Aberth Introduction to Precise Numerical Methods

2. Auflage 2007
ISBN: 978-0-08-047120-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 272 Seiten, Web PDF

Precise numerical analysis may be defined as the study of computer methods for solving mathematical problems either exactly or to prescribed accuracy. This book explains how precise numerical analysis is constructed. The book also provides exercises which illustrate points from the text and references for the methods presented.? Clearer, simpler descriptions and explanations of
the various numerical methods
? Two new types of numerical problems; accurately
solving partial differential equations with the included software and computing line integrals in the complex plane.

Oliver Aberth received his B.S. from City College of New York, his M.S. from Massachusetts Institute of Technology, and his Ph.D. from the University of Pennsylvania. He is also the author of Computable Analysis (McGraw-Hill, 1980) and Precise Numerical Methods Using C++ (Academic Press, 1998). He was professor of mathematics at Texas A & M University from 1970 to 1999. He is currently professor emeritus at Texas A & M University.

Autoren/Hrsg.

Aberth, Oliver

Inhaltsverzeichnis

1;Front cover;1
2;Title page;4
3;Copyright page;5
4;Table of contents;6
5;Preface;12
6;Acknowledgments;14
7;1 Introduction;16
7.1;1.1 Open-source software;16
7.2;1.2 Calling up a program;17
7.3;1.3 Log files and print files;18
7.4;1.4 More on log files;19
7.5;1.5 The tilde notation for printed answers;20
8;2 Computer Arithmetics;24
8.1;2.1 Floating-point arithmetic;24
8.2;2.2 Variable precision floating-point arithmetic;25
8.3;2.3 Interval arithmetic;26
8.4;2.4 Range arithmetic;28
8.5;2.5 Practical range arithmetic;30
8.6;2.6 Interval arithmetic notation;30
8.7;2.7 Computing standard functions in range arithmetic;32
8.8;2.8 Rational arithmetic;33
8.9;Software Exercises A;35
8.10;Notes and References;38
9;3 Classification of Numerical Computation Problems;40
9.1;3.1 A knotty problem;40
9.2;3.2 The impossibility of untying the knot;42
9.3;3.3 Repercussions from nonsolvable problem 3.1;42
9.4;3.4 Some solvable and nonsolvable decimal place problems;44
9.5;3.5 The solvable problems handled by calc;47
9.6;3.6 Another nonsolvable problem;47
9.7;3.7 The trouble with discontinuous functions;48
9.8;Notes and References;50
10;4 Real-Valued Functions;52
10.1;4.1 Elementary functions;52
10.2;Software Exercises B;54
11;5 Computing Derivatives;56
11.1;5.1 Power series of elementary functions;56
11.2;5.2 An example of series evaluation;63
11.3;5.3 Power series for elementary functions of several variables;64
11.4;5.4 A more general method of generating power series;67
11.5;5.5 The demo program deriv;69
11.6;Software Exercises C;69
11.7;Notes and References;69
12;6 Computing Integrals;72
12.1;6.1 Computing a definite integral;72
12.2;6.2 Formal interval arithmetic;74
12.3;6.3 The demo program integ for computing ordinary definite integrals;76
12.4;6.4 Taylor’s remainder formula generalized;78
12.5;6.5 The demo program mulint for higher dimensional integrals;79
12.6;6.6 The demo program impint for computing improper integrals;81
12.7;Software Exercises D;82
12.8;Notes and References;83
13;7 Finding Where a Function f(x) is Zero;84
13.1;7.1 Obtaining a solvable problem;84
13.2;7.2 Using interval arithmetic for the problem;87
13.3;7.3 Newton’s method;88
13.4;7.4 Order of convergence;90
13.5;Software Exercises E;92
14;8 Finding Roots of Polynomials;94
14.1;8.1 Polynomials;94
14.2;8.2 A bound for the roots of a polynomial;100
14.3;8.3 The Bairstow method for finding roots of a real polynomial;101
14.4;8.4 Bounding the error of a rational polynomial’s root approximations;105
14.5;8.5 Finding accurate roots for a rational or a real polynomial;107
14.6;8.6 The demo program roots;110
14.7;Software Exercises F;110
14.8;Notes and References;111
15;9 Solving n Linear Equations in n Unknowns;112
15.1;9.1 Notation;112
15.2;9.2 Computation problems;113
15.3;9.3 A method for solving linear equations;115
15.4;9.4 Computing determinants;117
15.5;9.5 Finding the inverse of a square matrix;119
15.6;9.6 The demo programs equat, r_equat, and c_equat;120
15.7;Software Exercises G;121
15.8;Notes and References;122
16;10 Eigenvalue and Eigenvector Problems;124
16.1;10.1 Finding a solution to Ax= 0 when det A= 0;125
16.2;10.2 Eigenvalues and eigenvectors;128
16.3;10.3 Companion matrices and Vandermonde matrices;133
16.4;10.4 Finding eigenvalues and eigenvectors by Danilevsky’s method;137
16.5;10.5 Error bounds for Danilevsky’s method;142
16.6;10.6 Rational matrices;149
16.7;10.7 The demo programs eigen, c_eigen, and r_eigen;150
16.8;Software Exercises H;151
17;11 Problems of Linear Programming;152
17.1;11.1 Linear algebra using rational arithmetic;152
17.2;11.2 A more efficient method for solving rational linear equations;155
17.3;11.3 Introduction to linear programming;156
17.4;11.4 Making the simplex process foolproof;160
17.5;11.5 Solving n linear interval equations in n unknowns;163
17.6;11.6 Solving linear interval equations via linear programming;167
17.7;11.7 The program linpro for linear programming problems;170
17.8;11.8 The program i_equat for interval linear equations;171
17.9;Software Exercises I;171
17.10;Notes and References;172
18;12 Finding Where Several Functions are Zero;174
18.1;12.1 The general problem for real elementary functions;174
18.2;12.2 Finding a suitable solvable problem;175
18.3;12.3 Extending the f(x) solution method to the general problem;178
18.4;12.4 The crossing parity;180
18.5;12.5 The crossing number and the topological degree;181
18.6;12.6 Properties of the crossing number;185
18.7;12.7 Computation of the crossing number;186
18.8;12.8 Newton’s method for the general problem;190
18.9;12.9 Searching a more general region for zeros;191
18.10;Software Exercises J;193
18.11;Notes and References;195
19;13 Optimization Problems;196
19.1;13.1 Finding a function’s extreme values;196
19.2;13.2 Finding where a function’s gradient is zero;199
19.3;13.3 The demo program;203
19.4;Software Exercises K;203
19.5;Notes and References;204
20;14 Ordinary Differential Equations;206
20.1;14.1 Introduction;206
20.2;14.2 Two standard problems of ordinary differential equations;208
20.3;14.3 Difficulties with the initial value problem;211
20.4;14.4 Linear differential equations;212
20.5;14.5 Solving the initial value problem by power series;213
20.6;14.6 Degree 1 interval arithmetic;216
20.7;14.7 An improved global error;220
20.8;14.8 Solvable two-point boundary-value problems;223
20.9;14.9 Solving the boundary-value problem by power series;225
20.10;14.10 The linear boundary-value problem;228
20.11;Software Exercises L;229
20.12;Notes and References;231
21;15 Partial Differential Equations;232
21.1;15.1 Partial differential equation terminology;232
21.2;15.2 ODE and PDE initial value problems;234
21.3;15.3 A power series method for the ODE problem;235
21.4;15.4 The first PDE solution method;238
21.5;15.5 A simple PDE problem as an example;242
21.6;15.6 A defect of the first PDE method;243
21.7;15.7 The revised PDE method with comparison computation;244
21.8;15.8 Higher dimensional spaces;245
21.9;15.9 Satisfying boundary conditions;246
21.10;Software Exercises M;247
21.11;Notes and References;248
22;16 Numerical Methods with Complex Functions;250
22.1;16.1 Elementary complex functions;250
22.2;16.2 The demo program c_deriv;252
22.3;16.3 Computing line integrals in the complex plane;252
22.4;16.4 Computing the roots of a complex polynomial;253
22.5;16.5 Finding a zero of an elementary complex function f(z);254
22.6;16.6 The general zero problem for elementary complex functions;257
22.7;Software Exercises N;260
22.8;Notes and References;262
23;The Precise Numerical Methods Program PNM;263
24;Index;264