Numerical Analysis Fall 2006, Math 165 (http://www.math.hmc.edu/math165) TTh 2:45-4:00 pm, Olin B143 (math computing lab) Prof. Darryl Yong (dyong@hmc.edu), Olin 1265, x72844 Office Hours: Wednesdays, 1-4 p.m. (open door policy) Grader: Matt Williams

Course description

Math 165 is an introduction to the analysis and implementation of basic numerical techniques. This is a course meant for anyone who will ever have to solve a mathematical problem with a computer. In this course, you will learn about fundamental issues that arise when searching for numerical solutions to equations, and whether you should trust your answer.

By the end of this course, I hope that you will be able to

• analyze the accuracy, stability, and efficiency of numerical algorithms, and
• use that information to help you select an appropriate numerical technique for a given problem.

Topics covered: root-finding, interpolation and polynomial approximation, numerical differentiation and integration, initial value problems and ordinary differential equations, direct and iterative methods for solving linear systems of equations.

Notes, handouts and suggested texts

• Burden and Faires, Numerical Analysis, 8th ed, Thomson/Brooks Cole (ISBN 0534392008)
• Gerald and Wheatley, Applied Numerical Analysis, 7th ed, Addison Wesley (ISBN 0321133048)
• Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, free online text at http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/pdetext.html.
  For convenience, here's the entire book as one PDF file.

• Computer arithmetic: Aug 29
• Solving nonlinear equations: Aug 31, Sep 5, MyNewton.m, f.m, fp.m, Proof of superlinear convergence of secant method, convergence_of_root_finding_methods.nb
• Interpolating, approximating, fitting: Sep 7, Sep 12, approximating_functions.nb, Sep 14
• Numerical integration: Sep 14, newton_cotes_formulas.nb, Sep 19
• Numerical differentiation: Sep 21, Sep 26, fddemo.nb, fddemo.m, Sep 28, spectralderiv.m
• Ordinary differential equations (IVPs): Oct 3, Oct 5, adams_deriv.nb, eulerdemo.m, ab4demo.m
  Oct 10 backeulerdemo.m, pendulumdemo.m
  Oct 12, bdfdemo.m
  Oct 24, pendulum.m, pendulum.nb
• Ordinary differential equations (BVPs): Oct 26, shootval.m, Oct 31, bvpexample.m, Nov 2
• Solving linear systems: Nov 7, poisson.m, Nov 9, Nov 14, poisson2.m
  Nov 16, steepdesc.m, conjgrad.m
• PDEs--method of lines for heat equation: Nov 28, molheat.m, molheatode1.m, molheatode2.m, method_of_lines.nb
  PDEs--finite difference methods for heat equation: Nov 30, heat1.m, heat2.m
  PDEs--finite difference methods for wave equation: Dec 5, wave1.m

Computing resources

• http://www.math.hmc.edu/computing -- everything you need to know about computing in the math dept
• http://www.math.hmc.edu/computing/support/tex/ -- lots of help on LaTeX
• LaTeX for PC try MiKTeX, LaTeX for Mac try TeXShop
• Wikibook on LaTeX

9/5/2006

Matlab: .m files can either be scripts (lists of commands, executed sequentially) or functions; [] and colon notation for matrices; format long to get more decimal places in answers; use of .* .^ for component-wise operations
Mathematica: To execute commands, remember to press shift-enter

9/7/2006

Matlab: To get help on commands, try "doc" or "lookfor [the thing you're looking for]"
Mathematica: To get help on commands, use help browser. Often you can find what you need using the "Master Index."
Matlab: You can have multiple function definitions in one .m file, but only the first one gets called (the name of the function matches the name of the .m file), and the other functions in that file are only accessible to that function at the top.

9/12/2006

LaTeX: vector-based graphics files (such as .eps) are preferable to raster-based graphics files (such as .jpg or .gif)
Matlab: Basic plotting commands; use "print -depsc2 filename.eps" or "File...Export" from figure's menu to export the current figure as an .eps file
Mathematica: Basic plotting commands; use Export["filename.eps", object to be exported, "eps"] to export a graphics object as an .eps file
LaTeX: Use the graphicx package to insert figures in your documents; .eps graphics are used when compiling using "latex", which generates a .dvi file; .edf graphics are used when compiling using "pdflatex", which generated a .pdf file directly from a .tex file; use the Unix command "epstopdf" to convert your .eps graphics into .pdf graphics.

9/14/2006

Mathematica: substitution rules (eg: a^2 /. a->1), Solve, FindRoot

9/21/2006

Mathematica: Using PrecisionPlot.m to use arbitrary precision arithmetic for plotting; Fit for finding interpolants and fitted curves
Matlab: polyfit and polyval for finding polynomial interpolants and evaluating them

10/5/2006

Mathematica: How to chop off tiny imaginary parts in floating-point calculations; reminder on how to use the output of Solve 2006-10-05tips.nb

10/24/2006

Matlab and Mathematica: How to use built-in ODE numerical solvers pendulum.m, pendulum.nb

Assignments and grading

Most of your learning will occur while completing your homework assignments, so take them seriously, and complete them thoroughly. Your homework should show clearly your solution processes. All code should be properly commented, graphs should be annotated. I encourage you to describe your solution process in words. Poor presentation may result in loss of credit.

All homeworks should be written up using LaTeX. Homeworks can be electronically submitted.

No late homeworks will be accepted except for family or medical emergencies. Your lowest homework grade will be dropped.

You are encouraged to work cooperatively on your homework assignments with your classmates. However, every student MUST write up his/her own homework separately. In addition, you must cite any sources of help that you use. If you work with one of your classmates on a problem, be sure to acknowledge that person in your homework write-up; if you use any published source besides our textbook, acknowledge that too. HMC's honor code is in effect for all students in this course.