The exam is based on APPM 5600-5610

## Texts

• K. AtkinsonIntroduction to Numerical Analysis (except Chapter 1).
• G. Golub and C. Van LoanMatrix Computations, Chapters 2-5, 7, 10.
• K. W. Morton and D. F. MayersNumerical 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. BulirschIntroduction to Numerical Analysis.

## Topics

The following topics are covered in APPM 5600-5610.
The prelim does NOT cover any additional APPM 6610 topics.

### Interpolation Theory

• 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.

### Numerical Integration

• 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.

### Linear Algebra

• 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 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.