COMPUTING AND ESTIMATING THE RATE OF
CONVERGENCE
JONATHAN R. SENNING
1. Rate of Convergence
These notes develop a formulation of the rate of convergence. After dening the
rate of convergence we will e
Gaussian Elimination Algorithm with Back Substitution (Cheney & Kincaid)
Standard form with pivots lik stored separately
for k = 1 to n 1 do
for i = k + 1 to n do
lik = aik /akk
for j = k + 1 to n do
EXAMPLE BOUNDING THE ERROR TERM IN TAYLORS
THEOREM
JONATHAN SENNING
Suppose we want to nd a Taylor Polynomial that approximates e2x to 7 decimal
places on the interval [0.5, 0.5]. How many terms of th
Solution of A x = b using LU factorization
and forward and backward solves. (Cheney & Kincaid)
Factor A into LU . Notice that no pivoting is done.
for k = 1 to n 1 do
for i = k + 1 to n do
aik = aik /
Gaussian Elimination Algorithm with Partial Pivoting
Algorithm that avoids switching rows (messy but ecient)
for k = 1 to n do
lk = k
endfor
for k = 1 to n 1 do
amax = |alk k |
r=k
for i = k + 1 to n
Gaussian Elimination Algorithm with Partial Pivoting
Algorithm that really switches rows (inecient)
for k = 1 to n 1 do
s = |akk |
r=k
for i = k + 1 to n do
if s < |aik | then
s = |aik |
r=i
endif
end
MAT342/CPS342 Lab Exercise 3: Romberg
Integration
Introduction
In this exercise we examine Romberg integration, both to better understand how it works and
also to explore convergence rates for iterati
MAT342/CPS342 Lab Exercise 4: Numerical
Solution of ODEs
Introduction
Today we explore using Python and some supporting tools for numerical work. For interactive
work IPython is a great choice. It has
MAT342/CPS342 Lab Exercise 1:
Introduction to Octave
Starting Octave
Octave is an interactive, command-line driven program. This means that you can type commands
directly into the program, which the m
MAT342/CPS342 Lab Exercise 2: Loss of
Significance
Introduction
Every number that a modern digital computer works with is rational and therefore has a finite or
repeating decimal representation. Even