For many years it has been an article of faith of numerical analysts that the evaluation of derivatives of complicated functions should be avoided. Derivatives were evaluated using finite differences or, more recently, using symbolic manipulation packages. The first has the disadvantage of limited accuracy. The second has disadvantages of being expensive and requiring considerable computer memory. The recent developments described in this text allow the evaluation of derivatives using simple automatic derivative evaluation subroutines proยญ grammed in FORTRAN or BASIC. These subroutines can even be programmed on a personal computer. The concept for the evaluation of the derivatives was originally developed by Wengert over 20 years ago. Significant imยญ provements have been made in Wengert's method and are utilized in this text. The purpose of this text is to familiarize computer users with a simple and practical method for obtaining the partial derivatives of complicated mathematical expressions. The text illustrates the use of automatic derivaยญ tive evaluation subroutines to solve a wide range of nonlinear least-squares, optimal control, system identification, two-point boundary value problems, and integral equations. The numerical values of the derivatives are evalũ ated exactly, except for roundoff, using simple FORTRAN or BASIC subยญ routines. These derivatives are derived automatically behind the scenes, from the equivalent of analytical expressions, without any effort from the user. The use of costly software packages is not required
CONTENT
1. Methods for Numerical Differentiation -- 1.1. Wengertโ{128}{153}s Method -- 1.2. FEED (Fast and Efficient Evaluation of Derivatives) -- 1.3. An Implementation of the FEED Procedure in BASIC -- 1.4. Wexlerโ{128}{153}s Approach -- 1.5. Higher-Order Methods for Finding Roots -- Exercises -- 2. Nonlinear Least Squares -- 2.1. Fitting the CES Production Function -- 2.2. Passive Ranging -- 2.3. Constrained Optimization -- 3. Optimal Control -- 3.1. Control Theory -- 3.2. Numerical Methods -- 3.3. Description of the Subroutines -- 3.4. Examples of Optimal Control Problems -- 3.5. Program Listings -- Exercises -- 4. System Identification -- 4.1. Quasilinearization -- 4.2. Fast and Efficient Evaluation of Derivatives (FEED) -- 4.3. Computer Program -- 4.4. Numerical Results -- 4.5. Program Listing -- 5. Sukhanovโ{128}{153}s Variable Initial Value Method for Boundary Value Problems -- 5.1. Sukhanovโ{128}{153}s Initial Value Equations -- 5.2. Automatic Derivative Evaluation -- 5.3. Examples Using Sukhanovโ{128}{153}s Method -- 5.4. Program Listing -- Exercises -- 6. Nonlinear Integral Equations -- 6.1. Derivation of the Imbedding Equations -- 6.2. Method of Computation -- 6.3. Automatic Derivative Evaluation -- 6.4. Examples of Integral Equation Problems -- 6.5. Program Listing -- Exercises -- References -- Author Index
