Mathematical Tools for Scientific Computing

Fall 2014

**GATEWAY_2014** is a course proposed for the Fall 2014 semester
in the Department of Scientific Computing at Florida State University.
The class is intended as a "gateway", that is, a review course (or
perhaps for some topics, an introductory course) that ensures that
students are familiar with the mathematical tools needed for other
courses in the field of scientific computing, and particularly those
offered by the Department.

The class can be appropriate for undergraduate students, but also for graduate students whose undergraduate training did not include a thorough presentation of these topics.

A syllabus for the class is available.

Lecture notes are available for each week:

- lecture_week01.pdf, introductory lecture; overview of linear algebra;
- lecture_week02.pdf, vectors and matrices.
- lecture_week03.pdf, triangular factorization and operation counts.
- lecture_week04.pdf, matrix norms, matrix condition, vector spaces, linear independence, spanning sets and basis, the null space and the range of a matrix.
- lecture_week05.pdf, four basis spaces (column space, null space, row space, null space of transpose)
- lecture_week06.pdf, linear least squares, orthonormal vectors, the QR decomposition.
- lecture_week07.pdf, eigenvalues.
- lecture_week08.pdf, numerical methods for finding an eigenvalue.
- lecture_week09.pdf, the singular value decomposition.
- lecture_week10.pdf, initial value problem for first order linear ODE's.
- lecture_week11.pdf, initial value problem for general first order ODE's, numerical methods.
- lecture_week12.pdf, single step and multistep methods for initial value problems.
- lecture_week13.pdf two point boundary value problems; functions of several variables.
- lecture_week14.pdf, the vector form of Taylor's series; integration in higher dimensions; Green's theorem.
- lecture_week15.pdf, line integrals; Green's theorem; an introduction to partial differential equations.