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Numerical Analysis 1 EE, NCKU Tien-Hao Chang (Darby Chang)
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In the previous slide Fixed point iteration scheme –what is a fixed point? –iteration function –convergence Newton’s method –tangent line approximation –convergence Secant method 2
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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
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2.6 4 Accelerating convergence
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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
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Aitken’s Δ 2 -method 8
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Aitken’s Δ 2 -method Accelerated? 10 which implies super- linearly convergence later answer
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Aitken’s Δ 2 -method Accelerated? 11 which implies super- linearly convergence later
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Aitken’s Δ 2 -method Accelerated? 12 which implies super- linearly convergence
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Any Questions? 14 About Aitken’s Δ 2 -method
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Accelerating convergence Anything to further enhance? 15
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Steffensen’s method 17
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18 Restoring quadratic convergence to Newton’s method
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Any Questions? 21
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Two disadvantages 22 answer
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Two disadvantages 23
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Any Questions? 24 Chapter 2 Rootfinding (2.7 is skipped)
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Exercise 25 Due at 2011/4/25 2:00pm Email to darby@ee.ncku.edu.tw or hand over in class. Note that the last problem includes a programming work.darby@ee.ncku.edu.tw
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30 (Programming)
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Chapter 3 31 Systems of equations
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Systems of equations Definition 32
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3.0 33 Linear algebra review (vectors and matrices)
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Matrix Definitions 34
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Any Questions? 35 m, n, i, j, E QUAL, S UM, S CALAR M ULTIPLICATION, P RODUCT …
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The inverse matrix 36 cannot be skipped
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Any questions? 38 answerquestion
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Any questions? 39 answer
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Any questions? 40
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The determinant 41 cannot be skipped, either
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42 cofactor
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Link the concepts All these theorems will be extremely important throughout this chapter Nonsingular matrices Determinants Solutions of linear systems of equations 44
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46 (Hard to prove)
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Any Questions? 47 3.0 Linear algebra review
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3.1 48 Gaussian elimination (I suppose you have already known it)
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An application problem 49
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Following Gaussian elimination 51
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Any Questions? 52 Gaussian elimination
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Gaussian elimination Operation counts 53
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Operation counts Comparison 54
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3.2 55 Pivoting strategy
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58 Compare to x 1 =1, x 2 =7, x 3 =1
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Pivoting strategy To avoid small pivot elements A scheme for interchanging the rows (interchanging the pivot element) Partial pivoting 59
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60 In action http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
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Any Questions? 63
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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 64 hint answer
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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 65 answer
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From the algorithm view 66
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67 In action http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
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68 Without pivoting
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72 Scaled partial pivoting
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Scaled partial pivoting An example 73
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Any Questions? 74
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Scaled partial pivoting A blind spot of partial pivoting 75 answer
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Scaled partial pivoting A blind spot of partial pivoting 76
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Scaled partial pivoting 77
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79 In action http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
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Any Questions? 83 3.2 Pivoting strategy
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