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Introduction to Numerical Analysis

Spring 2016

Instructor: Homer Reid

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Course Overview and Class Schedule

View this document in PDF form: Overview.pdf

Course Mechanics

Lectures: 2:30--4 pm, Tuesday / Thursdays, 2-105
PSets, Exams, etc: Approximately weekly PSets. Two in-class midterms. No final exams. Final project worth 25% of grade.
Grading: 25% homework, 25% midterm 1, 25% midterm 2, 25% final project.
Office hours:
  • 4-6 PM Tuesdays, 2-105
  • 5 PM Wednesdays, 2-449
  • By appointment (email me or stop by my office, 2-232c)

Note: At some point during the first two weeks of class I will hold special office hours for the purpose of ensuring that everybody is up and running with the programming language of your choice. Bring your laptop and we'll work through any issues you might be having. Stay tuned for the time, date, and place.

Course Syllabus

This course is an exploration of the art and science of extracting numbers from mathematical expressions. The material we will cover may be broadly divided into two units.

Unit 1 is all about basic numerical calculus. We will discuss elementary methods for obtaining accurate numerical estimates of integrals, derivatives, and infinite sums. This unit will include discussions of extrapolation, interpolation, root-finding, optimization, and evaluation of special functions. By the end of Unit 1, you will be in a position to implement and understand the properties of the most basic numerical methods for all of these tasks. However, in several places during the course of the elementary treatment of Unit 1, we will encounter phenomena that seem to be hinting at a deeper set of ideas.

This will set the stage for Unit 2 of our course, Fourier analysis and spectral methods. The overarching theme here is that we can often revolutionize the speed and accuracy of a calculation by approximating a function as an expansion in some convenient set of expansion functions -- often a set of orthogonal functions. Our discussion of orthogonal-function expansions will begin, as must any, with the granddaddy of them all: the Fourier series and its immediate descendants (the Fourier transform, Parseval's and related theorems, the FFT, etc.). Then we will broaden the setting to consider more general classes of functions and more general spectral methods: Gaussian quadrature, Chebyshev polynomials and Trefethen's chebfun, Nystrom solution of integral equations, and more.

Throughout Units 1 and 2 we will pepper the discussion with examples drawn from engineering and the sciences, including binding energies of solids, coding and modulation schemes for efficient use of the wireless communications spectrum, spherical Bessel functions for electromagnetic scattering and thermal engineering, and Ewald summation. The course will also include bird's-eye views of topics treated more thoroughly in other courses, such as numerical linear algebra, PDE solvers, and stochastic and Monte-Carlo methods.

Class Schedule

This is a rough schedule of the topics we will discuss and the order in which we will discuss them. The midterm dates will not change, but the order of some topics and the time we allocate to them may shift as the semester progresses.
Date Topic
2/2/2016 Tuesday Course invitation.
2/4/2016 Thursday Evaluation of infinite sums. Numerical integration of functions: Newton-Cotes quadrature. Application to improper integrals.
2/9/2016 Tuesday Heuristic convergence analysis of Newton-Cotes quadrature.
2/11/2016 Thursday Numerical integration of ODEs. Reduction of high-order ODEs to first-order form. Euler's method.
2/18/2016 Thursday Numerical integration of ODEs: Beyond Euler's method. Convergence analysis of the improved Euler method. Runge-Kutta methods. Adaptive stepsizing.
2/23/2016 Tuesday Pathologies in ODE solvers: Stability, stiffness, and implicit methods. Pathologies in ODEs themselves: non-uniqueness and non-existence. Boundary-value problems; shooting methods. The beam equation.
2/25/2016 Thursday Richardson extrapolation. Numerical differentiation: finite-difference stencils. Finite-difference approach to boundary-value problems.
3/1/2016 Tuesday Computer representation of numbers: fixed- and floating-point arithmetic. Exactly representable numbers and rounding errors. Catastrophic loss of floating-point precision.
3/3/2016 Thursday Computer representation of numbers: Accumulation of rounding errors. Numerical stability of forward and backward recurrence relations for special functions. Numerical root-finding: Sample problems and 1D methods.
3/8/2016 Tuesday Convergence of Newton's method. Newton's method in higher dimensions.
3/10/2016 Thursday Unit 1 summary and midterm review.
3/15/2016 Tuesday Midterm 1
3/17/2016 Thursday Monte-Carlo integration
3/29/2016 Tuesday Fourier analysis: The fourfold way. Parseval, Plancherel, and Poisson formulas.
3/31/2016 Thursday Paley-Wiener theorems. Convolution. Fourier series.
4/5/2016 Tuesday Applications of Fourier analysis, Part 1: Ewald summation.
4/7/2016 Thursday Applications of Fourier analysis, Part 2: Rigorous convergence analysis of Newton-Cotes quadrature. Fourier analysis in higher dimensions.
4/12/2016 Tuesday Clenshaw-Curtis quadrature. Discrete Fourier transforms.
4/14/2016 Thursday Applications of Fourier analysis, Part 3: the DFT and its uses. Signal processing, arbitrary-precision arithmetic, discrete convolution, PDE solvers.
4/21/2016 Thursday Final project proposal due. Orthogonal polynomials. Gauss-Legendre quadrature.
4/26/2016 Tuesday Guest lecture on fast Fourier transforms by Professor Steven Johnson, co-author of the industry-standard FFTW software package for speed-optimized FFTs.
4/28/2016 Thursday Midterm 2
5/3/2016 Tuesday Applications of Fourier analysis, Part 4: Modulation. Wireless communications and lock-in amplifiers. Spectral efficiency of telecommunications coding schemes.
5/5/2016 Thursday Integral equations. Nystrom's method.
5/10/2016 Tuesday Numerical linear algebra: dense-direct and sparse-iterative algorithms. Overview of PDE solvers.
5/12/2016 Thursday Numerical nonlinear algebra: Resultants and tensor eigenvalues.

A Note on Computer Programming

The PSets and the final project will require you to write simple computer programs. You may use any programming language you like, but we particularly recommend one of three choices: c (or c++), matlab, or julia.

If you are not sure which language to use, here are some thoughts to help guide your decision.

The c programming language, and its close cousin c++, are the most widely used programming languages in the world. Programming in these languages offers you maximal flexibility in using your computer's resources. c/c++ programs run significantly faster than programs written in matlab, python, or other languages. Moreover, almost every useful piece of software out there in the world that you might want to tie into your software is available in c/c++ form, so this option offers you the widest range of interoperability with existing codes. If you are planning to do much programming or to work with existing codes in your future, I strongly recommend you acquire at some point at least a working familiarity with c/c++. However, if you have not done much or any c/c++ programming, you may experience a rather steep learning curve, and you may not wish to make that time investment for 18.330.

matlab is a common choice for academic numerical programming. It is much easier to get started programming in matlab than in c/c++. Your programs will run much more slowly and will not be able to take full advantage of your computer's resources, but these factors will not be too significant for the programs you need to write in 18.330. The main difficulty with matlab is that it is closed-source software that is almost prohibitively expensive for anybody outside an academic or corporate environment. (An individual copy costs around $3000, and even we MIT faculty and instructors have to pay $100 every year if we want to use it.) This means not only that (a) you may find yourself without access to the tool once you are out of college, but also (b) codes you may write and want to share with others may be of limited utility, because most people in the world don't have matlab.

This brings us to julia, which combines many of the best features of the previous two options. julia is an exciting new language developed over the past several years at MIT. It is as easy to use as matlab (which means it is much easier to get started with numerical programming in julia than in c/c++), but it has the critical advantage of being free, open-source software, so there is no fear of ever losing access to it, nor of wanting to share your codes with someone who doesn't have it herself. It is significantly faster than matlab (again, this is not a crucial concern for 18.330) and has an enthusiastic and rapidly growing base of users and contributors. This means, in particular, that (a) if you request a new feature you have a decent chance of getting somebody to implement it for you, and (b) there are lots of opportunities for you to contribute to this cool new project. (For example, implementing some pieces of numerical software for inclusion in the julia distribution would be a great final project for 18.330.) The main difficulty with julia is that it is quite new and hence a little rough around the edges. Certain things that you would expect to just work in more mature packages may turn out to be incompletely implemented or (alas) buggy in julia. The good news is that the aforementioned enthusiatic developer base is vigilant about addressing bug reports, so any issues that you encounter will probably get fixed quickly; however, using julia has a bit of a scrappy startup / wild-wild-west feel to it which may not appeal to all students.

See below for programming examples and tutorial walkthroughs to get you started with numerical programming in each of these three languages. We will also hold special office hours during the first week of class to make sure you are up and running with the programming language of your choice.

If you are not sure which language to use, you should use julia, and in that case you should refer to the awesome resource Julia for Numerical Computation in MIT Courses maintained by MIT Professor Steven Johnson.

Collaboration policy

You may (and are encouraged to) work together on PSets, but you must write up and turn in your own solutions. For problems involving computer programs, you must submit a listing of your program and its output together with your PSet. If you work with other students to write your program, each student in your study group must separately type in and execute the program to generate a listing submitted with the PSet. This is to ensure that you become familiar with the mechanics and semantics (semicolons, parentheses, etc.) of programming.

Lecture Notes

Last Updated Topic julia code
2/04/2014 Invitation to 18.330 and Numerical Analysis GetIntegrationError.jl
2/06/2014 Convergence of Infinite Sums
2/18/2015 Numerical Integration, Part 1
2/13/2014 Ordinary differential equations PlanetaryMotion.jl
2/25/2014 Numerical differentiation
2/25/2014 Boundary value problems SolveBeamEquation.jl
2/27/2014 Richardson Extrapolation
3/1/2016 Machine Arithmetic RecursiveSum.jl
3/14/2014 Numerical Root-Finding PlotNewtonConvergence.jl
3/11/2014 Quick note on convergence terminology
3/20/2014 Monte Carlo Integration
4/8/2014 Fourier Analysis
4/03/2014 Modulation: Wireless Communications and Lock-in Amplifiers
4/10/2014 Ewald Summation Ewald1D.jl
4/27/2015 Clenshaw-Curtis Quadrature
4/24/2014 The Discrete Fourier Transform
4/29/2014 Chebyshev Spectral Methods
5/1/2014 Orthogonal Polynomials

Problem Sets

See the 18.330 PSet submission guidelines.

Date Due Problem Set Solutions
2/11/2016 PSet 1 PSet 1 Solutions
2/18/2016 PSet 2 PSet 2 Solutions
2/25/2016 PSet 3 PSet 3 solutions
3/03/2016 PSet 4 PSet 4 solutions
3/10/2016 PSet 5 PSet 5 solutions
3/17/2016 PSet 6 PSet 6 solutions
4/07/2016 PSet 7 PSet 7 solutions
4/14/2016 PSet 8 PSet 8 solutions
4/21/2016 PSet 9 PSet 9 solutions
5/13/2016 (Optional) PSet 10

Practice Exams

Practice Midterm 1
Practice Problems for Midterm 2

Note: I don't distribute written solutions of exam problems, but the answers to all parts of all questions may be found distributed throughout the lecture notes and the written PSet solutions.

Programming Examples

Other References

There is no assigned textbook for this course, but you may find the following references helpful. The links will direct you to online versions (in some cases you will need MIT certificates to access them).

About the Final Project

The final project for 18.330 is an exploration and implementation of a numerical algorithm of your choice. Typically this will be algorithm that we did not cover in class or an extended/improved version of an algorithm that we did cover, although these are not the only choices.

You have considerable leeway in choosing a topic of interest to you and structuring your report in a way that is appropriate for the topic you choose, but your project must incorporate all of the following elements:

A rough guideline for the magnitude of the writeup is that it should consist of 15--20 LaTeX pages (including plots but not including code listings).

The final project proposal

Before getting started on your final project, you must submit a brief proposal to let me know what you are planning to do and how you envision structuring your final report. This proposal is due by April 21, 2016 (you may email it to me directly or submit it in class). Below is a rough sample of what a reasonable proposal might look like.

Practical Evaluation of Lattice Sums: Is Ewald Summation Always Best?

Many problems in computational science require efficient and accurate evaluation of lattice sums---that is, sums that accumulate contributions to some physical quantity from sites in a one-, two-, or three-dimensional lattice. Brute-force evaluation of such sums is generally prohibitively expensive, and this fact has spurred the development of more sophisticated and efficient methods. In class we covered one such method---Ewald summation---but we discussed only the electrostatic (zero-frequency) case, and then only for the cases of one- and two-dimensional lattices. My final project will extend this discussion to a broader comparison of lattice-summation methods, including consideration of three-dimensional lattices and nonzero-frequency problems.

My paper will begin with a brief introduction (2--4 pages) to the problem. I will then discuss (3--5 pages) the history of lattice-summation methods, touching in particular on the original contributions of Paul Ewald himself and the physical problems that motivated his development of the Ewald-summation technique, but considering also the evolution of techniques spurred by the computer-aided-design (CAD) revolution of the late 20th century. Then I will explain (5-8 pages) the mathematical underpinnings of several sophisticated lattice-summation methods, including (1) Ewald summation, (2) Kummer decomposition, and (3) integral transforms. Finally, for the specific problem of a 2D lattice sum at nonzero frequency I will implement Ewald summation and Kummer decomposition and compare (3--5 pages) the accuracy and efficiency of these two methods.

How the final project is graded

Your most important task in the final project is to get your implementation of the algorithm(s) working and produce convincing numerical results (plots, data tables, etc.) that demonstrate that your implementation is successful. Thorough discussions of background, theory, and implementation are also important, but even the most thorough treatment of these topics will not not compensate for incomplete implementation or a failure to report results produced by your original implementation. The project is closer in spirit to a research paper than to a book report.

A rough grading rubric for the final project looks something like this.

If your project report contains Then your grade can be as high as
  • A code listing of your implementation
  • A thorough presentation (plots, data tables, etc.) of results produced by your code
  • A thorough discussion of background, theory, and implementation technicalities
  • Code listing
  • Thorough presentation of results
  • Sketchy discussion of background theory
  • Code listing
  • Sketchy presentation of results
  • Thorough discussion of background theory
  • Little or no presentation of results produced by your code
  • Lengthy discussion of background theory which you attempt to augment with lots of pictures copied from textbooks or websites

Some Possible Final Projects

Here are a number of possible final-project topics, with content spanning a variety of interesting calculations arising in various subfields of science and engineering.

Needless to say, this is not intended to be an exhaustive list, and in no way are you required to choose one of these topics for your project.

However, because these are all calculations that I have implemented myself over the years---usually just for my own fun and education, with no pretensions of competing with the pros (which means I carried them out to roughly the level of completeness and sophistication I expect from your final project)---they all have the advantage that I can work with you one-on-one at the beginning to help you get started and map out precisely what you will need to do. This may allow you to proceed slightly further down the road toward implementing a substantial numerical calculation than you might otherwise be able to travel on your own.

Electrical Engineering

Write a small-scale DC simulator for nonlinear electronic circuits. The input to your program will be a simple text file describing a circuit consisting of resistors, capacitors, inductors, transistors, diodes, and voltage sources. Your program will compute the DC operating point of the circuit and report (1) the DC voltage at each circuit node, and (2) the DC current through each circuit branch. (You will use Newton's method of root-finding, which we briefly studied in the first unit of the course.)

Electrical Engineering / Physics

Write a small-scale implementation of the boundary-element method of computational electrostatics to compute the capacitances of arbitrary-shaped objects. Given input files representing the surfaces of two conductors discretized into little triangles (like this), your program will compute and report the self- and mutual capacitances of the conductors. You will test your code by using it to reproduce the (analytically derivable) result for the capacitance between two spheres, then apply it to cases that are not analytically tractable---such as, for example, bus lines in microprocessors whose parasitic capacitance limits maximum clock frequencies.

Chemistry / Physics

Write a program that numerically solves the Hartree-Fock equations to compute the binding energy of atoms and small molecules. Your program will input a set of numbers describing, for a given atom or small molecule, the matrix elements of the one- and two-electron terms in the Hamiltonian. Your program will output the Hartree-Fock binding energy of the atom or molecule. You will use your program to compute curves of binding energy vs. bond length for some simple diatomic or triatomic molecules such as nitrogren and ozone; the minima of these curves will be the predictions of your code for the bond length of these molecules.

Here is a brief memo that I have put together on how the numerical Hartree-Fock procedure works. Note that I will supply you a code for computing the inputs to your Hartree-Fock calculation for an arbitrary molecule (that is, the one- and two-electron Hamiltonian matrix elements for a given molecule, with given bond lengths and angles, described by a given Gaussian basis set). Here is the binary data file referred to in the Example section of the memo: H2.sto-3g.hdf5.

Astrophysics / Gravitational Physics

Write a program that uses ODE stepper techniques to trace the trajectories of test particles (geodesics) moving in the presence of black holes. Given the initial position and velocity of a particle in a black-hole spacetime (either Schwarzschild or AdS), compute the (proper) time required for the particle to fall into the black hole. Using experimental data on the black hole at the center of our Milky Way galaxy, compute the time (in years) for the Earth to fall into this black hole.

Statistical Physics

Write a program that uses Monte-Carlo methods (the Metropolis / simulated-annealing algorithm or more sophisticated algorithms) to compute the critical temperature of the square-lattice Ising ferromagnet in 2 and/or 3 dimensions. Compare the results of your code to existing analytical results in 2D and to existing numerical results in 3D. Your report will include pictures showing typical configurations of the spins at temperatures well above, well below, and in the vicinity of the critical point.

Computer science / cryptography

Research the Lenstra-Lenstra-Lovasz (LLL) lattice-basis reduction algorithm and its use in proving the fundamental insecurity of knapsack cryptography. You may either (1) write your own simple implementation of the LLL algorithm and evaluate its performance on some relevant test cases, or (2) write code that calls a pre-existing implementation of the LLL algorithm to implement the steps an unscrupulous eavesdropper would take to decipher a communication that has been supposedly ``encrypted'' using knapsack cryptography.

Here are some references on the LLL algorithm (available electronically via MIT libraries if you have MIT certificates):

Pure mathematics

Write a program that computes nontrivial zeros of the Riemann zeta function ζ(z). Your code should input a height t off the critical line and determine---to arbitrary user-specified accuracy---the location of the nearest zero lying on the critical line at a height greater than or equal to t from the real axis. Use your code to determine the locations of the first 100 nontrivial zeros and compare their spacings to the predictions of random-matrix theory.

18.330 Spring 2016 Main course page