- Some disasters attributable to bad numerical computing.
- Application examples. Laplace BVP: SEM electron gun design, molecular electrostatic energy surface (linearized Poisson-Boltzmann equation is modified Helmholtz), Stokes BVP coupled to surface motion: fluid dynamic simulation of millions of vesicles. Helmholtz BVP: radar scattering from aircraft, electron micrograph of photonic crystal. Laplacian eigenmodes: car interior acoustic resonance, quantum chaos.
- Intro to floating-point number system by Cleve Moler (founder of Mathworks).
- Numerical Analysis Digest email discussion group archive.
- Runge phenomenon demo applet from Scott Sarra at Marshall.
- J. P. Boyd, "Fourier and Chebyshev Spectral Methods" (2001, Dover), available here
- D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition (Springer, 1998). [CK]

This has a good summary of scattering theory, potential theory, and numerical methods for scattering. -
W. H. Press, S. A. Teukolsky, W. H. Vetterling, and B. P. Flannery,
*Numerical Recipes in C*, 2nd Edition, available here as PDF files online, also worth buying for your shelf. Good all-round general overview of numerical methods, with lots of practical tips, good intuitive explanations, aimed at users. Not very rigorous, up-to-date or complete on PDEs (sometimes this book annoys numerical analysists, but physicists like it). - Anna-Karin Tornberg's notes on Gaussian quadrature
- Kress '91 paper on Helmholtz scattering, with singular product Martensen-Kussmaul style quadratures described to get spectral accuracy for D + i.eta.S in 2D.
- Marcos Capistran Ocampo's thesis on inverse 2D obstactle scattering, with Yu Chen (NYU, 2003).
- G. Still's paper on Moler-Payne theorem and generalizations, Numer. Math. (1988).
- Time Betcke's thesis (Oxford, 2005), a great tutorial, with many new results, on the MPS for eigenvalue problems.
- My paper with Timo on MPS for interior BVP, using fundamental solutions basis functions: MFS paper. My preprint (SIAM J. Numer. Anal.) on MPS for eigenvalue problems.
- My paper with Andrew Hassell on fast computation of eigenmodes using BIE and the Neumann-to-Dirichlet map.
- Fast Multipole Methods:
- A Short Course on Fast Multipole Methods by Rick Beatson and Leslie Greengard
- P.-G. Martinsson's course notes on fast algorithms, as of March 2011.
- Gumerov course notes
- A Fast Algorithm for Particle Simulations, L. Greengard and V. Rokhlin, J. Comput. Phys. 135, 280-292 (1987). The O(N) algorithm. Has proofs of multipole truncation errors for 2D Laplace kernel. Errata: Equation (2.10) has a divide by zero instead of by l. Also note that their monopole convention is log z not log 1/z, which changes the sign of the last term in (2.10).
- A Fast Adaptive Multipole Algorithm for Particle Simulations by J. Carrier, L. Greengard, and V. Rokhlin, SIAM J. Sci. Stat. Comput. 9, 669 (1988). Quad-tree goes to variable depth while still provably O(N).
- Demmel's course notes on Barnes-Hut treecode, from 1996.
- L. Ying et al, J. Comput. Phys. 196, 591-626 (2004) has a nice review of FMM including in App. B the translation operators for 2D Laplace kernel

- Geoffrey Hemion's notes on complex analysis.
- Paul Garabedian,
*Partial Differential Equations*(1964), is `functional analysis-free', simple and clear, has integral equations, compactness, minimax eigenvalues, Weyl's Law proof. - I. Stakgold,
*Boundary value problems of mathematical physics*, Vols. 1 & 2 (New York, Macmillan, 1967-68; SIAM republished corrected editions under Classics in Appl. Math #29, 2000). Berry/Cook seems to have 3 copies of the 1967 editions. Great general introduction to PDEs, Greens functions, spectral theory, Schwartz distribution theory, potential theory, scattering theory, applications. He also wrote the relevant*Green's Functions and Boundary Value Problems*, 2nd Edition (Wiley, 1998). - F. B. Hildebrand,
*Methods of Applied Mathematics, 2nd edition*(1965). Although old-looking, Chapter 3 on integral equations is a very clear introduction. - My Math 46 Introduction to Applied Math course, good for introduction to integral equations and Fourier transforms. Also see Logan course book used.
- R. Kress,
*Linear Integtral Equations, 2nd Edition*(Applied Mathematical Sciences vol. 82, Springer-Verlag, 1999). Rather formal functional-analytic background, potential theory, integral operators, numerical methods, more detail than we'll need. Excellent. On reserve at Berry/Cook Library. - D. Colton and R. Kress,
*Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition*, (Springer, 1998), is a good summary of (`forward') scattering theory and numerical methods. - D. Colton and R. Kress,
*Integral Equation Methods in Scattering Theory*(Wiley, 1983). Classic on boundary integral equations, formal style which proves everything from the ground up. Berry/Cook has 1 copy. - Scattering theory overview by Tilo Arens.
- Rainer Kress' forward and inverse scattering analysis lecture notes: go to Teil 1-3 under Inverse Scattering.
- Volume of d-dim ball, for Lecture 18.

- Always write on paper what you want to achieve, and then write out a pseduocode, before you start typing.
- Try to put all user-adjustable parameters
**once**at the top of your program (script). Make everything work from these parameters. - Write brief comments to explain what non-obvious lines do.
- Break down tasks which are repeated into
**subroutines**(in MATLAB these are called**functions**). Write out the interface (inputs and outputs) to your function on pencil and paper before you start to code it up. Document your function as you go. - For every function, write a test routine that goes along with it and verifies it does what it is supposed to do, on at least one known test case.

- Here is our department's LaTeX resources.
- If you have a UNIX account (you probably do as a grad student; ask your sysadmin) or linux OS then you can use a standard text editor and the latex command. If you have Mac OSX or Windows you need to install a LaTeX distribution, as described here.
- Please let me know if you are stuck with installing LaTeX. Sarunas Burdulis (our sysadmin, behing the math office) may also be able to help better than I.
- Once you have it installed, here are some simple sample files that you can edit for your homework. They produce PS which you can convert to a PDF file.
- Here is a quick guide to all the math symbols and brackets, etc. You will need to
`\usepackage{amssymb,amsmath}`

to get some of these symbols. - Beamer is a great latex package in which to make slides for your talks.
(Although I still use Prosper).

Here's my toy files:`a.tex`

which uses EPS figure`fig.eps`

. It produces this PDF.

- Get yourself a website account hosted at Dartmouth if you don't already have one (you probably do if you're a grad student; ask your department sysadmin).
- Dave Raggett's simple HTML guide is all you could ever need.
- Don't forget to Reload the page in your browser to check that the updates you have made worked.
- Here
is the simplest webpage I ever made (`View Page Source' on your
browser).
It contains text and one link; you could modify it and replace
the address in the
link by href="figure1.ps" if
`figure1.ps`

is a file in the same directory, for instance a figure file printed to file from Matlab.

- Download the software from Dartmouth (or here); we have 100 or so on-campus licenses.
- Susan A. Schwarz (email her if you have Mac OS 10.2 or earlier for install CDs) can help with installation issues.
- Dartmouth's Academic Computing Center's own Matlab tutorial sessions (sign up for them).
- Bent Petersen's Matlab starter page
- Robert Higdon's nice introductory notes
- My 1-page
`intro46.m`

code, and 1-page intro code from Linear Algebra (does loops). - Guide from Cambridge University Engineering Department.
- Simple intro from Utah, Hany Farid's intro reference, and Gilbert Strang's intro at MIT.
- Self-guided courses from Dartmouth academic computing: Introduction to Matlab, Programming in Matlab, and Introduction to Matlab Graphics