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Interdisciplinary Ph.D. Program in Computational Analysis and Modeling (CAM)::Qualifying Exams::Numerical Analysis Preliminaries Round-off errors and computer arithmetic, algorithms and convergence Solutions of Equations in One Variables 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 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, 5th ed., PWS-KENT publishing company, 1993. SAMPLE TEST 
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