Numerical Analysis EE NCKU TienHao Chang Darby Chang
Numerical Analysis EE, NCKU Tien-Hao Chang (Darby Chang) 1
In the previous slide n Fixed point iteration scheme – what is a fixed point? – iteration function – convergence n Newton’s method – tangent line approximation – convergence n Secant method 2
In this slide n Accelerating convergence – linearly convergent – Newton’s method on a root of multiplicity >1 – (exercises) n Proceed to systems of equations – linear algebra review – pivoting strategies 3
2. 6 Accelerating convergence 4
Accelerating convergence n 5
Accelerating convergence Linearly convergence n Thus far, the only truly linearly convergent sequence – false position – fixed point iteration n Bisection method is not according to the definition 6
7
Aitken’s Δ 2 -method n 8
9
Aitken’s Δ 2 -method Accelerated? which implies superanswer linearly convergence later 10
Aitken’s Δ 2 -method Accelerated? which implies superlinearly convergence later 11
Aitken’s Δ 2 -method Accelerated? which implies superlinearly convergence 12
13
Any Questions? About Aitken’s Δ 2 -method 14
Accelerating convergence Anything to further enhance? 15
16
Steffensen’s method 17
Restoring quadratic convergence to Newton’s method 18
19
20
Any Questions? 21
Two disadvantages n answer 22
Two disadvantages n 23
Any Questions? Chapter 2 Rootfinding (2. 7 is skipped) 24
Exercise Due at 2011/4/25 2: 00 pm Email to darby@ee. ncku. edu. tw or hand over in class. Note that the last problem includes a programming work. 25
26
27
28
29
(Programming) 30
Chapter 3 Systems of equations 31
Systems of equations Definition 32
3. 0 Linear algebra review (vectors and matrices) 33
Matrix Definitions 34
Any Questions? m, n, i, j, EQUAL, SUM, SCALAR MULTIPLICATION, PRODUCT… 35
The inverse matrix cannot be skipped 36
37
Any questions? question answer 38
Any questions? answer 39
Any questions? 40
The determinant cannot be skipped, either 41
cofactor 42
43
Link the concepts n All these theorems will be extremely important throughout this chapter n Nonsingular matrices n Determinants n Solutions of linear systems of equations 44
45
(Hard to prove) 46
Any Questions? 3. 0 Linear algebra review 47
3. 1 Gaussian elimination (I suppose you have already known it) 48
An application problem 49
n 50
Following Gaussian elimination 51
Any Questions? Gaussian elimination 52
Gaussian elimination Operation counts 53
Operation counts Comparison n 54
3. 2 Pivoting strategy 55
56
57
Compare to x 1=1, x 2=7, x 3=1 58
Pivoting strategy n n n To avoid small pivot elements A scheme for interchanging the rows (interchanging the pivot element) Partial pivoting 59
http: //thomashawk. com/hello/209/1017/1024/Jackson%20 Running. jpg In action 60
61
62
Any Questions? 63
From the algorithm view n How to implement the interchanging operation? hint – change implicitly n Introduce a row vector r – each time a row interchange is required, answer we need only swap the corresponding elements of the vector – number of operations from 3 n to 3 64
From the algorithm view n How to implement the interchanging operation? – change implicitly n Introduce a row vector r – each time a row interchange is required, answer we need only swap the corresponding elements of the vector – number of operations from 3 n to 3 65
From the algorithm view n 66
http: //thomashawk. com/hello/209/1017/1024/Jackson%20 Running. jpg In action 67
Without pivoting 68
69
70
n 71
Scaled partial pivoting 72
Scaled partial pivoting An example 73
Any Questions? 74
Scaled partial pivoting A blind spot of partial pivoting answer 75
Scaled partial pivoting A blind spot of partial pivoting 76
Scaled partial pivoting 77
78
http: //thomashawk. com/hello/209/1017/1024/Jackson%20 Running. jpg In action 79
80
81
n 82
Any Questions? 3. 2 Pivoting strategy 83
- Slides: 83