Lecture Summaries

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. (This resource may not render correctly in a screen reader.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. (This resource may not render correctly in a screen reader.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 (This resource may not render correctly in a screen reader.PDF)

32

Clenshaw-Curtis quadrature

"Clenshaw-Curtis Quadrature." Wikipedia.
33

Chebyshev approximation

No handouts