# MATH-T 640, Analysis of Numerical Methods II

### Course Description

This is the second part of a two-semester sequence of numerical analysis courses. Topics include approximation theory (e.g., least squares approximations); eigenvalue problems; numerical solutions of nonlinear systems of equations; initial-value problems and boundary-value problems for ordinary differential equations; numerical solutions to partial differential equations; stability of methods.

### Learning Outcomes

Upon completion of this course, students will be able to:

• Numerically solve nonlinear systems by Newton's method, quasi-Newton method, and the steepest Descent Techniques.
• Show that an initial value problem has a unique solution.
• Numerically solve initial value problems for ordinary differential equations by Euler's method, Runge-Kutta methods, and multistep methods.
• Perform stability analysis of numerical methods for initial value problems.
• Numerically solve boundary value problems for ordinary differential equations by the shooting method, Finite difference methods, and the Rayleigh-Ritz methods.
• Approximate eigenvalues and eigenvectors of a matrix by the power method, Householder's method, and the QR method.
• Categorize partial differential equations into elliptic, parabolic and hyperbolic partial differential equations.
• Numerically solve partial differential equations by finite difference methods, and finite element methods.
• Use error bounds to compute errors for above numerical methods.
• Derive error bounds for above numerical methods and use these error bounds to estimate errors.
• Write numerical programs, such as Python, C++, FORTRAN or Mathematica programs, to solve the above problems.

### Learning Materials

Numerical Analysis, 10th edition, by Richard L. Burden and J. Douglas Faires

### Assessment

13 Homework Assignments: 52%
15 Weekly Reflection: 3%
One Midterm: 15%
One Final Exam: 30%