Introduction to Precise Numerical Methods Book

Preface     xi
Acknowledgments     xiii
Introduction     1
Open-source software     1
Calling up a program     2
Log files and print files     3
More on log files     4
The tilde notation for printed answers     5
Computer Arithmetics     9
Floating-point arithmetic     9
Variable precision floating-point arithmetic     10
Interval arithmetic     11
Range arithmetic     13
Practical range arithmetic     15
Interval arithmetic notation     15
Computing standard functions in range arithmetic     17
Rational arithmetic     18
Software Exercises A     20
Notes and References     23
Classification of Numerical Computation Problems     25
A knotty problem     25
The impossibility of untying the knot     27
Repercussions from nonsolvable problem 3.1     27
Some solvable and nonsolvable decimal place problems     29
The solvable problems handled by calc     32
Another nonsolvable problem     32
The trouble with discontinuous functions     33
Notes andReferences     35
Real-Valued Functions     37
Elementary functions     37
Software Exercises B     39
Computing Derivatives     41
Power series of elementary functions     41
An example of series evaluation     48
Power series for elementary functions of several variables     49
A more general method of generating power series     52
The demo program deriv     54
Software Exercises C     54
Notes and References     54
Computing Integrals     57
Computing a definite integral     57
Formal interval arithmetic     59
The demo program integ for computing ordinary definite integrals     61
Taylor's remainder formula generalized     63
The demo program mulint for higher dimensional integrals     64
The demo program impint for computing improper integrals     66
Software Exercises D     67
Notes and References     68
Finding Where a Function f(x) is Zero     69
Obtaining a solvable problem     69
Using interval arithmetic for the problem     72
Newton's method     73
Order of convergence      75
Software Exercises E     77
Finding Roots of Polynomials     79
Polynomials     79
A bound for the roots of a polynomial     85
The Bairstow method for finding roots of a real polynomial     86
Bounding the error of a rational polynomial's root approximations     90
Finding accurate roots for a rational or a real polynomial     92
The demo program roots     95
Software Exercises F     95
Notes and References     96
Solving n Linear Equations in n Unknowns     97
Notation     97
Computation problems     98
A method for solving linear equations     100
Computing determinants     102
Finding the inverse of a square matrix     104
The demo programs equat, r_equat, and c_equat     105
Software Exercises G     106
Notes and References     107
Eigenvalue and Eigenvector Problems     109
Finding a solution to Ax = 0 when det A = 0     110
Eigenvalues and eigenvectors     113
Companion matrices and Vandermonde matrices     118
Finding eigenvalues and eigenvectors by Danilevsky's method     122
Error bounds for Danilevsky's method      127
Rational matrices     134
The demo programs eigen, c_eigen, and r_eigen     135
Software Exercises H     136
Problems of Linear Programming     137
Linear algebra using rational arithmetic     137
A more efficient method for solving rational linear equations     140
Introduction to linear programming     141
Making the simplex process foolproof     145
Solving n linear interval equations in n unknowns     148
Solving linear interval equations via linear programming     152
The program linpro for linear programming problems     155
The program i_equat for interval linear equations     156
Software Exercises I     156
Notes and References     157
Finding Where Several Functions are Zero     159
The general problem for real elementary functions     159
Finding a suitable solvable problem     160
Extending the f(x) solution method to the general problem     163
The crossing parity     165
The crossing number and the topological degree     166
Properties of the crossing number     170
Computation of the crossing number     171
Newton's method for the general problem      175
Searching a more general region for zeros     176
Software Exercises J     178
Notes and References     180
Optimization Problems     181
Finding a function's extreme values     181
Finding where a function's gradient is zero     184
The demo program extrema     188
Software Exercises K     188
Notes and References     189
Ordinary Differential Equations     191
Introduction     191
Two standard problems of ordinary differential equations     193
Difficulties with the initial value problem     196
Linear differential equations     197
Solving the initial value problem by power series     198
Degree 1 interval arithmetic     201
An improved global error     205
Solvable two-point boundary-value problems     208
Solving the boundary-value problem by power series     210
The linear boundary-value problem     213
Software Exercises L     214
Notes and References     216
Partial Differential Equations     217
Partial differential equation terminology     217
ODE and PDE initial value problems      219
A power series method for the ODE problem     220
The first PDE solution method     223
A simple PDE problem as an example     227
A defect of the first PDE method     228
The revised PDE method with comparison computation     229
Higher dimensional spaces     230
Satisfying boundary conditions     231
Software Exercises M     232
Notes and References     233
Numerical Methods with Complex Functions     235
Elementary complex functions     235
The demo program c_deriv     237
Computing line integrals in the complex plane     237
Computing the roots of a complex polynomial     238
Finding a zero of an elementary complex function f(z)     239
The general zero problem for elementary complex functions     242
Software Exercises N     245
Notes and References     247
The Precise Numerical Methods Program PNM     248
Index     249