Math 373: General Course Outline
Catalog Description

    373. Numerical Analysis I. Introduction to numerical methods with emphasis on algorithms, analysis of algorithms, and computer implementation issues. Solution of nonlinear equations. Numerical differentiation, integration, and interpolation. Solutions to Ordinary Differential Equations
Textbook
    R. Burden and J. Faires, Numerical Analysis, 8th Ed., Brooks/Cole.
Assignments
    Homework assignments in the course consist of both theoretical and computational work. For the computational component, the students should use a language/environment that possesses high level data types so that the students spend more time working with algorithms and not worrying about the details of writing computer code. Matlab and R are good choices (student versions are installed in the undergraduate computing laboratory). Fortran 77/90/95 and C++ with appropriate class libraries can also be used.
Schedule of Lectures

Lecture
Section
Topics
1
.
General course outline and Background for Programming projects.
2
2.1
Introduction to the solution of nonlinear equations and the bisection method.
3
2.1
Convergence estimates for the bisection method. Errors and residuals.
4
2.2
Fixed-Point methods.
5
2.3
Newton's Methods.
6
2.4
Error analysis for Newton-Raphson and the secant method. Rates of convergence.
7
3.1
Interpolation and the Lagrange Polynomial. (Quiz I)
8
3.2
Divided Difference
9
3.1
Polynomial interpolation error estimates. (Error estimate for equispaced nodes*)
10
3.3
Hermite Interpolations.
11
3.4
Cubic Spline Interpolation.
12
3.4
Cubic Spline Interpolation.
13
.
Midterm
14
4.1
Numerical Differentiation.
15
4.1
Numerical Differentiation based on Lagrange interpolating polynomials.
16
4.3
Numerical Integration.
17
4.3, 4.4
Numerical Integration, Newton-Cotes formulas. Composite Integration Formulas.
18
4.6
Adaptive Quadrature Methods.
19
4.7
Gauss Quadrature, derivation of 2 and 3 point formulas. Error Estimates. (Quiz II)
20
5.1
The Elementary Theory of Initial-Value Problems
21
5.2
Derivation of Euler's Method. Definition of Convergence.
22
5.2, 5.3
Error bounds and asymptotic error estimate for Euler's method. Local truncation error, global error. Higher-Order Taylor Methods.
23
5.3, 5.4
Higher-Order Taylor Methods and Runge-Kutta Methods. Derivation of the general second order Runge-Kutta methods.
24
5.4
Derivation of the general second order Runge-Kutta methods.
25
5.6
Liner Multistep methods.
26
5.7
Variable Step-Size Multistep Methods.
27
5.7
Variable Step-Size Multistep Methods.
28
Review.
Comments

* This topic is not in Burden and Faires. It can be found in Cheney-Kincaid, Numerical Mathematics and Computing, Brooks/Cole, section 4.2.

Topics listed above is temporary and may be modified.

Outline update: Y.J. Lee, 9/07
For more information, please contact , leeyoung@math.rutgers.edu.