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Chapter 12
COMPUTATIONAL PROBLEMS AND METHODS
RICHARD E. QUANDT*
Princeton University
Contents
1. Introduction
2. Matrix methods
2.1. Methods for solving ,4a = c
2.2. Singular value decomposition
2.3. Sparse matrix methods
3. Common functions requiring optimization
3. I. Likelihood functions
3.2. Generalized distance functions
3.3. Functions in optimal control
4. Algorithms for optimizing functions of many variables
4. I. Introduction
4.2. Methods employing no derivatives
4.3. Methods employing first and second derivatives
4.4. Methods employing first derivatives
5. Special purpose algorithms and simplifications
5.1. Jacobi and Gauss-Seidel methods
5.2. Parke’s Algorithm A
5.3. The EM algorithm
5.4. Simplified Jacobian computation
6. Further aspects of algorithms
6.1. Computation of derivatives
6.2. Linear searches
6.3. Stopping criteria
6.4. Multiple optima
7. Particular problems in optimization
7.1. Smoothing of non-differentiable functions
7.2. Unbounded likelihood functions and other false optima 742
7.3. Constraints on the parameters 744
8. Numerical integration 747
8. I. Monte Carlo integration 749
8.2. Polynomial approximations 750
8.3. Evaluation of multivariate normal integrals 751
8.4. Special cases of the multivariate normal integral 753
9. The generation of random numbers 755
9.1. The generation of uniformly distributed variables 756
9.2. The generation of normally distributed variables 757
References 760
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