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.

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)

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.