作者：John F. Monahan

• Hardcover: 464 pages
• Publisher: Cambridge University Press; 2 edition (April 18, 2011)
• Language: English
• ISBN-10: 0521191580
• ISBN-13: 978-0521191586
  

目录：
1 Algorithms and Computers 1
1.1 Introduction 1
1.2 Computers 3
1.3 Software and Computer Languages 5
1.4 Data Structures 8
1.5 Programming Practice 9
1.6 Some Comments on R 10
References 12
2 Computer Arithmetic 13
2.1 Introduction 13
2.2 Positional Number Systems 14
2.3 Fixed Point Arithmetic 17
2.4 Floating Point Representations 20
2.5 Living with Floating Point Inaccuracies 23
2.6 The Pale and Beyond 28
2.7 Conditioned Problems and Stable Algorithms 32
Programs and Demonstrations 34
Exercises 35
References 38
3 Matrices and Linear Equations 40
3.1 Introduction 40
3.2 Matrix Operations 41
3.3 Solving Triangular Systems 43
3.4 Gaussian Elimination 44
3.5 Cholesky Decomposition 50
3.6 Matrix Norms 54
3.7 Accuracy and Conditioning 55
3.8 Matrix Computations in R 60
Programs and Demonstrations 61
Exercises 63
References 65

4 More Methods for Solving Linear Equations
4.1 Introduction
4.2 Full Elimination with Complete Pivoting
4.3 Banded Matrices
4.4 Applications to ARMA Time-Series Models
4.5 Toeplitz Systems
4.6 Sparse Matrices
4.7 Iterative Methods
4.8 Linear Programming
Programs and Demonstrations
Exercises
References
5 Regression Computations 91
5.1 Introduction 91
5.2 Condition of the Regression Problem 93
5.3 Solving the Normal Equations 96
5.4 Gram–Schmidt Orthogonalization 97
5.5 Householder Transformations 100
5.6 Householder Transformations for Least Squares 101
5.7 Givens Transformations 104
5.8 Givens Transformations for Least Squares 105
5.9 Regression Diagnostics 107
5.10 Hypothesis Tests 110
5.11 Conjugate Gradient Methods 112
5.12 Doolittle, the Sweep, and All Possible Regressions 115
5.13 Alternatives to Least Squares 118
5.14 Comments 120
Programs and Demonstrations 122
Exercises 122
References 125
6 Eigenproblems 128
6.1 Introduction 128
6.2 Theory 128
6.3 Power Methods 130
6.4 The Symmetric Eigenproblem and Tridiagonalization 133
6.5 The QR Algorithm 135
6.6 Singular Value Decomposition 137
6.7 Applications 140
6.8 Complex Singular Value Decomposition 144
Programs and Demonstrations 146
Exercises 147
References 150

7 Functions: Interpolation, Smoothing, and Approximation 151
7.1 151
Introduction 153
7.2 156
Interpolation 159
7.3 163
Interpolating Splines 168
7.4 170
Curve Fitting with Splines: Smoothing and Regression 177
7.5 179
Mathematical Approximation 183
7.6
Practical Approximation Techniques
7.7
Computing Probability Functions
Programs and Demonstrations
Exercises
References
8 Introduction to Optimization and Nonlinear Equations 186
8.1 186
Introduction
8.2
Safe Univariate Methods: Lattice Search, Golden Section,
and Bisection
8.3
Root Finding
8.4
First Digression: Stopping and Condition
8.5
Multivariate Newton’s Methods
8.6
Second Digression: Numerical Differentiation
8.7
Minimization and Nonlinear Equations
8.8
Condition and Scaling
8.9
Implementation
8.10 A Non-Newton Method: Nelder-Mead
Programs and Demonstrations
Exercises
References
Maximum Likelihood and Nonlinear Regression 219
9.1 219
Introduction 220
9.2 226
Notation and Asymptotic Theory of Maximum Likelihood 228
9.3 230
Information, Scoring, and Variance Estimates 236
9.4 An Extended Example 237
9.5 242
Concentration, Iteration, and the EM Algorithm 246
9.6 251
Multiple Regression in the Context of Maximum Likelihood 252
9.7 255
Generalized Linear Models
9.8
Nonlinear Regression
9.9
Parameterizations and Constraints
Programs and Demonstrations
Exercises
References
Numerical Integration and Monte Carlo Methods 257
10.1 257
Introduction 258
10.2 264
Motivating Problems
10.3
One-Dimensional Quadrature

Contents
10.4 Numerical Integration in Two or More Variables
10.5 Uniform Pseudorandom Variables
10.6 Quasi–Monte Carlo Integration
10.7 Strategy and Tactics
Programs and Demonstrations
Exercises
References
11 Generating Random Variables from Other Distributions 303
11.1 303
Introduction 304
11.2 308
General Methods for Continuous Distributions 321
11.3 Algorithms for Continuous Distributions 325
11.4 330
General Methods for Discrete Distributions 334
11.5 Algorithms for Discrete Distributions 337
11.6 338
Other Randomizations 341
11.7 Accuracy in Random Number Generation
Programs and Demonstrations
Exercises
References
12 Statistical Methods for Integration and Monte Carlo 343
12.1 343
Introduction 343
12.2 350
Distribution and Density Estimation 353
12.3 359
Distributional Tests 361
12.4 363
Importance Sampling and Weighted Observations 365
12.5 Testing Importance Sampling Weights 370
12.6 372
Laplace Approximations 373
12.7
Randomized Quadrature
12.8
Spherical–Radial Methods
Programs and Demonstrations
Exercises
References
13 Markov Chain Monte Carlo Methods 375
13.1 375
Introduction 377
13.2 378
Markov Chains 383
13.3 386
Gibbs Sampling 390
13.4 394
Metropolis–Hastings Algorithm 398
13.5 Time-Series Analysis 398
13.6 Adaptive Acceptance / Rejection 400
13.7
Diagnostics
Programs and Demonstrations
Exercises
References

14
Sorting and Fast Algorithms
14.1
Introduction
14.2
Divide and Conquer
14.3
Sorting Algorithms
14.4
Fast Order Statistics and Related Problems
14.5
Fast Fourier Transform
14.6
Convolutions and the Chirp-z Transform
14.7
Statistical Applications of the FFT
14.8
Combinatorial Problems
Programs and Demonstrations
Exercises
References