MATH4500/6500: Numerical Analysis I

The course is designed to introduce the topics associated with numerical methods for mathematical problems in Calculus and Linear Algebra. We will start to understand how computers recognize numbers and do arithmetic in solving mathematical problems. We will consider methods for finding approximate numerical solutions to a variety of mathematical problems, featuring careful error analysis. A mathematical software package will be used to implement iterative techniques for nonlinear equations, polynomial interpolation, integration, and problems in linear algebra such as matrix inversion, eigenvalues, and eigenvectors.

Lecture Notes

Note 1.1: Review of Calculus (Post)
Note 1.2: Round-off Errors and Computer Arithmetic (Post)
Note 1.3: Algorithms and Convergence (Post)
Note 2.1: The Bisection Method (Post)
Note 2.2: Fixed-Point Iteration (Post)
Note 2.3: Newton’s Method and Its Extensions (Post)
Note 3.1: Interpolating and the Lagrange Polynomial (Post)
Note 3.3: Divided Differences (Post)
Note 3.4: Hermite Interpolation (Post)
Note 3.5: Cubic Spline Interpolation (Post)
Note 4.1: Numerical Differentiation (Post)
Note 4.3: Elements of Numerical Integration (Post)
Note 4.4: Composite Numerical Integration (Post)
Note 4.7: Gaussian Quadrature (Post)

Computer Codes

MATLAB Code 1: The Bisection Method / Fixed-Point Iteration
MATLAB Code 2: Newton's Method / The Secant Method
MATLAB Code 3: Interpolating and the Lagrange Polynomial / Divided Differences
MATLAB Code 4: Hermite Interpolation / Cubic Spline Interpolation
MATLAB Code 5: Elements of Numerical Integration / Composite Numerical Integration / Gaussian Quadrature

Final Project Topics

The Jacobi and Gauss-Seidel Iterative Techniques
The Power Method
Finite-Difference Methods for Linear Problems
Finite Difference Methods for Black-Scholes
Euler's and High-Order Taylor Methods
Steepest Descent Techniques
The Conjugate Gradient Method