Overview

Provides the basic numerical tools for understanding more advanced numerical methods. Topics for the course include: Sources and Analysis of Computational Error, Solution of Nonlinear Equations, Interpolation and Other Techniques for Approximating Functions, Numerical Integration and Differentiation, Numerical Solution of Initial and Boundary Value Problems in Ordinary Differential Equations, and Influences of Hardware and Software.

Prerequisites

• MA1115
• MA2121
• Ability to program in MATLAB and MAPLE

Learning Outcomes

A student who successfully completes this course will have developed/acquired the following skills/knowledge:

• Possess a basic working knowledge of floating-point numbers and arithmetic. Understand floating-point round-off and its consequences, including catastrophic cancellation. Be able to define the term “machine epsilon.”
• Understand the difference between absolute and relative error. Know when each is appropriate to use and why.
• Be able to implement basic numerical algorithms for root finding, including fixed-point iterations, Newton’s method, the secant method, and bisection.
• Be able to implement basic piecewise interpolation schemes.
• Be able to implement finite difference schemes for approximating derivatives of functions.
• Be able to implement composite Newton–Cotes quadrature methods, including Riemann sums, the trapezoid rule, and Simpson’s rule.
• Be able to implement various methods for solving initial-value problems (IVPs), including one-step methods (e.g., Euler’s method and Runge–Kutta methods), and multistep methods. Know the differences between one-step and multistep methods and those between implicit and explicit methods and the relative merits of each.
• Implement Lagrange schemes for polynomial interpolation, including barycentric schemes. Know the differences between Lagrange interpolation and piecewise interpolation and their relative merits. Implement Chebyshev interpolation.
• Implement trigonometric interpolation. Understand the connection between trigonometric interpolation and the discrete Fourier transform.
• Develop a basic understanding of orthogonal polynomials and their role in continuous least-squares fitting and Gauss quadrature.
• Implement basic methods for solving boundary-value problems, including finite difference methods and spectral collocation.