cep 2008/09
MT3802 - Numerical Analysis
Tutorial Sheet 5 1. Let p1 (x) be the linear interpolant of f (x) at the points (x0 , f (x0 ) and (x1 , f (x1 ), where x1 - x0 = h. Show that for x [x0 , x1 ] f (x) - p1 (x) where f (x)
M h2 8
M.
Suppose ex on [-
cep 2008/09
MT3802 - Numerical Analysis
Tutorial Sheet 2 1. For a sub-ordinate matrix norm and an invertible matrix A, where Ax = b, show that (i) (ii) As A
s
sN
1 x A-1 A b
2. Assume that (A + A)-1 is computed as an approximation to A-1 . Show that (A +
cep 2008/09
MT3802 - Numerical Analysis
Tutorial Sheet 1 1. Show that the following satisfy the requirements of a norm specified on a Vector Space: (i)
b
f =
a
|f (x)|dx
where f (x) C[a, b] ;
(ii) x = max |xi |
1in
where x Rn .
2. An inner product is defi
cep 2008/09
MT3802 - Numerical Analysis
SOLUTIONS - Tutorial Sheet 2 1. For a sub-ordinate matrix norm and an invertible matrix A, where Ax = b, show that (i) As A
s
sN
Using Rule 5 (1.9) of sub-ordinate matrix norms we have As A As-1 A
2
As-2 A
s-1
A A
s
cep 2008/09
MT3802 - Numerical Analysis
SOLUTIONS - Tutorial Sheet 1 1. Show that the following satisfy the requirements of a norm specified on a Vector Space: (i)
b
f =
a
|f (x)|dx
where f (x) C[a, b] ;
Proof that is satisfies Rule 1 (1.1) f = 0 & conti
Chapter 2
Iterative Methods
2.1 Introduction
In this section, we will consider three different iterative methods for solving a sets of equations. First, we consider a series of examples to illustrate iterative methods. To construct an iterative method, we
MT 3802 NUMERICAL ANALYSIS
2008/2009
Dr Clare E Parnell and Dr Stphane Rgnier e e
October 1, 2008
Chapter 0
Handout
0.1 Notation
Throughout this course we will be using scalars, vectors and matrices. It is essential that you know what they are and can tel