Math/CSCI 164: Scientific Computing

This course is meant to introduce you to computational techniques applied to problems in the sciences and engineering. Topics will include: modeling of physical problems, computer implementation, analysis of results; use of mathematical software; numerical methods chosen from: solutions of linear and nonlinear algebraic equations, solutions of ordinary and partial differential equations, finite elements, linear programming, optimization algorithms, and fast Fourier transforms.

Resources

There is no required textbook for this course. My lectures will be primarily based upon Fundamentals of Numerical Computation by Driscoll and Braun. Other excellent references include Numerical Methods for Scientific Computing by Novak, Numerical Linear Algebra with Julia by Darve and Wootters, Numerical Linear Algebra by Trefethen and Bau, and Applied Numerical Linear Algebra by Demmel.

Software

We will frequently program and run numerical experiments in this course. We will work in Julia.

Tentative Course Schedule

Week	Lecture	Topic	FNC Section
1	1	Floating point numbers, Conditioning Slides	1.1, 1.2
	2	Algorithms, Stability, Linear algebra review Slides	1.3, 1.4
2	3	Polynomial interpolation, Linear systems Slides	2.1, 2.2, 2.3
	4	LU factorization, Row pivoting Slides	2.4, 2.5, 2.6
3	5	Vector and matrix norms, Classical iterative methods Slides	2.7
	6	Conditioning of linear systems, Matrix structure Slides	2.8, 2.9, 8.1
4	7	Fitting to data, Normal equations, Space of linear systems Slides	3.1, 3.2
	8	QR factorizations Slides	3.3, 3.4
5	9	Rootfinding methods Slides	4.1, 4.2
	10	Newton's and interpolation-based method Slides	4.3, 4.4
6	11	Nonlinear systems, Quasi-Newton methods Slides	4.5, 4.6
	12	Nonlinear least-squares Slides	4.7
7	13	Interpolation and splines Slides	5.1, 5.2, 5.3
	14	Finite differences Slides	5.4, 5.5
8	15	Numerical integration Slides	5.6, 5.7
	16	Basic methods for IVPs Slides	6.1, 6.2
9	17	Runge-Kutta methods Slides	6.3, 6.4
	18	Adaptive and multistep methods Slides	6.5, 6.6
10	19	Implementation and zero-stability of multistep methods Slides	6.7, 6.8
	20	Data as matrices Slides	7.1
11	21	Matrix decompositions, NMF Slides	7.2, 7.3
	22	Dimension reduction Slides	7.4, 7.5
12	23	Power iteration Slides	8.2, 8.3
	24	Krylov methods, GMRES Slides	8.4, 8.5
13	25	MINRES, Conjugate gradients, Matrix-free iterations Slides	8.6, 8.7
	26	Preconditioning Slides	8.8