The exam is based on APPM 5600-5610
- K. Atkinson, Introduction to Numerical Analysis (except Chapter 1).
- G. Golub and C. Van Loan, Matrix Computations, Chapters 2-5, 7, 10.
- K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations,Chapters 2.2, 2.4, 2.6-2.9, 3.1, 3.2, 4.2, 5.1-5.5.
Recommended Supplemental Text
- J. Stoer and R. Bulirsch, Introduction to Numerical Analysis.
The following topics are covered in APPM 5600-5610.
The prelim does NOT cover any additional APPM 6610 topics.
- General aspects of polynomial interpolation theory.
- Formulations in different basis, e.g. Lagrange, Newton etc. and their approximation and computational properties (convergence, error bounds, conditioning, complexity, forward and backward error analysis etc.).
- Hermite interpolation and its properties.
- Piecewise polynomial interpolation and spline interpolation.
- Trigonometric interpolation, Fourier series, DFT and FFT.
Approximation of Functions
- The Weierstrass Theorem and Taylor's Theorem
- The minimax approximation problem
- The least squares approximation problem
- Orthogonal polynomials and their properties
Rootfinding for Nonlinear Equations
- Properties and formulations of basic rootfinding methods, e.g. the bisectionmethod, Newton's method and the secant method.
- Existence, uniqueness and convergence of one-step iteration methods.
- Aitken extrapolation for linearly convergent sequences.
- Systems of nonlinear equations.
- Newton's method for nonlinear systems.
- Basic unconstrained optimization.
- Properties and formulation of Newton-Cotes integration formulae, e.g. the trapezoidal rule and Simpson's rule.
- Properties and construction of Gaussian quadrature.
- Asymptotic error formulae for quadrature and their applications.
- Eigenvalues and canonical forms / factorizations of matrices.
- Vector and matrix norms, condition numbers.
- Sherman-Morrison type formulas.
Numerical Solution of Systems of Linear Equations, Direct methods
- Gaussian elimination, formulations, analysis and variations (pivoting strategies etc.).
- Forward and backward error analysis.
- Solution techniques for least squares problems.
Numerical Solution of Systems of Linear Equations, Iterative Methods
- Formulation and analysis of basic stationary methods e.g. Gauss-Jacobi, Gauss-Seidel.
- The numerical solution of Poisson's equation.
- The conjugate gradient method.
The Matrix Eigenvalue Problem
- Eigenvalue location, error, perturbation, and stability results.
- Formulation and analysis of the power method and its variations.
- Orthogonal transformations of Householder and Givens type.
- Properties of the eigenvalues of a symmetric tridiagonal matrix.
- The QR Method for eigenvalues and eigenvectors.
Numerical Methods for Ordinary Differential Equations
- Existence, uniqueness, and stability theory.
- Derivation and stability and convergence analysis of linear multistep methods (e.g. Euler’s, midpoint and trapezoidal method).
- Derivation and analysis of single-step and Runge-Kutta methods.
- Formulation of predictor corrector methods.
- Discretization of boundary value problems.
Introduction to Linear Parabolic and Hyperbolic PDEs
- Understanding of convergence, stability and consistency and their connection through the Lax Equivalence Theorem.
- Techniques for analyzing periodic finite difference methods including von Neumann, energy and CFL / domain of dependence techniques.
- Finite difference methods for parabolic problems in one space dimension: analysis of stability, convergence and consistency of explicit and implicit formulations.
- Methods and analysis techniques for Hyperbolic problems in one space dimension.