Numerical Methods - Tilley WPI - MA 510/CS 522

MA 510/ CS 522 - Numerical Methods

Fall 2019

Prof. B. S. Tilley

Department of Mathematical Sciences

Worcester Polytechnic Institute

Instructor: Prof. B.S. Tilley

Department of Mathematical Sciences

Worcester Polytechnic Institute

Office: SH 105A

Phone: (508) 831-6664

e-mail: tilley@wpi.edu

Tilley Home Page

Office Hours: MTR 1:00-1:50 or by appointment

Course Description: This course provides an introduction to a broad range of modern numerical techniques that are widely used in computational mathematics, science, and engineering. Subject areas include: numerical methods for systems of linear and nonlinear equations, interpolation and approximation, differentiation and integration, and differential equations. Specific topics include: basic direct and iterative methods for linear systems; classical root-finding methods; Newton’s method and related methods for nonlinear systems; fixed-point iteration; polynomial, piecewise polynomial, and spline interpolation methods; least-squares approximation; orthogonal functions and approximation; basic techniques for numerical differentiation; numerical integration, including adaptive quadrature; and methods for initial-value problems for ordinary differential equations. Additional topics may be included at the instructor’s discretion as time permits.

Both theory and practice are examined. Error estimates, rates of convergence, and the consequences of finite precision arithmetic are also discussed. Topics from linear algebra and elementary functional analysis will be introduced as needed. These may include norms and inner products, orthogonality and orthogonalization, operators and projections, and the concept of a function space.

The weekly topics list can be found on the Course Schedule Page.

Prerequisites: Knowledge of undergraduate linear algebra and differential equations is assumed, as is familiarity with MATLAB or a higher-level programming language. In our class, we will be using MATLAB for numerical computations and coding projects.

Recommended Text: Introduction to Numerical Analysis, Second Edition, F.B. Hildebrand (ISBN-13: 978- 0486653631) (1987).

Additional Resources:

Numerical Analysis, R.L. Burden and J.D. Faires (there are 10 editions)
Introduction to Numerical Analysis, J. Stoer and R. Bulirsch
An Introduction to Numerical Analysis, K.E. Atkinson
Numerical Linear Algebra, L.N. Trefethen and D. Bau.
Spectral Methods in Mathlab, L.N. Trefethen.

Course Organization

The course content is organized in the following themes

Systems of Equations: Floating-point arithmetic, errors, root-finding , systems of linear equations.
Calculus: Interpolation, Lagrange polynomials, splines, orthogonal polynomials, numerical differentiation, quadrature techniques
Ordinary Differential Equations: Forward/Backward Euler, Multistep Methods, Runge-Kutta, Adaptive Time-Stepping

To learn the material fully, we apply the following methods

Theory: Extending concepts from analysis and linear algebra to algorithms intended to approximate their computational analogues. Requires mathematical proof and rigor.
Practice: Developing scripts in MATLAB to implement these algorithms and verify that the results are consistent with the mathematical theory.

Classroom Culture

During class, please silence/turn off all cell phones. Laptops will be needed to work on the laboratory assignments at times during class, but otherwise laptops and tablet should be turned off. Side discussions and comments should be kept to a minimum, and should not be disruptive to other students during the class.

Course Grades and Policies

Grades will be determined based on homework assignments, coding projects and three in-class exams

Homework & Labs: 50%: These assignments will be a combination of written assignments along with programming assignments (labs). The labs will be uploaded onto Canvas (https://canvas.wpi.edu), and the format for the labs are MATLAB Scripts. Written homework will be turned in during class. All written assignments should be neatly written or typed (e.g. LaTeX or Microsoft Word). Illegible problems will receive no credit. Due dates for assignments can be found on the Canvas course site. Late assignments will not be accepted.
Three Exams: 50% Three in-class exams will take place at the dates below. The topics are the specific themes described above. Note that Exam 3 is on the final day of class, so plan accordingly. Plane tickets leaving for the break is not a sufficient reason to miss or reschedule Exam 3.
- Exam 1 (15%): October 3, 2019
- Exam 2 (15%): November 7, 2019
- Exam 3 (20%): December 12, 2019
Grading Policy Students have two business days to contact the instructor about potential errors in grading any assignment (homework, labs, exams) after receiving their graded work. Beyond this time, the grade on that assignment is final.
Make-Up Exam Policy Make-ups for Exams 1 & 2 require permission from the instructor at least two days prior to the exam, at which time a date for the make-up exam will be agreed upon by the student and instructor. No make-up exams are allowed after the scheduled exam is given (see the dates above). Students who cannot attend Exam 3 at the time above due need to inform the instructor at least two days prior to the exam. If the instructor feels that the reason to miss the exam is a valid one, the student will receive an incomplete (I) for the course, and the make-up will be taken in Spring 2020.
Accommodations: Reasonable accommodations will be made for students with verifiable disabilities. Students with disabilities who need to utilize accommodations in this class are encouraged to contact the Office of Disability Services (ODS) as soon as possible to ensure that such accommodations are implemented in a timely fashion. This office can be contacted via email: DisabilityServices@wpi.edu, via phone: (508) 831-4908, or in person: 124 Daniels Hall. If you have approved accommodations, please request your accommodation letters online through the Office of Disability Services Student Portal.

Academic Integrity and WPI Policies

All students are expected to be familiar with and adhere to WPI’s policy on academic integrity (i.e., no cheating, fabrication, facilitation, or plagiarism). Please refer to the WPI Academic Honesty Policy within the Student Code of Conduct (https://www.wpi.edu/about/policies/academic-integrity). Academic integrity violations will be prosecuted according to the university’s policy. For more details as to what constitutes academic dishonesty, please see https://www.wpi.edu/about/policies/academic-integrity/dishonesty.

Students are responsible to complying with all of WPI's policies at https://www.wpi.edu/about/policies