Introduction to Numerical Methods

Calendar

LEC #	TOPICS	KEY DATES
Week 1 covers Lectures 1 and 2.
1	Course overview, Newton’s method for root-finding
2	Floating-point arithmetic
Week 2 covers Lectures 3–5.
3	Floating-point summation and backwards stability
4	Norms on vector spaces
5	Condition numbers	Problem set 1 due
Week 3 covers Lectures 6–8.
6	Numerical methods for ordinary differential equations
7	The SVD, its applications, and condition numbers
8	Linear regression and the generalized SVD
Week 4 covers Lectures 9–11.
9	Solving the normal equations by QR and Gram-Schmidt
10	Modified Gram-Schmidt and Householder QR
11	Matrix operations, caches, and blocking	Problem set 2 due
Week 5 covers Lectures 12–14.
12	Cache-oblivious algorithms and spatial locality
13	LU factorization and partial pivoting
14	Cholesky factorization and other specialized solvers. Eigenproblems and Schur factorizations
Week 6 covers Lectures 15–17.
15	Eigensolver algorithms: Companion matrices, ill-conditioning, and Hessenberg factorization
16	The power method and the QR algorithm
17	Shifted QR and Rayleigh quotients	Problem set 3 due
Week 7 covers Lectures 18–20.
18	Krylov methods and the Arnoldi algorithm
19	Arnoldi and Lanczos with restarting
20	The GMRES algorithm and convergence of GMRES and Arnoldi	Final project proposal due
Week 8 covers Lectures 21–23.
21	Preconditioning techniques. The conjugate-gradient method
22	Convergence of conjugate gradient
23	Biconjugate gradient algorithms	Problem set 4 due
Week 9 covers Lectures 24–26.
24	Sparse-direct solvers
25	Overview of optimization algorithms	Take-home midterm exam due before Lec #25
26	Adjoint methods
Week 10 covers Lectures 27 and 28.
27	Adjoint methods for eigenproblems and recurrence relations
28	Trust-regions methods and the CCSA algorithm
Week 11 covers Lectures 29–31.
29	Lagrange dual problems
30	Quasi-Newton methods and the BFGS algorithm
31	Derivation of the BFGS update
Week 12 covers Lectures 32–34.
32	Derivative-free optimization by linear and quadratic approximations
33	Numerical integration and the convergence of the trapezoidal rule
34	Clenshaw-Curtis quadrature
Week 13 covers Lectures 35–37.
35	Chebyshev approximation
36	Integration with weight functions, and Gaussian quadrature
37	Adaptive and multidimensional quadrature
Week 14 covers Lectures 38 and 39.
38	The discrete Fourier transform (DFT) and FFT algorithms
39	FFT algorithms and FFTW	Final project due at the end of term