Numerical Derivatives and Nonlinear AnalysisFor 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 pro 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 im 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 deriva 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 evalu~ ated exactly, except for roundoff, using simple FORTRAN or BASIC sub 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. |
Contents
Methods for Numerical Differentiation | 1 |
Nonlinear Least Squares | 26 |
Appendix A Nonlinear Least Squares Using FEED | 32 |
Copyright | |
7 other sections not shown
Common terms and phrases
a₁ ALFA automatic derivative evaluation automatic solution boundary conditions boundary value problems calculation calculus of variations CALL ADD CALL MULT COMMON L,XID computed CONTINUE CONTINUE cost functional Defines the vector derivatives with respect differential constraints DIMENSION equal to zero Euler-Lagrange equations Euler's method example First-Order System Forms the sum FORTRAN FORTRAN program fourth-order given in Table GOSUB Gradient Method grid intervals Hamiltonian initial conditions INPUT subroutine integrand IROW iteration Kalaba linear main program matrix MMULT Multiplies Newton-Raphson method Nonlinear System nonlinear two-point boundary Number of Components numerical results numerical solution NVAR optimal control problem ordinary differential equations parameters partial derivatives Pontryagin's maximum principle program listing recurrence relation RETURN END SUBROUTINE Runge-Kutta method second derivatives Second-Order System solved subroutine INPUT subroutine LIN table method tions two-point boundary value variables and vectors vector components vectors corresponding x₁