Algorithms for Minimization Without

Outstanding text for graduate students and research workers proposes improvements to existing algorithms, extends their related mathematical theories, and offers details on new algorithms for approximating local and global minima. Many numerical examples, along with complete analysis of rate of convergence for most of the algorithms and error bounds that allow for the effect of rounding errors.

Preface to Dover Edition

viii

Preface

ix

Introduction and Summary

1

(8)

Introduction

1

(3)

Summary

4

(5)

Some Useful Results on Taylor Series, Divided Differences, and Lagrange Interpolation

9

(10)

Introduction

9

(1)

Notation and definitions

10

(1)

Truncated Taylor series

11

(1)

Lagrange interpolation

12

(1)

Divided differences

13

(2)

Differentiating the error

15

(4)

The Use of Successive Interpolation for Finding Simple Zeros of a Function and its Derivatives

19

(28)

Introduction

19

(2)

The definition of order

21

(1)

Convergence to a zero

22

(2)

Superlinear convergence

24

(2)

Strict superlinear convergence

26

(3)

The exact order of convergence

29

(5)

Stronger results for q = 1 and 2

34

(6)

Accelerating convergence

40

(3)

Some numerical examples

43

(2)

Summary

45

(2)

An Algorithm with Guaranteed Convergence for Finding a Zero of a Function

47

(14)

Introduction

47

(1)

The Algorithm

48

(5)

Convergence properties

53

(1)

Practical tests

54

(2)

Conclusion

56

(2)

ALGOL 60 procedures

58

(3)

An Algorithm with Guaranteed Convergence for Finding a Minimum of a Function of one Variable

61

(20)

Introduction

61

(2)

Fundamental limitations because of rounding errors

63

(2)

Unimodality and δ-unimodality

65

(7)

An algorithm analogous to Dekker's algorithm

72

(3)

Convergence properties

75

(1)

Practical tests

76

(2)

Conclusion

78

(1)

An ALGOL 60 procedure

79

(2)

Global Minimization Given an Upper Bound on the Second Derivative

81

(35)

Introduction

81

(3)

The basic theorems

84

(2)

An algorithm for global minimization

86

(11)

The rate of convergence in some special cases

97

(3)

A lower bound on the number of function evaluations required

100

(3)

Practical tests

103

(2)

Some extensions and generalizations

105

(2)

An algorithm for global minimization of a function of several variables

107

(4)

Summary and conclusions

111

(1)

ALGOL 60 procedures

112

(4)

A New Algorithm for Minimizing a Function of Several Variables Without Calculating Derivatives

116

(53)

Introduction and survey of the literature

116

(6)

The effect of rounding errors

122

(2)

Powell's algorithm

124

(4)

The main modification

128

(4)

The resolution ridge problem

132

(3)

Some further details

135

(2)

Numerical results and comparison with other methods

137

(17)

Conclusion

154

(1)

An ALGOL W procedure and test program

155

(14)

Bibliography

169

(18)

Appendix: Fortran subroutines

187

(6)

Index

193

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Algorithms for Minimization Without Derivatives

Summary

Table of Contents

Supplemental Materials