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CAM Ph.D. Qualifying Exam in Numerical Analysis
CONTENTS
Preliminaries
Round-off errors and computer arithmetic, algorithms and convergence
Solutions of Equations in One Variable
Bisection method, fixed-point iteration, the Newton-Raphson method, error analysis for iteration
methods, accelerating convergence, zeros of polynomials and Muller's method
Interpolation and Polynomial Approximation
Interpolation and the Lagrange polynomial, divided differences, Hermite interpolation, cubic
spline interpolation
Numerical Differentiation and Integration
Numerical differentiation, Richardson's extrapolation, element of numerical intergration,
composite numerical integration, Gaussian quadrature
Initial-Value Problems for ODEs
Elementary theory of initial value problems, Euler's method, higher-order Taylor methods,
Runge-Kutta method, multistep methods, higher-order equations and systems of differential
equations
Methods for Solving Linear Systems
Gaussian elimination (including pivoting strategies), matrix factorization, iterative techniques
(including Jacobi's, GS, SOR) and error estimate
Approximation Theory
Least square approximation, Orthogonal polynomials, Chebyshev polynomials, rational function
approximation, trigonometric polynomial approximation
Approximating Eigenvalues
Eigenvalue and eigenvectors, power methods
Numerical Solutions of Nonlinear Systems of Equations
Fixed points for functions of several variables, Newton's method
Boundary Value Problems for ODEs
The linear shooting method, finite difference methods for linear problems
REFERENCES
Burden, Richard L. and Faires, Douglas J., Numerical Analysis, 8th ed., Thomson Brooks/Cole,
2005.
Sample Exam Problem from Math 414
Sample Exam Problem from Math 415
CAM Ph.D. Qualifying Exam in Partial Differential Equations
CONTENTS
First Order Equations
Linear first-order equation; Quasi-linear equation; Method of characteristics
Second Order Equations
Classification of equations; Canonical forms of the hyperbolic, parabolic and elliptic equations;
Equations of mathematical physics: wave equation, heat equation and Laplace's equations
Elements of Fourier Analysis
Fourier series of a function; Convergence of Fourier series; Sine and Cosine expansions; Fourier
Sine and Cosine transforms; Fourier Integral; Fourier Transform
Wave Equation
Cauchy problem and d'Alembert's solution; d'Alembert's solution as a sum of forward and
backward waves; Domain of dependence and the characteristic triangle; Wave equation on a
half-line; Wave equation on a half-line with moving end; Nonhomogeneous wave equation on
the real line; Fourier series solutions on a closed interval; Nonhomogeneous problem on a closed
interval; Cauchy problem by Fourier integral; Wave equation in two space dimensions
Heat Equation
Weak maximum principle; Heat equation on bounded intervals; Heat equation on the real line;
Heat equation on the half-line; Nonhomogeneous heat equation; Heat equation in two space
variables
Dirichlet and Neumann Problems
Harmonic functions; Dirichlet problem for a rectangle; Dirichlet problem for a disk; Possion’s
integral representation for a disk; Dirichlet problem for the upper half-plane; Dirichlet problem
for a rectangular box; Neumann problem for a rectangle; Neumann problem for a disk
REFERENCES
P.V. O'Neil, Beginning Partial Differential Equations, 2nd Edition, Wiley, New York, 2008.
Sample Exam Problems
1. Use the method of characteristics to find a solution of
through the given curve defined by
,
0.
sec
that passes
2. Solve the Dirichlet problem
9.
,
0 for
9, and
,
for
CAM Ph.D. Qualifying Exam in Statistics
CONTENTS
Axioms of Probability
Random experiments, sample space and events, probability function, rules of probability
Combinatorial Methods
Permutations and combinations, ordered and unordered samples
Random Variables
Discrete random variables, continuous random variables, distribution functions, moments,
probability generating function, special distributions: binomial, Poisson, hypergeometric
geometric, negative Binomial, normal, Lognormal, negative exponential, uniform, Gamma and
Chi-square, Beta
Random Vectors
Bivariate and multivariate distributions, multinomial distribution, marginal distributions and
independence
Distributions of Functions of Random Variables
Sums of random variables, Jacobians, the t and F distributions, distributions of order statistics,
expectations of functions of random variables
Limit Theorems
Chebyshev’s inequality and Weak Law of Large Numbers, Strong Law of Large Numbers,
Central Limit Theorem, convergence in distribution and convergence in probability
Conditional Distributions and Expectations
Conditional densities and probability functions; conditional probability and independence;
Conditional expectations.
Estimation
Point estimation; Bayesian estimates, confidence intervals for means and variances, sufficient
statistics, maximum likelihood estimates, properties of maximum likelihood estimates, RaoCramer lower bound
Statistical Hypotheses
Certain best tests, uniformly most powerful tests, likelihood ratio tests, sequential probability
ratio test, Chi-square tests, T and F tests, noncentral F distributions, power of a test statistics,
least squares, simple and multiple regression, analysis of variance
Normal Distribution Theory
The multivariate normal distribution, the distribution of contain quadratic forms, the
independence of certain quadratic forms
REFERENCES
1. First Course in Probability, A, 7/Edition, Sheldon Ross, ISBN-10: 0131856626, ISBN-13:
9780131856622, Publisher: Prentice Hall.
2. Introduction to Mathematical Statistics, Robert V. Hogg, Allen Craig, Joseph W. McKean ,
Prentice Hall; 6th edition, ISBN-10: 0130085073, ISBN-13: 978-0130085078.
Sample Exam Problems
1. A bin of 50 manufactured parts contains three defective parts and 47 nondefective parts. A sample
of six parts is selected from the 50 parts. Selected parts are not replaced. That is, each part can
only be selected once and the sample is a subset of the 50 parts. How many different samples are the
of size six that contain exactly 2 defective parts?
2. In a certain assembly plant, three machines A, B and C make 30%, 45% and 25% respectively, of the
products. It is known from past experience that 2%, 3% and 2% of the products made by each machine
respectively are defective. Now suppose that a finished product is randomly selected. What is the
probability that the product is defective?