MATH 387: Honours Numerical Analysis Winter 2018

Final exam

The lab final exam is due Sunday April 22, 18:00 EST.
The final exam is scheduled on Thursday April 19, starting at 9am, in Burnside Hall 1B23.
It is a closed book, closed note exam. No calculators will be allowed or needed.
There will be 5–6 questions, with each question comparable in difficulty to an "easier" question from the assignments.
Please review the past assignments and the midterm exam.
Here are some practice problems.

The final course grade will be computed as Homework 30% + MAX { Midterm 20% + Final 50% , Final 70% }

In each of the categories (assignments, midterm, and final exam), the theory and lab parts will be equally weighted.

Announcements

Lab assignment 3 due Friday April 13, 18:00 EST
Practice problems for the final exam
Assignment 4 [tex] due Wednesday April 11 in class
Midterm exam [tex] due Saturday March 17, 18:00 EST
Assignment 3 [tex] due Wednesday March 14 in class
Lab assignment 2 due Sunday March 4, 18:00 EST
Assignment 2 [tex] due Friday February 23 in class
Lab assignment 1 due Friday February 16, 18:00 EST
Assignment 1 [tex] due Wednesday February 7 in class
Instructor's office hours are T 11:00–12:00, WF 13:00–13:05, Burnside Hall 1123

Lecture notes

Class schedule

WF 11:35–12:55, Burnside Hall 1205
Note: This schedule is subject to revision during the term.

Date	Topics
W 1/10	Floating point numbers. Axiom 0.
F 1/12	Floating point arithmetic. Axiom 1.
W 1/17	Propagation of error. Roundoff error analysis of summation methods.
F 1/19	Products. Condition number of a function. Elementary analytic functions.
W 1/24	Roundoff error analysis of power series. Babylonian method.
F 1/26	Fixed point iterations. Newton-Raphson method.
W 1/31	Modified Newton. Reciprocal iteration. Complexity of division. Bisection.
F 2/2	Secant method. Regula falsi. Gaussian elimination.
W 2/7	LU decomposition. GEPP. Condition number of a matrix.
F 2/9	Gram-Schmidt orthogonalization. QR decomposition.
W 2/14	Householder's elimination.
F 2/16	Backward error analysis.
W 2/21	Backward stability of Housholder's elimination.
F 2/23	Some examples. Cholesky factorization.
W 2/28	Lagrange interpolation. Vandermonde matrices.
F 3/2	Newton's divided differences. Lebesgue's lemma.
3/5–3/9	Reading week
W 3/14	Peano kernel theorem. Weierstrass approximation theorem.
F 3/16	Bézier curves. Existence of minimax polynomials.
W 3/21	Oscillation theorems of de la Vallée-Poussin and Chebyshev. Chebyshev polynomials.
F 3/23	Least squares approximation. Legendre polynomials.
W 3/28	Dirichlet kernel. Chebyshev kernel. Lebesgue constants for Chebyshev truncation.
F 3/30	Good Friday
W 4/4	Newton-Cotes quadrature. Romberg integration.
F 4/6	Gauss quadrature.
W 4/11	Euler-Maclaurin formula. Bernoulli numbers.
F 4/13	Fourier series. Jackson's inequality.
R 4/19	Final exam (9am, in department)

Online resources

Numerical analysis

Principles of Scientific Computing book draft by David Bindel (Cornell) and Jonathan Goodman (NYU)
Applied Numerical Computing by Prof. Lieven Vandenberghe (UCLA)
Significant figures. Suggestions to Authors of the Reports of the United States Geological Survey
David Goldberg. About floating-point arithmetic [pdf] [published version]
Notes, Presentations, Simulations in Maple, Mathcad, Matlab, Mathematica, and Self-Assessment Tests at Holistic Numerical Methods Institute
Lecture Notes and other resources by Professor Kendall Atkinson (University of Iowa)
Homepage of Lloyd N. Trefethen
Endre Suli's homepage (scroll down to see the "Lecture notes" section)

Python

Anaconda
Python
Introductory course and tutorials at DataCamp
Python Wiki
Getting started with SciPy

Recommended books

E. Süli. An introduction to numerical analysis. Cambridge University Press 2003
G. Dahlquist and A. Bjorck. Numerical methods. Dover 2003
L.R. Scott. Numerical analysis. Princeton University Press 2011

Course outline

Instructor: Dr. Gantumur Tsogtgerel

Prerequisite: MATH 325 (Honours ODE) or MATH 315 (ODE), COMP 202 or permission of instructor.

Corequisite: MATH 255 (Honours Analysis 2) or MATH 243 (Analysis 2).

Restriction: Intended primarily for Honours students.

Calendar description: Error analysis. Numerical solutions of equations by iteration. Interpolation. Numerical differentiation and integration. Introduction to numerical solutions of differential equations.

Topics to be covered:

Absolute and relative precision. Evaluation of Taylor series.
Root finding. Bisection. Fixed point iterations. Newton-Raphson method.
General theory of error analysis and concept of stability.
LU and QR factorizations.
Newton's method. Steepest descent. Simple iterative methods.
Interpolation. Newton's divided differences. Lagrange and Hermite polynomials.
Least squares and uniform approximations.
Numerical differentiation. Quadrature.
Eigenvalue solvers and SVD (if time permits).
Computation with polynomials. Fast Fourier transform (if time permits).
Introduction to ODE integrators (if time permits).

Homework: We will have 3–4 homework assignments. Each assignment will consist of theoretical and programming components.

Exams: A midterm exam and a final exam. The midterm will be a take-home exam, and the final will consist of a theory exam in class, and a programming ("lab") exam to be done at home.

Grading: Homework 30% + MAX { Midterm 20% + Final 50% , Final 70% }.