75
Numerical Analysis 1 EE, NCKU Tien-Hao Chang (Darby Chang)
Numerical Analysis
1
EE, NCKUTien-Hao Chang (Darby Chang)
In the previous slide Fixed point iteration scheme
– what is a fixed point?
– iteration function
– convergence
Newton’s method– tangent line approximation
– convergence
Secant method2
In this slide Accelerating convergence
– linearly convergent
– Newton’s method on a root of multiplicity >1
– (exercises)
Proceed to systems of equations– linear algebra review
– pivoting strategies
3
2.6
4
Accelerating Convergence
Accelerating convergence Having spent so much time discussing
convergence– is it possible to accelerate the convergence?
How to speed up the convergence of a linearly convergent sequence?
How to restore quadratic convergence to Newton’s method?– on a root of multiplicity > 1
5
Accelerating convergence
Linearly convergence Thus far, the only truly linearly
convergent sequence– false position
– fixed point iteration
Bisection method is not according to the definition
6
7
Aitken’s Δ2-method Substituting Eq. (2) into Eq. (1) Substituting Eq. (4) into Eq. (3)
The above formulation should be a
better approximation to p than pn
8
12
21
2
)(
nnn
nnn ppp
pppp
9
Aitken’s Δ2-method
Accelerated?
10
which implies super-linearly convergence
later
answer
11
Any Questions?
12
About Aitken’s Δ2-method
Accelerating convergence
Anything to further enhance?
13
14
Why not use p-head instead of p?
Steffensen’s method
15
16
Restoring quadratic convergence to Newton’s method
17
18
Any Questions?
19
Two disadvantages
Both the first and the second derivatives of f are needed
Each iteration requires one more function evaluations
20
)()()]([
)()()(
2 xfxfxf
xfxfxxg
answer
Any Questions?
21
Chapter 2 Rootfinding (2.7 is skipped)
Exercise
22
2010/4/21 9:00amEmail to [email protected] or hand over in class. You may arbitrarily pick one problem among the first three, which means this exercise contains only five problems.
23
24
25
26
27
(Programming)
Chapter 3
28
Systems of Equations
Systems of Equations
Definition
29
3.0
30
Linear Algebra Review(vectors and matrices)
Matrix
Definitions
31
Any Questions?
32
m, n, m, i, j, EQUAL, SUM,SCALAR MULTIPLICATION, PRODUCT…
The Inverse Matrix
33
(cannot be skipped)
34
Any questions?
35
answerquestion
The Determinant
36
(cannot be skipped, too)
37
cofactor
38
Link the concepts– All these theorems will be extremely
important throughout this chapter
Nonsingular matrices Determinants Solutions of linear systems of
equations
39
40
41
(Hard to prove)
Any Questions?
42
3.0 Linear Algebra Review
3.1
43
Gaussian Elimination(I suppose you have already known it)
An application problem
44
I1-I2-I3=0
I2-I4-I5=0
I3+I4-I6=0
2I3+I6=7
I2+2I5=13
-I2+2I3-3I4=0
45
Following Gaussian elimination
46
Any Questions?
47
Gaussian elimination
Gaussian elimination
Operation Counts
48
Operation Counts
Comparison Gaussian elimination
– forward elimination
– back substitution
Gauss-Jordan elimination
Compute the inverse matrix
49
nnn 672
213
32
2n
nnn 672
233
32
nnn 23
nnn 23 22
3.2
50
Pivoting Strategy
51
52
53
Compare to x1=1, x2=7, x3=1
Pivoting strategy To avoid small pivot elements A scheme for interchanging the rows
(interchanging the pivot element) Partial pivoting
54
55
In action
http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
56
57
Compare to x1=1, x2=7, x3=1
Any Questions?
58
From the algorithm view How to implement the interchanging
operation?– change implicitly
Introduce a row vector r– each time a row interchange is required,
we need only swap the corresponding elements of the vector
– number of operations from 3n to 3
59
hint
answer
60
In action
http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
61
Without pivoting
62
63
x = [1.000, -0.9985, 0.9990, -1.000]T
– exact solution x = [1,-1,1,-1]T
– no r x = [1.131, -0.7928, 0.8500, -0.9987]T
64
Scaled Partial Pivoting
65
Scaled partial pivoting
An example
66
Any Questions?
67
Scaled partial pivoting
A blind spot of partial pivoting
68
answer
Scaled partial pivoting
69
70
71
In action
http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
72
73
x = [1.000, -1.000, 1.000, -1.000]T
– exact solution x = [1,-1,1,-1]T
– no s x = [1.000, -0.9985, 0.9990, -1.000]T
– no r x = [1.131, -0.7928, 0.8500, -0.9987]T
74
Any Questions?
75
3.2 Pivoting Strategy
