Prerequisites: MATH 2205 and COSC 1030

Catalog Description: Machine arithmetic, analysis of rounding errors, direct and iterative methods for linear systems of equations, iteration, interpolation, numerical differentiation and integration, numerical solution of differential equations. Programming exercises using a procedural language. Offered spring semester only. Cross listed with COSC 4340.

Objectives: Math 4340 is an undergraduate course for applied numerical analysis. The objective of this course is to equip the student with the fundamental numerical tools to solve practical problems arising from engineering or physical sciences. This course will provide engineering or science majors with a basic knowledge of numerical methods including: root-finding, elementary numerical linear algebra, solving systems of linear equations, curve fitting, optimization, and numerical solution to ordinary and partial differential equations. MATLAB is the software environment used for implementation and application of these numerical methods. The numerical techniques learned in this course enable students to work with mathematical models of technology and systems. This course is intended to be a mathematical introduction to the theory and practical use of basic numerical methods. While the emphasis of the course is placed solidly on applications, we will discuss some of the mathematical theory behind the methods we study. Some theoretical understanding is critical to the proper practice of numerical analysis, because novel numerical methods are developed based on a solid grab of mathematical theories.

Course Content:

• Review of pertinent concepts from calculus.
• Computer arithmetic and round-off errors.
• Linear algebraic systems (Gauss-Jordan elimination, LU decomposition, iterative methods).
• Eigenvalues and eigenvectors (Power methods, Jacobi's method, QR method).
• Interpolation and curve fitting (Lagrange interpolation polynomials, cubic splines, least squares).
• Root finding (bisection method, Newton's method, contraction mapping method).
• Numerical differentiation and integration (Richardson's extrapolation, Newton-Cotes formulae for integration, Romberg integration, Gaussian quadrature).
• Optimization (line searches, steepest descent method, conjugate-gradient method).
• Initial-value problems for ordinary differential equations (Euler's method, higher-order Taylor methods, Runge-Kutta methods).

Basic concepts:

• A 33-page Practical Introduction to Matlab, by Mark Gockenbach (Michigan Technological University); the same Introduction, in html format.
• A 16-page Matlab Tutorial by Peter Blossey (University of Washington).
• A 8-page Introduction to Plotting with Matlab from the University of Washington.
• A very detailed (125 pages) Matlab Overview by Ed Overman (Ohio State University).
• A 35-page Matlab Primer by Kermit Sigmon (University of Florida).
• University of Florida (General Introduction)
• Michigan Tech (A Practical Introduction to Matlab)
• University of Utah (Matlab Tutorial)
• University of New Hampshire (Matlab Tutorial)
• University of Colorado (Linear Algebra Using Matlab)
• University of Colorado (Introduction to Linear Algeba in Matlab)

• Materials for Linear Algebra: Matlab
• Solving ODE
• Higher order ODEs and Systems
• Solving Elliptic PDEs with the PDE Toolbox

Homework: Homework and/or programming projects will be given approximately bi-weekly and posted on the course website. Collaboration on homework is allowed and encouraged but copying from another person is prohibited.

• Homework 1 (due February 1): 1.1, 1.2(b), 1.5(b), 1.9, 1.13, 1.14, 1.18 (also plot P(x) for a=1,b=-2, and c=1).
• Homework 2 (due February 17): 2.3, 2.6, 2.11, 2.12, 2.17, 2.18, 2.22
• Homework 3 (due March 9): 4.3, 4.6, 4,7, 4,13, and the problems assigned in class.
• Homework 4 (due April 26): 7.3, 7.4, 7.10, 7.11, 7.18, 7.19, 8.3, 8.5, 8.6

Final grade: Based on homework and exams.