Dear Students,
We are very much looking forward to seeing you all here at the beginning of the new academic year in August. Many of you have asked about background reading to help prepare for the core classes. Here are some ideas.
(i) Linear algebra. You can never know too much linear algebra! In the past we have recommended taking a text-book such as Strang’s “Linear Algebra and its Applications” and reading it from beginning to end. The book is about matrices and also a good read with a flavour of its own. A more theoretical approach can be achieved with a text such as “Linear Algebra” by Friedberg, Insel, and Spence. This is a good text and is used at the University of Arizona for an undergraduate senior level linear algebra course (which covers most of the "unstarred" section in the textbook). Overall, you should be familiar with matrix and determinant operations; the basics of vector spaces and linear transformations; eigenvalues and eigenvectors; inner products; and elementary Jordan forms. If you only review one thing this summer, it should be linear algebra. This will help you with all three core courses.
(ii) Methods. A very useful general source of background material is Kreyszig’s “Advanced Engineering Mathematics” It covers just about all the basics, and you can use it to brush up on basic complex variables, elementary ideas about partial differential equations, Fourier series, review vector calculus, optimization, probability, etc. However, it is an undergraduate level text and the Methods core course (Math 583) is taught at a much more sophisticated level that also ties in with material taught in the Math 527 and 575 core courses. Nonetheless, it is probably worth owning a copy of Kreyszig (or an equivalent text) as a general reference. Complex variables are an important theme in the fall and a portion of the first semester will be spent reviewing the key ideas, in particular the techniques of contour integration. If you have not taken a course in complex variables (or have forgotten it!) you should review the basics. The chapters in Kreyszig are a good place to start doing this. Kreyszig is also a good source to prepare for discussions of optimization and stochastic processes in the spring.
(iii) Theory. Some familiarity with elementary basic analysis (sometimes taught under the guise of “advanced calculus with theory”) would be useful background for the Theory core course (Math 527). You will find it helpful to be familiar with ideas of continuity; convergence of sequences and series; integration, differentiation and mean value theorems; elementary ideas about metric spaces; and the basics of theorem proving. A nice text is “Advanced Calculus” by Patrick M. Fitzpatrick (ITP press) that is used at the University of Arizona in an undergraduate senior level analysis course.
(iv) Algorithms. You will be allowed to use your favourite programming language in the numerical analysis core course (Math 575). Matlab, Python and Julia are most popular among our students and faculty. If you have not used any of the programming languages before and have a chance to install and play with them over the summer, that could be helpful. In terms of theoretical background for the fall semester, linear algebra is again the key. The text book for the first module of Math 575a "Numerical Linear Algebra" by L.N. Trefethen and D. Bau, published by SIAM. This is an excellent book and you may want to own a copy. It may also be useful to review books by Boyd and Vandenberghe on Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares and Convex Optimization which will be used both in the Algorithm and the Methods class throughout the year and especially in the spring.