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Java Number Cruncher: The Java Programmers Guide to Numerical Computing

ISBN: 0130460419
EAN: 2147483647

Year: 2001
Pages: 141

Authors: Ronald Mak

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• Java Number Cruncher: The Java Programmer s Guide to Numerical Computing
• Table of Contents
• Copyright
• Preface
• How to Download the Source Code
• Part I: Why Good Computations Go Bad
• Chapter 1. Floating-Point Numbers Are Not Real
• 1.1 Roundoff Errors
• 1.2 Error Explosion
• 1.3 Real Numbers versus Floating-Point Numbers
• 1.4 Precision and Accuracy
• 1.5 Disobeying the Laws of Algebra
• 1.6 And What about Those Integers?
• References
• Chapter 2. How Wholesome Are the Integers?
• 2.1 The Integer Types and Operations
• 2.2 Signed Magnitude versus Two s-Complement
• 2.3 Whole Numbers versus Integer Numbers
• 2.4 Wrapper Classes
• 2.5 Integer Division and Remainder
• 2.6 Integer Exponentiation
• References
• Chapter 3. The Floating-Point Standard
• 3.1 The Floating-Point Formats
• 3.2 Denormalized Numbers
• 3.3 Decomposing Floating-Point Numbers
• 3.4 The Floating-Point Operations
• 3.5 0, , and NaN
• 3.6 No Exceptions
• 3.7 Another Look at Roundoff Errors
• 3.8 Strict or Nonstrict Floating-Point Arithmetic
• 3.9 The Machine Epsilon
• 3.10 Error Analysis
• References
• Part II: Iterative Computations
• Chapter 4. Summing Lists of Numbers
• 4.1 A Summing Mystery?athe Magnitude Problem
• 4.2 The Kahan Summation Algorithm
• 4.3 Summing Numbers in a Random Order
• 4.4 Summing Addends with Different Signs
• 4.5 Insightful Computing
• 4.6 Summation Summary
• References
• Chapter 5. Finding Roots
• 5.1 Analytical versus Computer Solutions
• 5.2 The Functions
• 5.3 The Bisection Algorithm
• 5.4 The Regula Falsi Algorithm
• 5.5 The Improved Regula Falsi Algorithm
• 5.6 The Secant Algorithm
• 5.7 Newton s Algorithm
• 5.8 Fixed-Point Iteration
• 5.9 Double Trouble with Multiple Roots
• 5.10 Comparing the Root-Finder Algorithms
• References
• Chapter 6. Interpolation and Approximation
• 6.1 The Power Form versus the Newton Form
• 6.2 Polynomial Interpolation Functions
• 6.3 Divided Differences
• 6.4 Constructing the Interpolation Function
• 6.5 Least-Squares Linear Regression
• 6.6 Constructing the Regression Line
• References
• Chapter 7. Numerical Integration
• 7.1 Back to Basics
• 7.2 The Trapezoidal Algorithm
• 7.3 Simpson s Algorithm
• References
• Chapter 8. Solving Differential Equations Numerically
• 8.1 Back to Basics
• 8.2 A Differential Equation Class
• 8.3 Euler s Algorithm
• 8.4 A Predictor-Corrector Algorithm
• 8.5 The Fourth-Order Runge-Kutta Algorithm
• References
• Part III: A Matrix Package
• Chapter 9. Basic Matrix Operations
• 9.1 Matrix
• 9.2 Square Matrix
• 9.3 Identity Matrix
• 9.4 Row Vector
• 9.5 Column Vector
• 9.6 Graphic Transformation Matrices
• 9.7 A Tumbling Cube in 3-D Space
• References
• Chapter 10. Solving Systems of Linear Equations
• 10.1 The Gaussian Elimination Algorithm
• 10.2 Problems with Gaussian Elimination
• 10.3 Partial Pivoting
• 10.4 Scaling
• 10.5 LU Decomposition
• 10.6 Iterative Improvement
• 10.7 A Class for Solving Systems of Linear Equations
• 10.8 A Program to Test LU Decomposition
• 10.9 Polynomial Regression
• References
• Chapter 11. Matrix Inversion, Determinants, and Condition Numbers
• 11.1 The Determinant
• 11.2 The Inverse
• 11.3 The Norm and the Condition Number
• 11.4 The Invertible Matrix Class
• 11.5 Hilbert Matrices
• 11.6 Comparing Solution Algorithms
• References
• Part IV: The Joys of Computation
• Chapter 12. Big Numbers
• 12.1 Big Integers
• 12.2 A Very Large Prime Number
• 12.3 Big Integers and Cryptography
• 12.4 Big Decimal Numbers
• 12.5 Big Decimal Functions
• References
• Chapter 13. Computing p
• 13.1 Estimates of p and Ramanujan s Formulas
• 13.2 Arctangent Formulas That Generate p
• 13.3 Generating Billions of Digits
• References
• Chapter 14. Generating Random Numbers
• 14.1 Pseudorandom Numbers
• 14.2 Uniformly Distributed Random Numbers
• 14.3 Normally Distributed Random Numbers
• 14.4 Exponentially Distributed Random Numbers
• 14.5 Monte Carlo, Buffon s Needle, and p
• References
• Chapter 15. Prime Numbers
• 15.1 The Sieve of Eratosthenes and Factoring
• 15.2 Congruences and Modulo Arithmetic
• 15.3 The Lucas Test
• 15.4 The Miller-Rabin Test
• 15.5 A Combined Primality Tester
• 15.6 Generating Prime Numbers
• 15.7 Prime Number Patterns
• References
• Chapter 16. Fractals
• 16.1 Fixed-Point Iteration and Orbits
• 16.2 Bifurcation and the Real Function f(x)x2 c
• 16.3 Julia Sets and the Complex Function f(z)z2 c
• 16.4 Newton s Algorithm in the Complex Plane
• 16.5 The Mandelbrot Set