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Java™ Number Cruncher: The Java Programmer's Guide to Numerical Computing

Name: Java™ Number Cruncher: The Java Programmer's Guide to Numerical Computing
Author: Ronald Mak
ISBN: 0130460419

by Ronald Mak

October 2002

Intermediate to advanced

480 pages

9h 6m

English

Pearson

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Copyright
Dedication
Preface
How to Download the Source Code
I. Why Good Computations Go Bad
1. Floating-Point Numbers Are Not Real!
1.1. Roundoff Errors1.2. Error Explosion1.3. Real Numbers versus Floating-Point Numbers1.4. Precision and Accuracy1.5. Disobeying the Laws of Algebra1.6. And What about Those Integers?References
Bibliography
2. How Wholesome Are the Integers?
2.1. The Integer Types and Operations2.2. Signed Magnitude versus Two's-Complement2.3. Whole Numbers versus Integer Numbers2.4. Wrapper Classes2.5. Integer Division and Remainder2.6. Integer ExponentiationReferences
Bibliography
3. The Floating-Point Standard
3.1. The Floating-Point FormatsNormalized NumberDenormalized Number3.2. Denormalized Numbers3.3. Decomposing Floating-Point Numbers3.4. The Floating-Point Operations3.5. ±0, ±∞, and NaN3.6. No Exceptions!3.7. Another Look at Roundoff Errors3.8. Strict or Nonstrict Floating-Point Arithmetic3.9. The Machine Epsilon ∊3.10. Error AnalysisReferences
Bibliography

II. Iterative Computations
4. Summing Lists of Numbers
4.1. A Summing Mystery—the Magnitude Problem4.2. The Kahan Summation Algorithm4.3. Summing Numbers in a Random Order4.4. Summing Addends with Different Signs4.5. Insightful Computing4.6. Summation SummaryReferences
Bibliography
5. Finding Roots
5.1. Analytical versus Computer Solutions5.2. The Functions5.3. The Bisection Algorithm5.4. The Regula Falsi Algorithm5.5. The Improved Regula Falsi Algorithm5.6. The Secant Algorithm5.7. Newton's Algorithm5.8. Fixed-Point Iteration5.9. Double Trouble with Multiple Roots5.10. Comparing the Root-Finder AlgorithmsReferences
Bibliography
6. Interpolation and Approximation
6.1. The Power Form versus the Newton Form6.2. Polynomial Interpolation Functions6.3. Divided Differences6.4. Constructing the Interpolation Function6.5. Least-Squares Linear Regression6.6. Constructing the Regression LineReferences
Bibliography
7. Numerical Integration
7.1. Back to Basics7.2. The Trapezoidal Algorithm7.3. Simpson's AlgorithmReferences
Bibliography
8. Solving Differential Equations Numerically
8.1. Back to Basics8.2. A Differential Equation Class8.3. Euler's Algorithm8.4. A Predictor-Corrector Algorithm8.5. The Fourth-Order Runge-Kutta AlgorithmReferences
Bibliography
III. A Matrix Package
9. Basic Matrix Operations
9.1. Matrix9.2. Square Matrix9.3. Identity Matrix9.4. Row Vector9.5. Column Vector9.6. Graphic Transformation Matrices9.7. A Tumbling Cube in 3-D SpaceReferences
Bibliography
10. Solving Systems of Linear Equations
10.1. The Gaussian Elimination Algorithm10.2. Problems with Gaussian Elimination10.3. Partial Pivoting10.4. Scaling10.5. LU Decomposition10.6. Iterative Improvement10.7. A Class for Solving Systems of Linear Equations10.8. A Program to Test LU Decomposition10.9. Polynomial RegressionReferences
Bibliography
11. Matrix Inversion, Determinants, and Condition Numbers
11.1. The Determinant11.2. The Inverse11.3. The Norm and the Condition Number11.4. The Invertible Matrix Class11.5. Hilbert Matrices11.6. Comparing Solution AlgorithmsReferences
Bibliography
IV. The Joys of Computation
12. Big Numbers
12.1. Big Integers12.2. A Very Large Prime Number12.3. Big Integers and Cryptography12.4. Big Decimal Numbers12.5. Big Decimal FunctionsReferences
Bibliography
13. Computing π
13.1. Estimates of π and Ramanujan's Formulas13.2. Arctangent Formulas That Generate π13.3. Generating Billions of DigitsReferences
Bibliography
14. Generating Random Numbers
14.1. Pseudorandom Numbers14.2. Uniformly Distributed Random Numbers14.3. Normally Distributed Random Numbers14.4. Exponentially Distributed Random Numbers14.5. Monte Carlo, Buffon's Needle, and πReferences
Bibliography
15. Prime Numbers
15.1. The Sieve of Eratosthenes and Factoring15.2. Congruences and Modulo Arithmetic15.3. The Lucas Test15.4. The Miller-Rabin Test15.5. A Combined Primality Tester15.6. Generating Prime Numbers15.7. Prime Number PatternsReferences
Bibliography
16. Fractals
16.1. Fixed-Point Iteration and Orbits16.2. Bifurcation and the Real Function f(x) = x2 + c16.3. Julia Sets and the Complex Function f(z) = z2 + c16.4. Newton's Algorithm in the Complex Plane16.5. The Mandelbrot SetReferences
Bibliography

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Java™ Number Cruncher: The Java Programmer's Guide to Numerical Computing

by Ronald Mak

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