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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



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