Date | Topics |
Tu 1/12 | Number systems and formats. |
Th 1/14 | Floating point numbers. Basic operations. Absolute and relative error. |
Tu 1/19 | Error propagation. Cancellation and loss of precision. |
Th 1/21 | Simple iterations. Lagrange's theorem on Taylor series. Trigonometric functions. π. |
Tu 1/26 | Logarithms. Roundoff error analysis of Taylor series. |
Th 1/28 | Pairwise summation. Euler transform. Aitken extrapolation. |
Tu 2/2 | Conditioning. Backward error analysis. |
Th 2/4 | Backward stability. Stability. Accuracy. |
Tu 2/9 | Gaussian elimination. LU decomposition. |
Th 2/11 | Backward stability of LU decomposition. Conditioning of linear systems. |
Tu 2/16 | Gram-Schmidt orthogonalization. QR decomposition. |
Th 2/18 | Least squares. Householder reflection and Givens rotation. |
Tu 2/23 | Lagrange interpolation. Lagrange coefficients. Pointwise error estimate. |
Th 2/25 | Midterm exam |
2/29–3/4 | Reading week |
Tu 3/8 | Lebesgue constants for Lagrange interpolation. Numerical differentiation. |
Th 3/10 | Uniform approximation. Weierstrass approximation theorem. Bernstein polynomials. |
Tu 3/15 | De la Vallée Poussin's theorem. Chebyshev's equioscillation theorem. |
Th 3/17 | Chebyshev polynomials. Lagrange interpolation with Chebyshev nodes. |
Tu 3/22 | Least squares approximation. Legendre polynomials. |
Th 3/24 | Lebesgue constants for Chebyshev truncation. Simple quadrature rules. |
Tu 3/29 | Newton-Cotes quadrature. Euler-Maclaurin formula. |
Th 3/31 | Bernoulli numbers. Romberg integration. |
Tu 4/5 | Gauss quadrature. Improper integrals. |
Th 4/7 | Fixed point iterations. Newton-Raphson method. Multiple roots. |
Tu 4/12 | Bisection. Secant method. Regula Falsi. |
Th 4/14 | Newton's method. Gradient descent. Line search. |
Fr 4/22 | Final exam (2pm, Currie Gym) |
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:
Homework: We will have 3–4 homework assignments. Each assignments will consist of theoretical and programming components.
Exams: A midterm exam and a final exam. Each exam will consist of a theory exam in class, and a programming ("lab") exam to be done at home.
Grading: Homework 30%, Midterm 20%, Final 50%.
The following topics will be covered.