Elsevier S & T

Introduction to Precise Numerical Methods

Aberth, Oliver

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Table of contents
  • Cover
  • Copyright Pageiv
  • Contentsv
  • Prefacexi
  • Acknowledgmentsxiii
  • Chapter 1 Introduction1
  • 1.1 Open-source software1
  • 1.2 Calling up a program2
  • 1.3 Log files and print files3
  • 1.4 More on log files4
  • 1.5 The tilde notation for printed answers5
  • Chapter 2 Computer Arithmetics9
  • 2.1 Floating-point arithmetic9
  • 2.2 Variable precision floating-point arithmetic10
  • 2.3 Interval arithmetic11
  • 2.4 Range arithmetic13
  • 2.5 Practical range arithmetic15
  • 2.6 Interval arithmetic notation15
  • 2.7 Computing standard functions in range arithmetic17
  • 2.8 Rational arithmetic18
  • Software Exercises A20
  • Notes and References23
  • Chapter 3 Classification of Numerical Computation Problems25
  • 3.1 A knotty problem25
  • 3.2 The impossibility of untying the knot27
  • 3.3 Repercussions from nonsolvable problem 3.127
  • 3.4 Some solvable and nonsolvable decimal place problems29
  • 3.5 The solvable problems handled by calc32
  • 3.6 Another nonsolvable problem32
  • 3.7 The trouble with discontinuous functions33
  • Notes and References35
  • Chapter 4 Real-Valued Functions37
  • 4.1 Elementary functions37
  • Software Exercises B39
  • Chapter 5 Computing Derivatives41
  • 5.1 Power series of elementary functions41
  • 5.2 An example of series evaluation48
  • 5.3 Power series for elementary functions of several variables49
  • 5.4 A more general method of generating power series52
  • 5.5 The demo program deriv54
  • Software Exercises C54
  • Notes and References54
  • Chapter 6 Computing Integrals57
  • 6.1 Computing a definite integral57
  • 6.2 Formal interval arithmetic59
  • 6.3 The demo program integ for computing ordinary definite integrals61
  • 6.4 Taylor’s remainder formula generalized63
  • 6.5 The demo program mulint for higher dimensional integrals64
  • 6.6 The demo program impint for computing improper integrals66
  • Software Exercises D67
  • Notes and References68
  • Chapter 7 Finding Where a Function f(x) is Zero69
  • 7.1 Obtaining a solvable problem69
  • 7.2 Using interval arithmetic for the problem72
  • 7.3 Newton’s method73
  • 7.4 Order of convergence75
  • Software Exercises E77
  • Chapter 8 Finding Roots of Polynomials79
  • 8.1 Polynomials79
  • 8.2 A bound for the roots of a polynomial85
  • 8.3 The Bairstow method for finding roots of a real polynomial86
  • 8.4 Bounding the error of a rational polynomial’s root approximations90
  • 8.5 Finding accurate roots for a rational or a real polynomial92
  • 8.6 The demo program roots95
  • Software Exercises F95
  • Notes and References96
  • Chapter 9 Solving n Linear Equations in n Unknowns97
  • 9.1 Notation97
  • 9.2 Computation problems98
  • 9.3 A method for solving linear equations100
  • 9.4 Computing determinants102
  • 9.5 Finding the inverse of a square matrix104
  • 9.6 The demo programs equat, r_equat, and c_equat105
  • Software Exercises G106
  • Notes and References107
  • Chapter 10 Eigenvalue and Eigenvector Problems109
  • 10.1 Finding a solution to Ax=0 when det A=0110
  • 10.2 Eigenvalues and eigenvectors113
  • 10.3 Companion matrices and Vandermonde matrices118
  • 10.4 Finding eigenvalues and eigenvectors by Danilevsky’s method122
  • 10.5 Error bounds for Danilevsky’s method127
  • 10.6 Rational matrices134
  • 10.7 The demo programs eigen, c_eigen, and r_eigen135
  • Software Exercises H136
  • Chapter 11 Problems of Linear Programming137
  • 11.1 Linear algebra using rational arithmetic137
  • 11.2 A more efficient method for solving rational linear equations140
  • 11.3 Introduction to linear programming141
  • 11.4 Making the simplex process foolproof145
  • 11.5 Solving n linear interval equations in n unknowns148
  • 11.6 Solving linear interval equations via linear programming152
  • 11.7 The program linpro for linear programming problems155
  • 11.8 The program i_equat for interval linear equations156
  • Software Exercises I156
  • Notes and References157
  • Chapter 12 Finding Where Several Functions are Zero159
  • 12.1 The general problem for real elementary functions159
  • 12.2 Finding a suitable solvable problem160
  • 12.3 Extending the f(x) solution method to the general problem163
  • 12.4 The crossing parity165
  • 12.5 The crossing number and the topological degree166
  • 12.6 Properties of the crossing number170
  • 12.7 Computation of the crossing number171
  • 12.8 Newton’s method for the general problem175
  • 12.9 Searching a more general region for zeros176
  • Software Exercises J178
  • Notes and References180
  • Chapter 13 Optimization Problems181
  • 13.1 Finding a function’s extreme values181
  • 13.2 Finding where a function’s gradient is zero184
  • 13.3 The demo program extrema188
  • Software Exercises K188
  • Notes and References189
  • Chapter 14 Ordinary Differential Equations191
  • 14.1 Introduction191
  • 14.2 Two standard problems of ordinary differential equations193
  • 14.3 Difficulties with the initial value problem196
  • 14.4 Linear differential equations197
  • 14.5 Solving the initial value problem by power series198
  • 14.6 Degree 1 interval arithmetic201
  • 14.7 An improved global error205
  • 14.8 Solvable two-point boundary-value problems208
  • 14.9 Solving the boundary-value problem by power series210
  • 14.10 The linear boundary-value problem213
  • Software Exercises L214
  • Notes and References216
  • Chapter 15 Partial Differential Equations217
  • 15.1 Partial differential equation terminology217
  • 15.2 ODE and PDE initial value problems219
  • 15.3 A power series method for the ODE problem220
  • 15.4 The first PDE solution method223
  • 15.5 A simple PDE problem as an example227
  • 15.6 A defect of the first PDE method228
  • 15.7 The revised PDE method with comparison computation229
  • 15.8 Higher dimensional spaces230
  • 15.9 Satisfying boundary conditions231
  • Software Exercises M232
  • Notes and References233
  • Chapter 16 Numerical Methods with Complex Functions235
  • 16.1 Elementary complex functions235
  • 16.2 The demo program c_deriv237
  • 16.3 Computing line integrals in the complex plane237
  • 16.4 Computing the roots of a complex polynomial238
  • 16.5 Finding a zero of an elementary complex function f(z)239
  • 16.6 The general zero problem for elementary complex functions242
  • Software Exercises N245
  • Notes and References247
  • The Precise Numerical Methods Program PNM248
  • Index249
Book details
  • Vendor Elsevier S & T
  • SKU 9780123738592
  • ISBN-13 9780080471204
  • Author Aberth, Oliver
  • Edition 2nd
  • Category Mathematics
  • Subject Number Theory

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Description eBook delivery & access Reviews

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.

All disc-based content for this title is now available on the Web.



· 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.