In Java Number Cruncher, author Ronald Mak explains how to spot-and how to avoid-the subtle programming miscues that can cause vexing calculation errors in your applications. An authority on mapping pure math to computer math, he explains how to use the often-overlooked computational features of Java, and does so in a clear, non-theoretical style.
Without getting lost in mathematical detail, you'll learn practical numerical algorithms for safely summing numbers, finding roots of equations, interpolation and approximation, numerical integration, solving differential equations, matrix operations, and solving sets of simultaneous equations. You'll also enjoy intriguing topics such as searching for patterns in prime numbers, generating random numbers, computing thousands of digits of pi, and creating intricately beautiful fractal images.
Java Number Cruncher includes:
* Practical information all Java programmers should know
* Popular computational algorithms in Java-without excessive mathematical theory
* Interactive graphical programs that bring the algorithms to life on the computer screen
* Rounding errors, the pitfalls of integer arithmetic, Java's implementation of the IEEE 754 floating-point standard, and more
This book is useful to all Java programmers, especially for those who want to learn about numerical computation, and for developers of scientific, financial, and data analysis applications.
Copyright
Preface
How to Download the Source Code
Part I. Why Good Computations Go Bad
Chapter 1. Floating-Point Numbers Are Not Real!
Section 1.1. Roundoff Errors
Section 1.2. Error Explosion
Section 1.3. Real Numbers versus Floating-Point Numbers
Section 1.4. Precision and Accuracy
Section 1.5. Disobeying the Laws of Algebra
Section 1.6. And What about Those Integers?
References
Chapter 2. How Wholesome Are the Integers?
Section 2.1. The Integer Types and Operations
Section 2.2. Signed Magnitude versus Two's-Complement
Section 2.3. Whole Numbers versus Integer Numbers
Section 2.4. Wrapper Classes
Section 2.5. Integer Division and Remainder
Section 2.6. Integer Exponentiation
References
Chapter 3. The Floating-Point Standard
Section 3.1. The Floating-Point Formats
Section 3.2. Denormalized Numbers
Section 3.3. Decomposing Floating-Point Numbers
Section 3.4. The Floating-Point Operations
Section 3.5. ±0, ±, and NaN
Section 3.6. No Exceptions!
Section 3.7. Another Look at Roundoff Errors
Section 3.8. Strict or Nonstrict Floating-Point Arithmetic
Section 3.9. The Machine Epsilon
Section 3.10. Error Analysis
References
Part II. Iterative Computations
Chapter 4. Summing Lists of Numbers
Section 4.1. A Summing Mystery—the Magnitude Problem
Section 4.2. The Kahan Summation Algorithm
Section 4.3. Summing Numbers in a Random Order
Section 4.4. Summing Addends with Different Signs
Section 4.5. Insightful Computing
Section 4.6. Summation Summary
References
Chapter 5. Finding Roots
Section 5.1. Analytical versus Computer Solutions
Section 5.2. The Functions
Section 5.3. The Bisection Algorithm
Section 5.4. The Regula Falsi Algorithm
Section 5.5. The Improved Regula Falsi Algorithm
Section 5.6. The Secant Algorithm
Section 5.7. Newton's Algorithm
Section 5.8. Fixed-Point Iteration
Section 5.9. Double Trouble with Multiple Roots
Section 5.10. Comparing the Root-Finder Algorithms
References
Chapter 6. Interpolation and Approximation
Section 6.1. The Power Form versus the Newton Form
Section 6.2. Polynomial Interpolation Functions
Section 6.3. Divided Differences
Section 6.4. Constructing the Interpolation Function
Section 6.5. Least-Squares Linear Regression
Section 6.6. Constructing the Regression Line
References
Chapter 7. Numerical Integration
Section 7.1. Back to Basics
Section 7.2. The Trapezoidal Algorithm
Section 7.3. Simpson's Algorithm
References
Chapter 8. Solving Differential Equations Numerically
Section 8.1. Back to Basics
Section 8.2. A Differential Equation Class
Section 8.3. Euler's Algorithm
Section 8.4. A Predictor-Corrector Algorithm
Section 8.5. The Fourth-Order Runge-Kutta Algorithm
References
Part III. A Matrix Package
Chapter 9. Basic Matrix Operations
Section 9.1. Matrix
Section 9.2. Square Matrix
Section 9.3. Identity Matrix
Section 9.4. Row Vector
Section 9.5. Column Vector
Section 9.6. Graphic Transformation Matrices
Section 9.7. A Tumbling Cube in 3-D Space
References
Chapter 10. Solving Systems of Linear Equations
Section 10.1. The Gaussian Elimination Algorithm
Section 10.2. Problems with Gaussian Elimination
Section 10.3. Partial Pivoting
Section 10.4. Scaling
Section 10.5. LU Decomposition
Section 10.6. Iterative Improvement
Section 10.7. A Class for Solving Systems of Linear Equations
Section 10.8. A Program to Test LU Decomposition
Section 10.9. Polynomial Regression
References
Chapter 11. Matrix Inversion, Determinants, and Condition Numbers
Section 11.1. The Determinant
Section 11.2. The Inverse
Section 11.3. The Norm and the Condition Number
Section 11.4. The Invertible Matrix Class
Section 11.5. Hilbert Matrices
Section 11.6. Comparing Solution Algorithms
References
Part IV. The Joys of Computation
Chapter 12. Big Numbers
Section 12.1. Big Integers
Section 12.2. A Very Large Prime Number
Section 12.3. Big Integers and Cryptography
Section 12.4. Big Decimal Numbers
Section 12.5. Big Decimal Functions
References
Chapter 13. Computing p
Section 13.1. Estimates of p and Ramanujan's Formulas
Section 13.2. Arctangent Formulas That Generate p
Section 13.3. Generating Billions of Digits
References
Chapter 14. Generating Random Numbers
Section 14.1. Pseudorandom Numbers
Section 14.2. Uniformly Distributed Random Numbers
Section 14.3. Normally Distributed Random Numbers
Section 14.4. Exponentially Distributed Random Numbers
Section 14.5. Monte Carlo, Buffon's Needle, and p
References
Chapter 15. Prime Numbers
Section 15.1. The Sieve of Eratosthenes and Factoring
Section 15.2. Congruences and Modulo Arithmetic
Section 15.3. The Lucas Test
Section 15.4. The Miller-Rabin Test
Section 15.5. A Combined Primality Tester
Section 15.6. Generating Prime Numbers
Section 15.7. Prime Number Patterns
References
Chapter 16. Fractals
Section 16.1. Fixed-Point Iteration and Orbits
Section 16.2. Bifurcation and the Real Function f(x) = x2 + c
Section 16.3. Julia Sets and the Complex Function f(z) = z2 + c
Section 16.4. Newton's Algorithm in the Complex Plane
Section 16.5. The Mandelbrot Set
References
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