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