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Previous: MC224 Vector Calculus
MC226 Numerical Linear Algebra
Credits: 10 
Convenor: Prof. B. Leimkuhler 
Semester: 2 
Prerequisites: 
essential: MC146, MC147 

Assessment: 
Coursework: 20% 
One and a half hour exam: 80% 
Lectures: 
18 
Classes: 
5 
Tutorials: 
none 
Private Study: 
47 
Labs: 
5 
Seminars: 
none 
Project: 
none 
Other: 
none 
Total: 
75 


Explanation of Prerequisites
All the material in the course MC147 (Introductory Linear
Algebra) will be absolutely essential for this module. From
the module MC146, the idea of a supremum will be required,
as well as the material on convergence of sequences and
the Intermediate Value Theorem.
Course Description
Solving linear systems Ax = b or finding the eigenvalues
of a matrix may appear to be trivial tasks.
In fact, the systems of linear equations which arise in applications
are often large, involving thousands or even millions of unknowns, and
may be illconditioned (the solution may be very sensitive to the
value of the right hand side). It is often then necessary to use an
iterative method, which generates a sequence of approximations to
the solution of the system. The rate of convergence of this sequence
is very important.
For finding eigenvalues, even of a symmetric real matrix, it is certainly
not feasible to locate the zeroes of the polynomial :the work involved in doing this would be of the order of n! for an
matrix A. Instead there are transformations which can be
used to reduce the matrix to a simpler form, and these may be used
in conjunction with an iterative or other method to locate the
eigenvalues.
This course will describe some of these algorithms, which lie behind the
methods used by many linear algebra software packages.
Aims
This course will present the mathematical algorithms and analysis behind
software for numerical solution of linear systems of equations and
matrix eigenvalue problems.
Objectives
By the end of this module a diligent student should be able to
 form the LU and/or LL^{T} decompositions of appropriate
matrices, and prove theorems on their existence under suitable
hypotheses;
 know how to define the Jacobi, GaussSeidel and SOR iterative
schemes for solving linear systems; state and prove theorems
on the circumstances in which these methods may be expected to
converge;
 prove the convergence of the power method and its variants for
finding the eigenvalues of appropriate matrices; use the Sturm Sequence
method for eigenvalues of tridiagonal matrices.
Transferable Skills
This course will impart a basic understanding of numerical linear
algebra which is useful throughout science and engineering.
Syllabus
Gaussian Elimination; LU and Cholesky factorizations; Matrix
norms; Iterative methods for linear equations, including Jacobi,
GaussSeidel and Successive OverRelaxation.
The Schur Decomposition; the Power method and its variants; the
Sturm Sequence method for tridiagonal matrix eigenproblems;
Householder transformations for tridiagonalisation of a
symmetric matrix (if time allows).
Reading list
Recommended:
D. Kincaid and W. Cheney,
Numerical Analysis,
Brooks/Cole.
R. L. Burden and J. Douglas Faires,
Numerical Analysis,
PWSKent.
Details of Assessment
The final assessment of this module will consist of 10% coursework, 10%
from a short series of MATLAB projects
and 80% from a one and a half hour examination during the Summer exam
period. The 10% coursework contribution will be determined by students'
solutions to coursework problems. The examination paper will contain
4 questions with full marks on the paper obtainable from 3 complete answers.
Next: MC227 Fluids and Waves
Up: Year 2
Previous: MC224 Vector Calculus
S. J. Ambler
11/20/1999