Iterative Methods
Simple Iteration
Numerical Analysis
1
Tuesday, October 29, 13
1
Direct Methods
We have so far focused on solving Ax = b using direct
methods.
Gaussian Elimination
LU Decomposition
Variants of LU, including Crout and Doolittle
Other de
Matrix Inverses
1
Monday, October 7, 13
1
Gaussian Elimination
Gaussian Elimination is an algorithm for linear systems
like the following
5x1
x 2 + 2x 3 = 7
2x1 + 6x2 + 9x3 = 0
7x1 + 5x2
3x3 = 5
In this set of slides, we want to understand how the
matrix
Polynomial interpolation
Vandermonde matrix systems
1
Sunday, November 24, 13
1
Polynomial interpolation
Up to this point, we have been considering only nding
curves that in some sense approximate the data. The
assumption is that the data is noisy or impr
Iterative Methods
Splitting Methods
Numerical Analysis
1
Thursday, November 7, 13
1
Direct Methods
We have so far focused on solving Ax = b using direct
methods.
Gaussian Elimination
LU Decomposition
Variants of LU, including Crout and Doolittle
Other
Elimination using matrices
Recall : Multiplication on the left by a row vector results
in a row vector.
a
a
b
2
c
4
2
2
1
2
4
5
+b
3
4
9
1
1
5
3 5=
7
9
3
Result is a row vector
+c
2
17
Multiplication on the left can be thought of as a linear
combination o
Numerical Analysis
Professor Donna Calhoun
Ofce : MG241A
Ofce Hours : Wednesday 10:00-12:00 and 1:00-3:00
Fall 2013
Math 465/565
http:/math.boisestate.edu/~calhoun/teaching/Math565_Fall2013
Friday, September 6, 13
What is Numerical Analysis
Friday, Septem
Matrix decompositions
How can we solve
Ax = b
?
1
Monday, October 7, 13
1
Linear algebra
Typical linear system of equations :
5x1
x 2 + 2x 3
=
7
2x 1 + 6 x 2 + 9 x 3
=
0
7x 1 + 5 x 2
=
5
3x3
The variables x1 , x2 , and x3 only appear as linear terms
(no p
Numerical Root Finding
Bisection
Fixed point iteration
Newtons Method
Secant Method
Math 465/565
1
Thursday, August 29, 13
Motion with air resistance
h( t)
Vertical height of a rocket with air resistance
t
h ( t) =
Thursday, August 29, 13
33t + 784(1
e
0
Order and Rates of Convergence
Numerical Analysis
Math 465/565
1
Saturday, September 14, 13
Speed of convergence
We now have two algorithms which we can compare bisection and the xed-point method. Which is faster?
Hard to answer : Depends on what interval
Fixed point iteration
Numerical Analysis
Math 465/565
1
Monday, September 9, 13
Fixed Point Iteration
Suppose we wanted to solve :
f (x) = cos(x)
x=0
or
cos(x) = x
We might consider a iteration of this type :
xk+1 = cos(xk )
Try this : Enter any number in