Welcome to the class Introduction to Numerical Analysis. Computational approaches play an important role in the study of differential equations arising in a variety of fields, including basic scientific research, engineering, and finance. This class will discuss both the mathematical foundations and the practical implementation of modern numerical methods. Examples will also be discussed from the above application areas. For more details see the syllabus and the topics listed below.
Please be sure to read the prerequisites and grading policies for the class.
Selection of Topics
- Adaptive Quadrature Methods
- Gaussian Quadrature
- Monte-Carlo Methods
- Numerical Differentiation
- Initial Value Problems for ODE's
- Euler's Method
- Higher-Order Methods (Explicit / Implicit)
- Multistep Methods
- Runge-Kutta Methods
- Stability
- Stiff Differential Equations
- Stochastic Differential Equations
- Weak and Strong Convergence of SDEs
- Euler-Maruyama Method, Milstein Method
- Direct Methods for Solving Linear Systems
- Matrix Factorizations
- Iterative Methods in Linear Algebra
- Conjugate Gradient Method
For more information see the class syllabus Δ.
Prerequisites:
Ordinary Differential Equations, Linear Algebra, and some programming experience.
Grading:
The grade for the class will be based on the homework assignments (see policy below), midterm exam, and final project as follows:
Homework Assignments 30%
Midterm Exam 30%
Final Project 40%
Homework Policy:
Assignments will be made weekly and posted on the class website. Prompt submission of the homework assignments is required. While no late homework submissions will be accepted, one missed assignment will be allowed without penalty. While it is permissible for you to discuss materials with classmates, the submitted homework must be your own work. The assignments will consist of a combination of analytic problems and numerical calculations. Basic programming in Matlab/Octave may also be required for some assignments.
Exams:
A midterm exam will be given in the class on Thursday, September 2nd.
Solutions to the midterm exam [PDF] Δ.
Final projects will be announced toward the end of the quarter.
Supplemental Class Notes:
Below are suggestions for final projects. If another topic interests you, please feel free to write a one paragraph proposal to send to me by e-mail for approval. Please feel free after lecture to see me to discuss any issues you may be having with your specific projects.
GNU Octave Software and Documentation
Octave Software (Binaries from SourceForge.net)
GNU Octave Tutorial
GNU Octave Links (Tutorials and other Information)
Ubuntu (Linux Operating System) [Maybe useful to install dual-boot for running Octave]
Matlab Software and Documentation
Explosion of the Ariane 5 Rocket (Consequence of Faulty Numerics)
The Sinking of the Sleipner A Offshore Platform (Consequence of Faulty Numerics)
Class Annoucements:
Homework Assignments:
Turn all homeworks into my mailbox in South Hall on the 6th Floor by 5pm on the due date. Graded homeworks will be returned in class. Numbered exercises are from Numerical Analysis, Burden & Faires, 8th edition.
HW1: (Due Monday, August, 9th) 4.1: 1a, 3b, 5c, 7c, 22; 4.3: 1bh, 13, 15, 24.
HW2: (Due Thursday, August, 19th) 5.1: 1acd, 3abd, 7, 8ad, 9; 5.2: 1ad, 2ac, 3ad, 11, 16; 5.3: 1b, 3b, 11, 12.
HW3: (Due Monday, August, 30th) 5.6: 1acd, 5, 9, 12; 5.11: 1abc, 2ad, 12; 5.4: 1bc, 2ad, 27, 28.
HW4: (Due Tuesday, September, 14th) [PDF] Δ
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