An Introduction to Numerical Methods in C++

by Flowers, B. H.

• ISBN13: 9780198506935
• ISBN10: 0198506937
• Additional ISBN(s): 9780198538639
• Edition: Revised
• Format: Paperback
• Copyright: 2000-06-22
• Publisher: Oxford University Press
• More Book Details

Summary

Designed for the many applied mathematicians and engineers who wish to explore computerized numerical methods, this text explores the power of C++ as a tool for work in numerical methods. This revision of the successful first edition includes for the first time information on programming in Windows-based environments. In addition it includes new topics and methods throughout the text that clarify and enhance the treatment of the subject.

Table of Contents

Preliminaries	1	(20)
Elementary input and output	1	(4)
Strings	1	(2)
Numbers	3	(1)
Standard error output	4	(1)
Comments	5	(1)
Basic data types	6	(2)
Integer types	6	(1)
Floating point types	7	(1)
Void type	8	(1)
Derived types	8	(3)
Constants	9	(1)
Enumerations	9	(1)
Structures	10	(1)
Pointers	11	(5)
Pointer to void	13	(1)
Constants and pointers	13	(1)
Pointers to structures	14	(1)
Arrays	14	(2)
References	16	(1)
Preprocessor directives	17	(2)
Token replacement	17	(1)
Conditional directives	18	(1)
File inclusion	18	(1)
Keywords and identifiers	19	(1)
Expressions, statements and functions	20	(35)
Expressions	21	(4)
Arithmetic expressions	21	(1)
Relational expressions	22	(1)
Logical expressions	22	(1)
Bitwise expressions	22	(1)
Comma expressions	23	(1)
Other expressions	23	(1)
Operator precedence	23	(2)
Statements	25	(3)
Declaration and initialization	25	(1)
Assignments	26	(1)
Compound assignments	26	(1)
Increments and decrements	26	(2)
Compound statements	28	(2)
Scope	29	(1)
Lifetime	30	(1)
Conditional statements	30	(7)
if-else statements	30	(3)
while statements	33	(1)
do statements	34	(1)
for statements	34	(1)
break and continue statements	34	(1)
goto statements	35	(1)
switch statements	36	(1)
Functions	37	(9)
Inline functions	41	(1)
Passing arguments to functions	42	(1)
Functions returning void	42	(1)
Reference arguments	43	(1)
Pointer arguments	44	(1)
Overloading function names	44	(1)
Function templates	45	(1)
Recursive functions	46	(3)
Static local variables	49	(1)
Pointers and functions	50	(1)
Function types	51	(1)
The function main	52	(3)
Errors, theorems and speed	55	(14)
Truncation	55	(1)
Rounding errors	55	(2)
Quadratic equations	57	(1)
Floating point equality	58	(1)
Conditioning and stability	59	(1)
Local and global errors	60	(1)
Basic theorems	60	(2)
Rates of convergence	62	(1)
Reciprocals without division	63	(1)
Speed of computation	64	(1)
Speed and recursion	65	(3)
Mistakes	68	(1)
Roots of non-linear equations	69	(25)
Bisection method	69	(5)
Simple recursive procedure	69	(1)
Refined recursive procedure	70	(3)
Non-recursive procedure	73	(1)
Regula falsi method	74	(2)
Secant method	76	(1)
Convergence	77	(2)
Fixed point method	79	(2)
Choice of fixed point function	81	(1)
Newton's method	82	(1)
Multiple roots	83	(1)
Aitken's extrapolation	84	(1)
Roots of polynomials	85	(3)
Horner's algorithm	86	(1)
Stability	87	(1)
Maxima and minima of functions	88	(3)
Some standard things	91	(3)
Classes	94	(26)
Complex numbers	94	(10)
Declaration	95	(1)
Implementation	96	(3)
Friend functions	99	(1)
Operator functions	100	(3)
Complex roots of quadratic equations	103	(1)
Class string	104	(9)
Declaration	104	(1)
Implementation	105	(2)
Assignment	107	(1)
Indexing	108	(2)
Concatenation	110	(2)
Input and output	112	(1)
Example	112	(1)
Static members	113	(2)
Class arrays	115	(4)
Array of complex numbers	116	(1)
Array of strings	117	(1)
Array template	118	(1)
Classes and header files	119	(1)
Derived classes and streams	120	(25)
The base class duple	120	(5)
The derived class point	121	(3)
The derived class complex	124	(1)
Derivation and access	125	(2)
Class string revisited	127	(4)
Streams	131	(8)
The stream buffer	131	(1)
Output streams	132	(1)
Input streams	133	(1)
Formatted input and output	133	(4)
Tabulation	137	(2)
File input and output	139	(6)
Text files	141	(2)
Binary files	143	(2)
Integer arithmetic	145	(26)
Prime numbers	145	(6)
A table of primes	146	(2)
Prime factors	148	(2)
The greatest common divisor	150	(1)
Rational numbers	151	(9)
The class rational	152	(5)
The harmonic numbers	157	(1)
Bernoulli's numbers	158	(2)
Congruences and residues	160	(2)
Fermat's theorem	161	(1)
Periodic sequences	162	(1)
Random numbers	162	(9)
Linear congruence generators	163	(1)
Choice of parameters	164	(2)
Sequential potency	166	(2)
Practical generators	168	(1)
Shuffling	169	(1)
Example---Monte Carlo integration	170	(1)
Tests of randomness	171	(17)
Even distribution test	173	(3)
Serial correlation test	176	(1)
The spectral test	177	(11)
Minimization process	179	(3)
The case k = 2	182	(1)
The case k = 3	183	(4)
Results	187	(1)
Vectors and matrices	188	(16)
The class vector	188	(7)
The class matrix	195	(5)
Operations of linear algebra	200	(4)
Direct solution of linear equations	204	(23)
Gaussian elimination	205	(3)
Algorithms	205	(2)
Speed of computation	207	(1)
Refinement	208	(5)
Pivoting	208	(2)
The case of the vanishing determinant	210	(1)
Minimizing the arithmetic	210	(1)
Refined gauss	211	(1)
Improved speed	212	(1)
Scaling	212	(1)
Matrix decomposition	213	(2)
Crout algorithm	215	(5)
Simple Crout	215	(2)
Speed	217	(1)
Crout with pivoting	218	(2)
The inverse matrix	220	(2)
Tridiagonal equations	222	(3)
Modular programming	225	(2)
Errors in matrix manipulation	227	(9)
Norms	227	(3)
Convergence	230	(1)
Error estimation	231	(3)
Random perturbations	233	(1)
Iterative refinement	234	(2)
Iterative solutions of systems of equations	236	(14)
Gauss-Jacobi iteration	236	(3)
Gauss-Seidel iteration	239	(2)
Jacobi and Seidel compared	241	(1)
Successive over-relaxation	242	(1)
Speed of computation	243	(3)
Roots of systems of non-linear equations	246	(4)
Matrix eigenvalue problems	250	(30)
General theory	251	(4)
Locating eigenvalues	252	(2)
Eigenvalue stability and errors of computation	254	(1)
The power method	255	(6)
Inverse power method	261	(2)
Jacobi rotation method	263	(5)
Householder's method	268	(12)
Reduction to tridiagonal form	269	(4)
Francis diagonalization	273	(7)
Interpolation and data fitting	280	(25)
Lagrangian interpolation	281	(8)
Lagrangian algorithm	282	(1)
Errors of interpolation	283	(3)
Neville's algorithm	286	(2)
Divided differences	288	(1)
Inverse interpolation	289	(1)
Cubic spline piecewise interpolation	289	(4)
Cubic spline routine	290	(2)
Lagrange polynomial and cubic spline compared	292	(1)
Data fitting	293	(4)
Least squares approximation	294	(2)
Other approximations	296	(1)
Sorting	297	(8)
Quicksort	299	(3)
Heapsort	302	(3)
Graphics	305	(29)
Text mode	305	(5)
Graphics mode	310	(11)
Classes point and pixel	311	(3)
Block size	314	(2)
Opening and closing graphics mode	316	(1)
Presentation	316	(2)
Drawing a function	318	(2)
Scanning the screen	320	(1)
The complex plane: quadratic Julia sets	321	(4)
Dynamic use of colour: the Mandelbrot set	325	(3)
Bezier curves	328	(6)
Differentiation and integration	334	(23)
Differentiation	334	(4)
Richardson extrapolation	338	(1)
Integration	339	(4)
Composite formulae	343	(3)
Asymptotic errors	346	(1)
Adaptive integration	347	(2)
Romberg integration	349	(5)
Improper integrals	354	(1)
Multiple integrals	355	(2)
Orthogonal polynomials	357	(26)
The Gram-Schmidt process	357	(2)
Zeros of orthogonal polynomials	359	(1)
Particular systems	360	(4)
Legendre polynomials	360	(2)
Chebyshev polynomials	362	(1)
Laguerre polynomials	363	(1)
Hermite polynomials	364	(1)
Approximation of functions	364	(5)
Taylor approximation	366	(1)
Legendre approximation	366	(2)
Chebyshev approximation	368	(1)
Minimax approximation	369	(3)
Gaussian quadrature	372	(11)
Computation of nodes	374	(1)
Computation of weights	375	(1)
Nodes and weights combined	376	(4)
An example	380	(3)
Differential equations	383	(30)
Existence and uniqueness	383	(2)
Stability and conditioning	385	(2)
Euler's method	387	(4)
Higher order methods	391	(1)
Runge-kutta methods	392	(7)
Convergence	395	(1)
Adaptation	396	(3)
Multiple-step methods	399	(10)
Two-step midpoint method	399	(5)
Other two-step methods	404	(5)
Predictor-corrector methods	409	(2)
Domain of existence	411	(2)
More about differential equations	413	(25)
Systems of ordinary differential equations	413	(5)
Runge-Kutta method	414	(1)
Lotka-Volterra equations	415	(3)
Second-order initial value problems	418	(2)
Second-order boundary value problems	420	(8)
The shooting method	420	(3)
The linear finite difference method	423	(3)
Extrapolated linear finite difference method	426	(1)
The non-linear finite difference method	426	(2)
Variational methods	428	(3)
Eigenvalue problems	431	(2)
Partial differential equations	433	(5)
Recursive data types---lists	438	(33)
The base class list	441	(7)
List of class template	448	(3)
The travelling salesman	451	(10)
Multiple byte integers	461	(10)
Elements of Fourier analysis	471	(10)
Fourier series	471	(2)
Periodic functions	473	(2)
Discrete Fourier transforms	475	(1)
Fast Fourier transforms	476	(5)
Addendum Programming in Windows	481	(52)
A1 The Window classes TApplication and TWindow	483	(4)
A2 The class TWindow and the function Paint	487	(4)
A3 Programming mouse buttons	491	(1)
A4 An input dialogue box	492	(2)
A5 A multiple document interface	494	(7)
A6 The class Tscreen	501	(10)
A7 Drawing a function	511	(4)
A8 Mapping the screen	515	(1)
A9 Ordinary differential equations	516	(7)
A10 Two simultaneous differential equations	523	(10)
Bibliography	533	(2)
Appendix A standard.h	535	(2)
Appendix B restring.h	537	(1)
Appendix C rational.h	538	(1)
Appendix D random.h	539	(1)
Appendix E vecmat.h	540	(2)
Appendix F stdgraph.h	542	(2)
Appendix G list.h	544	(2)
Appendix H screen.h	546	(3)
Index	549