Math-327/397: (Honours) Matrix Numerical Analysis

Course Outline, Winter 2013

• Coordinates:
  Tues, Thur, 2:30pm to 4:00pm in ARTS 150 starting Tuesday 8th January.
  Prof Humphries, BH-1112, Tel: 398-3821, E-mail: humphries@math.mcgill.ca
  Office Hours: Mon 9:30-10:30, Wed 3:00-4:00, Thurs 4:00-5:00 and by arrangement
  

• Description
  This is a first course on matrix numerical analysis. We will consider the main basic problems, of linear systems including under and over determined problems, and eigenvalue problems and the main direct and iterative methods to solve them. Derivation and analysis of the algorithms will be an essential part of the course. Does the algorithm give the exact answer? If not, is the error small, and in what sense? We will use matlab to implement some of these algorithms and experiment with others.

• Syllabus
  An overview of numerical methods for linear algebra applications and their analysis. Problem classes include linear systems, least squares problems and eigenvalue problems.

• Topics
  Topics to be covered, subject to time constraints, and maybe not in this order are
  1. Basics : Matrices, vectors, norms, orthogonality.
  2. Orthogonal Factorizations: QR, Gram-Schmidt procedure, Householder triangularization, SVD.
  3. Least Squares Problems: set-up, normal equations, applications.
  4. Linear Systems: Gaussian elimination, LU factorization, pivoting, Cholesky
  5. Eigenvalue problems: properties of eigenvalues, power method, inverse iteration, Rayleigh quotient, QR iteration
  6. Iterative Methods: Arnoldi, GMRES, Lanczos, Conjugate-Gradient Method

• Differences in the Courses
  Math327 is intended primarily for students in majors programs. Math397 is intended for students in Honours programs. The courses share the same lectures, but the assignments, midterm and final exam for the two courses will be different. Honours students should expect some analytical questions.

• References:
  There is no compulsory textbook for this course.

  Optional text:

  • Fundamentals of Matrix Computations, by David S. Watkins, published by Wiley. The second edition of this text is available on campus for free as an e-book.

  Additional texts which contain some useful information and may be worth consulting are

  • Numerical Linear Algebra, by L.N. Trefethen and D. Bau III, ISBN#: 978-0-898713-61-9.
  • Numerical Analysis, 8th Ed. by R.L. Burden and J.D. Faires, ISBN#: 0-534-39200-8.
  • Numerical Mathematics, 2nd Ed. by Quarteroni, Sacco & Saleri. Springer, ISBN#: 978-3540346586. (Also available as an e-book on campus).

• Prerequisites:
  The course is intended for all students with an interest in matrix numerical analysis. The class will combine theory and practice, with the emphasis towards theory. Nevertheless we will also use matlab to perform computations.

  The prerequisites are
  MATH 223 or MATH 236 or MATH 247 or MATH 251,
  and
  COMP 202
  or
  consent of instructor.

  Familiarity with matlab would also be useful. Information on matlab appears on my webpage for this course and on MyCourses.

• Assignments:
  There will be regular assignments . These will count for 20% of the final mark.
• Midterm:
  The midterm will be in class on Tuesday 26^thFebruary, and will be worth 20% of the final mark.

• Final Exam:
  There will be a formal 3 hour final exam which will count 60% of the final mark.

• Policies:
  • No make up tests will be given.
  • No late homework will be accepted under any circumstances, except for an absence approved in advance by the instructor.
  • Important announcements (including posting of assignments and due dates) will be made to registered students by e-mail (via MyCourses to your official McGill e-mail address) and/or posted on MyCourses. You are responsible for reading your McGill e-mail.
  • I attempt to reply to e-mail in a timely fashion, but do not expect immediate responses. I usually will not reply to email sent the day before a test.
  • Should you miss a lecture, you are responsible for making up any material covered in class.

• Academic Integrity
  The work you hand in should be your own effort; any collaboration must be acknowledged.

  McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/students/srr/honest/ for more information).

• Language
  In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded.