Introduction, Systems of Linear Equations, Iterative Methods, Linear Least Squares, Eigenvalue Problems, Nonlinear Equations, Optimization; Coursework and Programming Projects.

For further information see the academic catalog: IAM561

#### Course Objectives

At the end of this course, the student will learn:

- how computer arithmetics is significant in computational methods
- how to solve linear systems of equations using Gaussian elimination techniques including pivoting
- how to decompose and factorise matrices, including Cholesky decomposition
- how to define a linear least squares problem and solve it using various factorisation methods
- how to find specific eigenvalue and eigenvectors of matrices using both variants of Power iterations and QR iterations
- how to solve nonlinear (root finding) systems of equations using iterative methods, including Newton's method and fixed-point algorithms
- how to apply root finding algorithms in nonlinear optimisation problems

#### Course Learning Outcomes

Student, who passed the course satisfactorily will be able to:

- characterise a problem based on condition number as well as backward and forward error analysis
- solve linear systems of equations using classical factorisation techniques
- solve linear equations iteratively when the speed and storage is important
- define and solve linear least squares problem when (experimental) data is available for the model
- locate and approximately calculate eigen-pairs (eigenvalues and eigenvectors) of matrices
- solve nonlinear (systems) of equations and root-finding problems
- apply nonlinear system solvers in optimisation problems

#### Tentative (Weekly) Outline

- Computer arithmetic
- Error and Stability Analysis
- Systems of Linear Equations
- Linear Least Squares
- Eigenvalue Problems
- Eigenvalue Problems (QR Iterations)
- Iterative Methods for Nonlinear Equations
- Fixed Point Algorithms

#### Course Textbook(s)

- M.T. Heat, "Scientific Computing", McGraw Hill, 1997
- A. Quarteroni, R. Sacco, F. Saleri, " Numerical Mathematics", Springer, 2000
- C. F. van Loan, Introduction to Scientific Computing, Prentice Hall, 1999
- A. Quarterioni, F. Saleri, Scientific Computing with MATLAB and Octave, Springer-Verlag, 2006