M471

M471 Introduction to Numerical Methods (scheduled)
Winter 2020
Instructor: Dr. Shixu Meng
Office: East Hall 3060
When:

Section 001: MoWeFr 8:00AM – 9:00AM

Section 002: MoWeFr 9:00AM – 10:00AM

Where: 1084 EH
Office Hours: MoWe: 10:00 AM-11:30 AM

Course Description: This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications.  Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming.

Textbook: Richard L. Burden and J. Douglas Faires, Numerical Analysis, 10th Edition.

Grades are based on homework, computer projects, mid and final exams.

Main Topics:

  • Preliminaries of Computing
    • Basic concepts: round-off errors, floating point arithmetic, Convergence.
  • Numerical solution of Nonlinear Equations
    • Bisection method, fixed-point iteration, Newton’s method.
    • Error analysis for Iterative Methods.
    • Computing roots of polynomials*.
  • Interpolation and Polynomial Approximation
    • Lagrange Polynomial.
    • Divided Differences.
    • Hermite Interpolation and Cubic Spline Interpolation.
  • Numerical differentiation and integration
    • Numerical differentiation.
    • Numerical integration (Quadrature rules).
  • Initial-Value Problems for ODEs
    • Euler’s, Taylor, Runge-Kutta, and multistep methods, Stability.
  • Numerical linear algebra
    • Direct methods for solving linear systems.
    • Iterative methods.
  • Approximation theory
    • Least square approximation*.
  • Approximating Eigenvalues
    • Power method, Householder’s method*.
  • BVP for ODEs
    • Shooting methods*.

*: if time permits