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 examine the rate of convergence of several root-nding
me
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
aij = aij lik akj
endfor
bi = bi lik bk
endfor
endfor
f
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 the Taylor Series for e2x should
we include?
According to
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 /akk
for j = k + 1 to n do
aij = aij aik akj
endfor
endf
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 do
if amax < |ali k | then
amax = |ali k |
r=i
endif
en
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
endfor
if r = k then
(switch rows r and k of A)
(switch br
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 iterative numerical integration methods.
The Exercise
1. Save
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 several tools and libraries bundled with it; of primar
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 most efficient way to use it's many functions. For examp
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 though we often think of computers as working with real