Introduction to Numerical Analysis

Upon completion of MATH 3316 the student should be able to:

• understand decimal, binary and floating point representation of numbers
• understand error propagation and how to estimate numerical error
• implement methods to solve nonlinear equations including, but not limited to, bisection method, secant method, fixed point iteration and Newton’s method
• understand how to accelerate convergence in different solution methods for nonlinear equations
• define and determine rate of convergence for solution methods to nonlinear equations
• solve directly a system of linear equations using Gaussian elimination
• solve linear systems numerically using matrix factorization, partial pivoting and matrix inverse
• understand computational complexity of direct methods for Gaussian elimination
• define the norm, determinant and condition number of a matrix
• solve linear systems numerically using iterative methods
• understand how iterative methods for solving linear systems differ from direct methods for solving linear systems
• recognize eigenvalue problems and understand the issues involved in obtaining eigenvalues numerically
• understand interpolating polynomials: Lagrange form and error formula
• implement spline interpolation
• understand the concept of trigonometric interpolation and Fourier series and Chebyshev polynomials
• use the method of least squares to deal with inconsistent linear systems
• use the QR factorization to solve least squares problems
• understand and implement numerical differentiation techniques including finite differences and Richardson extrapolation
• understand and implement numerical quadrature, using methods such as Romberg integration and composite rules
• solve numerically initial value problems using Euler’s method and Runge-Kutta methods
• use and understand the concepts of convergence, stability and stiffness when solving initial value problems numerically
• solve numerically systems of ordinary differential equations
• optional: Discrete Fourier representation and transforms, Singular Value Decomposition