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I will be teaching a one-semester course on numerical methods at a liberal arts college. The students will be primarily math, physics, and engineering majors. Note that there is no computer science department at this institution.

This is my first time teaching such a course and I rarely use numerical methods in my research. What are some of the most essential topics that should appear in the course?

I would like to treat the course as a baby introduction to programming using Python. There is no assumption that the students will have had previous programming experience. The only prerequisite is Calculus II.

I should probably include the following topics, as they are mentioned in the course catalog:

  • round-off errors
  • computer arithmetic with algorithm and convergence
  • solutions of equations in one variable with polynomial approximation
  • numerical differential equations
  • linear systems of equations

I’ve chosen Hamming’s Numerical Methods for Scientists and Engineers as the text, as it is quite inexpensive and has excellent reviews. It’s very thick though, and includes far more than I could cover in a single semester!

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    $\begingroup$ To follow up on @Wrzlprmft's questions, has this course been taught in the past? Can you talk to previous instructors? Is there a syllabus? What textbook? "Numerical Methods" could mean a lot of things, from "Let's play with mathematica" to a computationally heavy PDE course. $\endgroup$ – Steven Gubkin Jun 11 '15 at 21:51
  • $\begingroup$ Thanks for the comments! I've clarified my question. $\endgroup$ – Doug Torrance Jun 11 '15 at 21:57
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Not in any order of importance (but probably as outline):

  • introduction to programming languages (Python how does it work)
  • modeling and use of data (types)
  • Numerical evaluation (computable numbers, approximations and round off errors )
  • Resolution of equations (Bisection, Newton Raphson, open and closed methods convergence criteria)
  • Numerical integration ( Simpson and trapezoid rule)
  • Numerical differentiation
  • Matrix computations (determinant , stiffness)
  • ODEs
  • System of ODE
  • PDE
    • individual topics and further reading (high performance computing, pandas...)
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Introduction to Approximation: Approximation, Error. Decimal rounding. Sampling. Differentiation, Integration.

Numerical Algorithms Bisection Method. Newton Raphson Method. Gaussian Elimination with Partial Pivoting. Trapezoidal Rule. Euler Method.

Taylor Series Taylor Series in several variables. Propogation of Errors. Differential Equations.

Curve Fitting Polynomial Interpolation. Least Squares. Weierstrass Approximation Theorem. Boundary Layer Regularisation.

Normal Approximation Histograms. Binomial and Poisson Distributions.

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