COURSE NUMBER AND TITLE: MATH 4350 Numerical Analysis

CREDIT HOURS: 3

CATALOG DESCRIPTION: A study of non-linear equations, numerical integration and differentiation and numerical solution of initial value problems in ordinary differential equations.

PREREQUISITE(S): CSCI 1301 or CSCI 2060, and MATH 3020, or permission of instructor.

SUGGESTED TEXT(S): Numerical Methods, by Faires and Burden

COURSE OUTLINE:

1. Mathematical Preliminaries and Error Analysis
   • Review of Calculus
   • Round-off Error and Computer Arithmetic
   • Errors in Scientific Computation
2. Solutions of Equations of One Variable
   • The Bisection Method
   • The Secant Method
   • Newton’s Method
   • Error Analysis and Accelerating Convergence
   • Muller’s Method
3. Interpolation and Polynomial Approximation
   • Lagrange Polynomials
   • Divided Differences
   • Hermite Interpolation
   • Spline Interpolation
   • Parametric Curves
4. Approximation Theory
   • Discrete Least Squares Approximation
   • Continuous Least Squares Approximation
   • Chebyshev Polynomials
   • Rational Function Approximation
   • Trigonometric Polynomial Approximation
   • Fast Fourier Transforms
5. Numerical Integration and Differentiation
   • Basic Quadrature Rules
   • Composite Quadrature Rules
   • Romberg Integration
   • Gaussian Quadrature
   • Adaptive Quadrature
   • Multiple Integrals
   • Improper Integrals
   • Numerical Differentiation
6. Numerical Solution of Initial-Value Problems
   • Taylor Methods
   • Runge-Kutta Methods
   • Predictor-Corrector Methods
   • Extrapolation Methods
   • Methods for Systems of Equations
7. Direct Methods for Solving Linear Systems
   • Gaussian Elimination
   • Pivoting Strategies
   • Linear Algebra and Matrix Inversion
   • Matrix Factorization
8. Iterative Methods for solving Linear Systems
   • Eigenvalues and Eigenvectors
   • The Jacobi and Gauss-Seidel Methods
   • The SOR Method
9. Solutions of Systems of Nonlinear Equations
   • Newton’s Method for Systems
   • Quasi-Newton Methods
   • The Steepest Descent Method