAM Frontiers in Applied Mathematics, 2000.

Contents

Name	Title
exp_2	Second Order Exponential Approximation
exp_eps	An Epsilon Accurate Exponential Approximation
exp_apx.cpp	Correctness Tests For Exponential Approximation in Introduction