COSC 3121.03
Introduction to Numerical Computations I
(same as AS/SC/MATH 3241.03)
Section A Fall Tue, Thu 10:00-11:30
Section B Fall Wed 19:00-22:00
Section B is an Atkinson course. Students may complete this course for Arts
or Science credit. However, Section B may differ slightly from Section A.
For instance, Section B may have a different examination time which is not
posted in the regular examination schedule.
This course is concerned with an introduction to matrix
computations in linear algebra for solving the problems
of linear equations, interpolation and linear least
squares. Errors due to representation, rounding and
finite approximation are studied. Ill-conditioned
problems versus unstable algorithms are discussed. The
Gaussian elimination with pivoting for general system
of linear equations, and the Cholesky factorization for
symmetric systems are explained. Orthogonal
transformations are studied for computations of the QR
decomposition and the Singular Values Decompositions
(SVD). The use of these transformations in solving
linear least squares problems that arise from fitting
linear mathematical models to observed data is
emphasized. Finally, polynomial interpolation by
Newton's divided differences and spline interpolation are
discussed as special cases of linear equations. The
emphasis of the course is on the development of
numerical algorithms, the use of intelligent
mathematical software and the interpretation of the
results obtained on some assigned problems.
Topics covered may include the following.
- Preliminaries - linear algebra, computer
programming and mathematical software
- Number Systems and Errors - machine representation
of numbers, floating-point arithmetic, simple error
analysis, ill-conditioned problems and unstable
algorithms
- Solution of Systems of Linear Equations - Gaussian
elimination and its computational complexity,
pivoting and stability, special structures (Cholesky's
factorization for positive definite systems, banded
systems, storage and computational complexities)
error analysis, condition number and iterative
refinement
- Solution of Overdetermined Systems of Linear
Equations by Linear Least Squares Approximations -
linear least squares problems, normal equations,
orthogonal transformations (Given's and
Householder's), QR and Singular Values
Decompositions (SVD), SVD and rank-deficient
problems, computational complexities versus
robustness
- Interpolation - Newton's divided differences spline
interpolation; banded linear systems, error analysis
for interpolation. Other interpolations (rational, B-
splines)
Texts: t.b.a.
Prerequisites: for Computer Science majors - general prerequisites, including MATH2221.03; for others - COSC1030.03 or COSC1530.03 or
COSC1540.03; MATH1010.03 or MATH1014.03 or
Math1310.03; MATH2221.03
Degree Credit Exclusion: MATH3241.03