CAM Ph.D. Qualifying Exam Contents


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, stability

  • Methods for Solving Linear Systems

    Gaussian elimination (including pivoting strategies), matrix factorization, iterative techniques (including Jacobi's, GS, SOR) and error estimate, conjugate graduate method

  • 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, 8th ed., Thomson Brooks/Cole, 2005.

    SAMPLE TEST

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