Lecture Summaries | Introduction to Numerical Methods | Mathematics

Lecture notes files.
SES #	Lecture summaries	HANDOUTs
1	Key concerns of numerical methods	No handouts
2	Performance: Arithmetic vs. memory	Performance experiments with matrix multiplication (PDF) Ideal-cache terminology (PDF)
3	Memory optimization and cache obliviousness	Experiments with cache-oblivious matrix-multiplication (PDF)
4	Accuracy and floating-point arithmetic	Notes on floating-point (PDF) *Note: Files in this section are from a previous version of the course.
5	Floating-point and numerical stability	No handouts
6	Backwards stability of summation, norms	No handouts
7	Condition numbers and eigenvalues	No handouts
8	The singular-value decomposition	No handouts
9	Least-square problems and QR factors	No handouts
10	Gram-Schmidt stability and householder QR	Gram-Schmidt notes (PDF) Householder notes (PDF) *Note: Files in this section are from a previous version of the course.
11	Householder, Gaussian, and Cholesky factorization	No handouts
12	Eigenproblems, characteristic polynomials, and Shur factors	No handouts
13	Hessenberg factorization and its applications, power methods	Hessenberg handout (PDF) *Note: Files in this section are from a previous version of the course.
14	QR iteration for eigenproblems	No handouts
15	Overview of iterative and sparse solvers	No handouts
16	Arnoldi and Lanczos iterations	No handouts
17	Restarting Lanczos iterations	No handouts
18	GMRES and MINRES	No handouts
19	Steepest-descent and conjugate-gradient methods	Shewchuk, Jonathan Richard. "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain." August 4, 1994. Pages 8, and 20. (PDF)
20	Preconditioning and condition numbers of PDE-like matrices	No handouts
21	Biconjugate-gradient methods, sparse-direct solvers	Summary of options for solving linear systems (PDF) Notes on sparse-direct solvers (PDF) *Note: Second file in this section is from a previous version of the course.
22	Nonlinear conjugate gradient, and conjugate-gradient eigensolvers	No handouts
23	Overview of optimization problems	Overview of optimization (PDF) Notes on adjoint methods (PDF)
24	Adjoint methods and sensitivity analysis	No handouts
25	Adjoint methods for recurrences, CCSA algorithms	Adjoint methods for recurrence relations (PDF) Svanberg, Krister. "A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations." SIAM Journal on Optimization 12, no. 2 (2002): 555-573. Pages 1-10.
26	Lagrange dual functions and KKT conditions	No handouts
27	Quasi-Newton methods	No handouts
28	BFGS updates	No handouts
29	Derivative-free optimization: Linear and quadratic models	No handouts
30	Global optimization and the DIRECT algorithm	Jones, D. R., C. D. Perttunen, and B. E. Stuckman. "Lipschitzian Optimization Without the Lipschitz Constance." Journal of Optimization Theory and Application 79 no. 1 (1993): 157. First few pages. (PDF - 1.5MB)
31	Numerical integration and accuracy of the trapezoidal rule	Notes on error analysis of the trapezoidal rule and Clenshaw-Curtis quadrature in terms of Fourier series. (PDF) Two numerical experiments with trapezoidal rule (PDF)
32	Clenshaw-Curtis quadrature	"Clenshaw-Curtis Quadrature." Wikipedia.
33	Chebyshev approximation	No handouts