OBJECT-ORIENTED IMPLEMENTATION OF NUMERICAL METHODS
An Introduction with Java & Smalltalk To order this title, and for more information, click here
By Didier Besset, Consultant, Geneva, Switzerland.
Description
Numerical methods naturally lend themselves to an object-oriented approach. Mathematics builds high- level ideas on top of previously
described, simpler ones. Once a property is demonstrated for a given concept, it can be applied to any new concept sharing the same premise
as the original one, similar to the ideas of reuse and inheritance in object-oriented (OO) methodology.
Few books on numerical methods
teach developers much about designing and building good code. Good computing routines are problem-specific. Insight and understanding
are what is needed, rather than just recipes and black box routines. Developers need the ability to construct new programs for different
applications.
Object-Oriented Implementation of Numerical Methods reveals a complete OO design methodology in a clear and systematic
way. Each method is presented in a consistent format, beginning with a short explanation and following with a description of the general
OO architecture for the algorithm. Next, the code implementations are discussed and presented along with real-world examples that the
author, an experienced software engineer, has used in a variety of commercial applications.
On the enclosed CD-ROM, you'll find
files containing tested source code implementations of all the algorithms discussed in the book in both Java and Smalltalk. This includes
repository files for VisualAge for Java and VisualAge for Smalltalk under ENVY.
Contents Code Listings
Preface
1 Introduction
1.1 Object-Oriented Paradigm and Mathematical Objects
1.2 Object-Oriented Concepts in a
Nutshell
1.3 Dealing with Numerical Data
1.3.1 Floating Point Representation
1.3.2 Rounding Errors
1.3.3 Real Example of Rounding
Error
1.3.4 Outsmarting Rounding Errors
1.3.5 Wisdom from the Past
1.4 Finding the Numerical Precision of a Computer
1.4.1 Computer
Numerical Precision?General Implementation
1.4.2 Computer Numerical Precision?Smalltalk Implementation
1.4.3 Computer Numerical Precision?Java
Implementation
1.5 Comparing Floating-Point Numbers
1.5.1 Comparing Floating-Point Numbers?Smalltalk Code
1.6 Speed Consideration
1.6.1 Smalltalk Particular
1.7 Conventions
1.7.1 Class Diagrams
1.7.2 Smalltalk Code
1.7.3 Java Code
1.8 Road Map
2 Function
Evaluation
2.1 Function Concept
2.1.1 Function?Smalltalk Implementation
2.1.2 Function?Java Implementation
2.2 Polynomials
2.2.1
Mathematical Definitions
2.2.2 Polynomials?General Implementation
2.2.3 Polynomials?Smalltalk Implementation
2.2.4 Polynomials?Java
implementation
2.3 Error Function
2.3.1 Mathematical Definitions
2.3.2 Error Function?Smalltalk Implementation
2.3.3 Error Function?Java
Implementation
2.4 Gamma Function
2.4.1 Mathematical Definitions
2.4.2 Gamma Function?Smalltalk Implementation
2.4.3 Gamma Function?Java
Implementation
2.5 Beta Function
2.5.1 Mathematical Definitions
2.5.2 Beta Function?Smalltalk Implementation
2.5.3 Beta Function?Java
Implementation
3 Interpolation
3.1 General Remarks
3.1.1 Interpolation Concepts and Examples
3.2 Lagrange Interpolation
3.2.1
Lagrange Interpolation?Smalltalk Implementation
3.2.2 Lagrange Interpolation?Java Implementation
3.3 Newton Interpolation
3.3.1
Newton Interpolation?General Implementation
3.3.2 Newton Interpolation?Smalltalk Implementation
3.3.3 Newton Interpolation?Java Implementation
3.4 Neville Interpolation
3.4.1 Neville Interpolation?General Implementation
3.4.2 Neville Interpolation?Smalltalk Implementation
3.4.3 Neville Interpolation?Java Implementation
3.5 Bulirsch-Stoer Interpolation
3.5.1 Bulirsch-Stoer Interpolation?General Implementation
3.5.2 Bulirsch-Stoer Interpolation?Smalltalk Implementation
3.5.3 Bulirsch-Stoer Interpolation?Java Implementation
3.6 Cubic Spline
Interpolation
3.6.1 Cubic Spline Interpolation?General Implementation
3.6.2 Cubic Spline Interpolation?Smalltalk Implementation
3.6.3 Cubic Spline Interpolation?Java Implementation
3.7 Which Method to Choose?
4 Iterative Algorithms
4.1 Successive Approximations
4.1.1 Iterative Process?Smalltalk Implementation
4.1.2 Iterative Process?JAVA Implementation
4.2 Evaluation with Relative Precision
4.2.1 Relative Precision?Smalltalk Implementation
4.2.2 Relative Precision?Java Implementation
4.3 Examples
5 Finding the Zero
of a Function
5.1 Introduction
5.2 Finding the Zeroes of a Function?Bisection Method
5.2.1 Bisection Algorithm?General Implementation
5.2.2 Bisection Algorithm?Smalltalk Implementation
5.2.3 Bisection Algorithm?Java Implementation
5.3 Finding the Zero of a Function?Newton's
Method
5.3.1 Newton's Method?Smalltalk Implementation
5.3.2 Newton's Method?Java Implementation
5.4 Example of Zero Finding?Roots
of Polynomials
5.4.1 Roots of Polynomials?Smalltalk Implementation
5.4.2 Roots of Polynomials?Java Implementation
5.5 Which Method
to Choose?
6 Integration of Functions
6.1 Introduction
6.2 General Framework?Trapeze Integration Method
6.2.1 End Game Strategy
6.2.2 Trapeze Integration?General Implementation
6.2.3 Trapeze Integration?Smalltalk Implementation
6.2.4 Trapeze Integration?Java
Implementation
6.3 Simpson Integration Algorithm
6.3.1 Simpson Integration?General Implementation
6.3.2 Simpson Integration?Smalltalk
Implementation
6.3.3 Simpson Integration?Java Implementation
6.4 Romberg Integration Algorithm
6.4.1 Romberg Integration?General
Implementation
6.4.2 Romberg Integration?Smalltalk Implementation
6.4.3 Romberg Integration?Java Implementation
6.5 Evaluation of
Open Integrals
6.5.1 Bag of Tricks
6.6 Which Method to Choose?
6.6.1 Smalltalk Comparison
6.6.2 Java Comparison
7 Series
7.1 Introduction
7.2 Infinite Series
7.2.1 Infinite Series?Smalltalk Implementation
7.2.2 Infinite Series?Java Implementation
7.3
Continued Fractions
7.3.1 Continued Fractions?Smalltalk Implementation
7.3.2 Continued Fractions?Java Implementation
7.4 Incomplete
Gamma Function
7.4.1 Mathematical Definitions
7.4.2 Incomplete Gamma Function?Smalltalk Implementation
7.4.3 Incomplete Gamma Function?Java
Implementation
7.5 Incomplete Beta Function
7.5.1 Mathematical Definitions
7.5.2 Incomplete Beta Function?Smalltalk Implementation
7.5.3 Incomplete Beta Function?Java Implementation
8 Linear Algebra
8.1 Vectors and Matrices
8.1.1 Vector and Matrix?Smalltalk
Implementation
8.1.2 Vector and Matrix?Java Implementation
8.2 Linear Equations
8.2.1 Backward Substitution
8.2.2 Gaussian Elimination
8.2.3 Fine Points
8.2.4 Linear Equations?General Implementation
8.2.5 Linear Equations?Smalltalk Implementation
8.2.6 Linear
Equations?Java Implementation
8.3 LUP Decomposition
8.3.1 LUP Decomposition?General Implementation
8.3.2 LUP Decomposition?Smalltalk
Implementation
8.3.3 LUP Decomposition?Java Implementation
8.4 Computing the Determinant of a Matrix
8.4.1 Computing the Determinant
of Matrix?General Implementation
8.4.2 Computing the Determinant of Matrix?Smalltalk Implementation
8.4.3 Computing the Determinant
of Matrix?Java Implementation
8.5 Matrix Inversion
8.5.1 Implementation Strategy
8.5.2 Matrix Inversion?Smalltalk Implementation
8.5.3 Matrix Inversion?Java Implementation
8.5.4 Matrix Inversion?Rounding Problems
8.6 Matrix Eigenvalues and Eigenvectors of
a Nonsymmetrical Matrix
8.6.1 Finding the Largest Eigenvalue?General Implementation
8.6.2 Finding the Largest Eigenvalue?Smalltalk
Implementation
8.6.3 Finding the Largest Eigenvalue?Java Implementation
8.7 Matrix Eigenvalues and Eigenvectors of a Symmetrical
Matrix
8.7.1 Jacobi's Algorithm
8.7.2 Jacobi's Algorithm?General Implementation
8.7.3 Jacobi's Algorithm?Smalltalk Implementation
8.7.4 Jacobi's Algorithm?Java Implementation
9 Elements of Statistics
9.1 Statistical Moments
9.1.1 Statistical Moments?General
Implementation
9.1.2 Statistical Moments?Smalltalk Implementation
9.1.3 Statistical Moments?Java Implementation
9.2 Robust Implementation
of Statistical Moments
9.2.1 Robust Central Moments?General Implementation
9.2.2 Robust Central Moments?Smalltalk Implementation
9.2.3 Robust Central Moments?Java Implementation
9.3 Histograms
9.3.1 Histograms?General Implementation
9.3.2 Histograms?Smalltalk
Implementation
9.3.3 Histograms?Java Implementation
9.4 Random Number Generator
9.4.1 Linear Congruential Random Generators
9.4.2
Additive Sequence Generators
9.4.3 Bit-Pattern Generators
9.4.4 Random Number Generator?Smalltalk Implementation
9.4.5 Random Number
Generator?Java Implementation
9.5 Probability Distributions
9.5.1 Probability Distributions?General Implementation
9.5.2 Probability
Distributions?Smalltalk Implementation
9.5.3 Probability Distributions?Java Implementation
9.6 Normal Distribution
9.6.1 Normal
Distribution?Smalltalk Implementation
9.6.2 Normal Distribution?Java Implementation
9.7 Gamma Distribution
9.7.1 Gamma Distribution?Smalltalk
Implementation
9.7.2 Gamma Distribution?Java Implementation
9.8 Experimental Distribution
9.8.1 Experimental Distribution?General
Implementation
9.8.2 Experimental Distribution?Smalltalk Implementation
9.8.3 Experimental Distribution?Java Implementation
10
Statistical Analysis
10.1 F-Test and the Fisher-Snedecor Distribution
10.1.1 Fisher-Snedecor Distribution?Smalltalk Implementation
10.1.2 Fisher-Snedecor Distribution?Java Implementation
10.2 t-Test and the Student Distribution
10.2.1 Student Distribution?Smalltalk
Implementation
10.2.2 Student Distribution?Java Implementation
10.3 . 2 -Test and . 2 Distribution
10.3.1 . 2 Distribution?Smalltalk
Implementation
10.3.2 . 2 Distribution?Java Implementation
10.3.3 Weighted-Point Implementation
10.4 . 2 -Test on Histograms
10.4.1
. 2 -Test on Histograms?Smalltalk Implementation
10.4.2 . 2 -Test on Histograms?Java Implementation
10.5 Definition of Estimation
10.5.1 Maximum-Likelihood Estimation
10.5.2 Least-Square Estimation
10.6 Least-Square Fit with Linear Dependence
10.7 Linear Regression
10.7.1 Linear Regression?General Implementation
10.7.2 Linear Regression?Smalltalk Implementation
10.7.3 Linear Regression?Java
Implementation
10.8 Least-Square Fit with Polynomials
10.8.1 Polynomial Least-Square Fits?Smalltalk Implementation
10.8.2 Polynomial
Least-Square Fits?Java Implementation
10.9 Least-Square Fit with Nonlinear Dependence
10.9.1 Nonlinear Fit?General Implementation
10.9.2 Nonlinear Fit?Smalltalk Implementation
10.9.3 Nonlinear Fit?Java Implementation
10.10 Maximum-Likelihood Fit of a Probability
Density Function
10.10.1 Maximum-Likelihood Fit?General Implementation
10.10.2 Maximum-Likelihood Fit?Smalltalk Implementation
10.10.3
Maximum-Likelihood Fit?Java Implementation
11 Optimization
11.1 Introduction
11.2 Extended Newton Algorithms
11.3 Hill-Climbing
Algorithms
11.3.1 Optimizing?General Implementation
11.3.2 Common Optimizing Classes?Smalltalk Implementation
11.3.3 Common Optimizing
Classes?Java Implementation
11.4 Optimizing in One Dimension
11.4.1 Optimizing in One Dimension?Smalltalk Implementation
11.4.2
Optimizing in One Dimension?Java Implementation
11.5 Bracketing the Optimum in One Dimension
11.5.1 Bracketing the Optimum?Smalltalk
Implementation
11.5.2 Bracketing the Optimum?Java Implementation
11.6 Powell's Algorithm
11.6.1 Powell's Algorithm?General Implementation
11.6.2 Powell's Algorithm?Smalltalk Implementation
11.6.3 Powell's Algorithm?Java Implementation
11.7 Simplex Algorithm
11.7.1
Simplex Algorithm?General Implementation
11.7.2 Simplex Algorithm?Smalltalk Implementation
11.7.3 Simplex Algorithm?Java Implementation
11.8 Genetic Algorithm
11.8.1 Mapping the Search Space on Chromosomes
11.8.2 Genetic Algorithm?General Implementation
11.8.3 Genetic
Algorithm?Smalltalk Implementation
11.8.4 Genetic Algorithm?Java Abstract Implementation
11.8.5 Genetic Algorithm?Java Implementation
with Vectors
11.9 Multiple Strategy Approach
11.9.1 Multiple Strategy Approach?General Implementation
12 Data Mining
12.1 Data
Server
12.1.1 Data Server?Smalltalk Implementation
12.1.2 Data Server?Java Implementation
12.2 Covariance and Covariance Matrix
12.2.1 Using Covariance Information
12.2.2 Covariance Matrix?General Implementation
12.2.3 Covariance Matrix?Smalltalk Implementation
12.2.4 Covariance Matrix?Java Implementation
12.3 Multidimensional Probability Distribution
12.4 Covariance Data Reduction
12.5
Mahalanobis Distance
12.5.1 Examples of Use
12.5.2 Mahalanobis Distance?General Implementation
12.5.3 Mahalanobis Distance?Smalltalk
Implementation
12.5.4 Mahalanobis Distance?Java Implementation
12.6 Cluster Analysis
12.6.1 Algorithm Details
12.6.2 Cluster Analysis?General
Implementation
12.6.3 Cluster Analysis?Smalltalk Implementation
12.6.4 Cluster Analysis?Java Implementation
12.7 Covariance Clusters
12.7.1 Covariance Clusters?General Implementation
App. A Decimal Floating-Point Simulation
App. B Smalltalk Primer for Java Programmers
B.1 Syntax in a Nutshell
B.1.1 Smalltalk Expressions
B.1.2 Precedence
B.1.3 Assignment, Equality, and Identity
B.2 Class and
Methods
B.2.1 Instance Methods
B.2.2 Class Methods
B.2.3 Block Context
B.3 Iterator Methods
B.3.1 do:
B.3.2 collect:
B.3.3
inject:into:
B.4 Double Dispatching
B.5 Multiple Dispatching
App. C Java Primer for Smalltalk Programmers
C.1 Remarks on the
Syntax
C.1.1 Classes
C.1.2 Instance Variables
C.1.3 Method Declaration and Method Calling
C.1.4 Objects and Nonobjects
C.1.5
Packages
C.1.6 Scope qualifiers
C.1.7 Static Qualifier
C.2 Abstract Class and Interface
C.3 Exception Handling
C.4 Collections
and Related Topics
App. D Additional Probability Distributions
D.1 Beta Distribution
D.1.1 Beta Distribution?Smalltalk Implementation
D.1.2 Beta Distribution?Java Implementation
D.2 Cauchy Distribution
D.2.1 Cauchy Distribution?Smalltalk Implementation
D.2.2 Cauchy
Distribution?Java Implementation
D.3 Exponential Distribution
D.3.1 Exponential Distribution?Smalltalk Implementation
D.3.2 Exponential
Distribution?Java Implementation
D.4 Fisher-Tippett Distribution
D.4.1 Fisher-Tippett Distribution?Smalltalk Implementation
D.4.2
Fisher-Tippett Distribution?Java Implementation
D.5 Laplace Distribution
D.5.1 Laplace Distribution?Smalltalk Implementation
D.5.2
Laplace Distribution?Java Implementation
D.6 Log Normal Distribution
D.6.1 Log Normal Distribution?Smalltalk Implementation
D.6.2
Log Normal Distribution?Java Implementation
D.7 Triangular Distribution
D.7.1 Triangular Distribution?Smalltalk Implementation
D.7.2
Triangular Distribution?Java Implementation
D.8 Uniform Distribution
D.8.1 Uniform Distribution?Smalltalk Implementation
D.8.2 Uniform
Distribution?Java Implementation
D.9 Weibull Distribution
D.9.1 Weibull Distribution?Smalltalk Implementation
D.9.2 Weibull Distribution?Java
Implementation
App. E Accurate Accumulation of Expectation Values
E.1 Accurate Accumulation of Central Moments
E.2 Accurate Accumulation
of the Covariance
App. F Accompanying CD-ROM
F.1 Smalltalk code
F.1.1 non-ENVY users
F.1.2 ENVY users
F.2 Java code
F.2.1
Visual Age for Java
F.2.2 Other Java systems
F.3 Utilities
F.3.1 Distribution demo
F.3.2 File reader
References
Index
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