1996-97 Department of Computer Science Course directors: B. Guo (Section A) M. McNamee (Section B) Implementation details

## 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