Numerical Methods-Lecture VIII: Interpolation
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1 Numerical Methods-Lecture VIII: Interpolation (See Judd Chapter 6) Trevor Gallen Fall, / 113
2 Motivation Most solutions are functions Many functions are (potentially) high-dimensional Want a way to simplify A cloud of points and connecting the dots is one way How should we connect the dots (and choose where they are?) 2 / 113
3 The most basic The basics: Evenly-spaced grid, Connect the dots Linearly or quadratically summarize the line 3 / 113
4 Polynomial Basis We can summarize functions with different bases The most common is the polynomial basis: f (x) = [ ] a 0 a 1 a 2 a n 1 x x 2. x n This is really just the taylor polynomial: f (x) = a 0 + a 1 x + a 2 x a n x n But the polynomials are really colinear This is pretty inefficient coding 4 / 113
5 Monomial Basis 5 / 113
6 Example: Regression Have a cloud of n points (data is Y ), location in grid summarized by X Regress cloud on polynomial basis of dimension i, i < n: ˆβ = (X X ) 1 X Y Then your interpolation is always just: Ŷ = ˆβX This is okay for a lot of problems 6 / 113
7 Example: Regression 7 / 113
8 Example: Regression 8 / 113
9 Example: Regression 9 / 113
10 Example: Regression 10 / 113
11 Example: Regression 11 / 113
12 Example: Regression 12 / 113
13 Runge s Phenomenon: f (x) = 1 1+x / 113
14 Approx: f (x) = 1 1+x. 2 : 1-deg. approx 14 / 113
15 Approx: f (x) = 1 1+x. 2 : 2-deg. approx 15 / 113
16 Approx: f (x) = 1 1+x. 2 : 3-deg. approx 16 / 113
17 Approx: f (x) = 1 1+x. 2 : 4-deg. approx 17 / 113
18 Approx: f (x) = 1 1+x. 2 : 5-deg. approx 18 / 113
19 Approx: f (x) = 1 1+x. 2 : 6-deg. approx 19 / 113
20 Approx: f (x) = 1 1+x. 2 : 7-deg. approx 20 / 113
21 Approx: f (x) = 1 1+x. 2 : 8-deg. approx 21 / 113
22 Approx: f (x) = 1 1+x. 2 : 9-deg. approx 22 / 113
23 Approx: f (x) = 1 1+x. 2 : 10-deg. approx 23 / 113
24 Approx: f (x) = 1 1+x. 2 : 11-deg. approx 24 / 113
25 Approx: f (x) = 1 1+x. 2 : 12-deg. approx 25 / 113
26 Approx: f (x) = 1 1+x. 2 : 13-deg. approx 26 / 113
27 Errors: f (x) = 1 1+x. 2 : 1-deg. approx 27 / 113
28 Errors: f (x) = 1 1+x. 2 : 2-deg. approx 28 / 113
29 Errors: f (x) = 1 1+x. 2 : 3-deg. approx 29 / 113
30 Errors: f (x) = 1 1+x. 2 : 4-deg. approx 30 / 113
31 Errors: f (x) = 1 1+x. 2 : 5-deg. approx 31 / 113
32 Errors: f (x) = 1 1+x. 2 : 6-deg. approx 32 / 113
33 Errors: f (x) = 1 1+x. 2 : 7-deg. approx 33 / 113
34 Errors: f (x) = 1 1+x. 2 : 8-deg. approx 34 / 113
35 Errors: f (x) = 1 1+x. 2 : 9-deg. approx 35 / 113
36 Errors: f (x) = 1 1+x. 2 : 10-deg. approx 36 / 113
37 Errors: f (x) = 1 1+x. 2 : 11-deg. approx 37 / 113
38 Errors: f (x) = 1 1+x. 2 : 12-deg. approx 38 / 113
39 Errors: f (x) = 1 1+x. 2 : 13-deg. approx 39 / 113
40 Errors: f (x) = 1 1+x. 2 : 1-deg. approx 40 / 113
41 Errors: f (x) = 1 1+x. 2 : 2-deg. approx 41 / 113
42 Errors: f (x) = 1 1+x. 2 : 3-deg. approx 42 / 113
43 Errors: f (x) = 1 1+x. 2 : 4-deg. approx 43 / 113
44 Errors: f (x) = 1 1+x. 2 : 5-deg. approx 44 / 113
45 Errors: f (x) = 1 1+x. 2 : 6-deg. approx 45 / 113
46 Errors: f (x) = 1 1+x. 2 : 7-deg. approx 46 / 113
47 Errors: f (x) = 1 1+x. 2 : 8-deg. approx 47 / 113
48 Errors: f (x) = 1 1+x. 2 : 9-deg. approx 48 / 113
49 Errors: f (x) = 1 1+x. 2 : 10-deg. approx 49 / 113
50 Errors: f (x) = 1 1+x. 2 : 11-deg. approx 50 / 113
51 Errors: f (x) = 1 1+x. 2 : 12-deg. approx 51 / 113
52 Errors: f (x) = 1 1+x. 2 : 13-deg. approx 52 / 113
53 Chebychev Polynomials Less violent and colinear polynomials are desirable We also want something that avoids Runge s phenomenon if possible We also want something that s pretty easy to integrate It just so happens Chebychev polynomials are all of these things Defined recursively: T 0 (x) = 1 T 1 (x) = x T n (x) = 2xT n 1 (x) T n 2 (x) Defined over [-1,1]! Needs transformation. 53 / 113
54 Chebychev Basis 54 / 113
55 Properties Also can show that: T n (x) = cosh(n arccosh(x)) And that the roots xj T n are: ( ) 2j 1 x j = cos 2n π j = 1,..., n It just so happens that sampling at Chebychev nodes help minimize the maximum error and make sure it s monotonically decreasing as n increases 55 / 113
56 Interpolation: 1-d You have some function f (x) that you want to interpolate over [a, b]. Define a bridge between your range and the interpolating polynomial s range: Define: z = 2(x a)/(b a) 1, or x = a + (b a)(z+1) 2 Then x = a z = 1 x = b z = 1 x = (a + b)/2 z = 0. Our goal will be, for interpolating polynomial g(z) over [ 1, 1] and f (x) over [a, b], that: g(2(x a)/(b a) 1) = f (x) x 56 / 113
57 Interpolation: 1-d 1. Pick fcn. f (x), degree n, bounds a, b for interpolation. 2. Find Chebychev nodes for f (x) in z space. ( ) 2j 1j z j = cos π j = 1,..., n 2n 3. Convert nodes into x space: x j = a + (b a)(z j + 1) 2 4. Calculate y j = f (x j ) at each x j. This is the value of g(z j ) at each of the corresponding nodes. 5. Given y j, calculate the Chebychev coefficients c j as: c j = 2 n y j T j (z j ) n Your approximation is given by: ˆf (x) = j k=0 c j T j (x) f (x) 57 / 113
58 Fit: f (x) = 1 1+x. 2 : 2-deg. approx 58 / 113
59 Fit: f (x) = 1 1+x. 2 : 3-deg. approx 59 / 113
60 Fit: f (x) = 1 1+x. 2 : 4-deg. approx 60 / 113
61 Fit: f (x) = 1 1+x. 2 : 5-deg. approx 61 / 113
62 Fit: f (x) = 1 1+x. 2 : 6-deg. approx 62 / 113
63 Fit: f (x) = 1 1+x. 2 : 7-deg. approx 63 / 113
64 Fit: f (x) = 1 1+x. 2 : 8-deg. approx 64 / 113
65 Fit: f (x) = 1 1+x. 2 : 9-deg. approx 65 / 113
66 Fit: f (x) = 1 1+x. 2 : 10-deg. approx 66 / 113
67 Fit: f (x) = 1 1+x. 2 : 11-deg. approx 67 / 113
68 Fit: f (x) = 1 1+x. 2 : 12-deg. approx 68 / 113
69 Fit: f (x) = 1 1+x. 2 : 13-deg. approx 69 / 113
70 Errors: f (x) = 1 1+x. 2 : 2-deg. approx 70 / 113
71 Errors: f (x) = 1 1+x. 2 : 3-deg. approx 71 / 113
72 Errors: f (x) = 1 1+x. 2 : 4-deg. approx 72 / 113
73 Errors: f (x) = 1 1+x. 2 : 5-deg. approx 73 / 113
74 Errors: f (x) = 1 1+x. 2 : 6-deg. approx 74 / 113
75 Errors: f (x) = 1 1+x. 2 : 7-deg. approx 75 / 113
76 Errors: f (x) = 1 1+x. 2 : 8-deg. approx 76 / 113
77 Errors: f (x) = 1 1+x. 2 : 9-deg. approx 77 / 113
78 Errors: f (x) = 1 1+x. 2 : 10-deg. approx 78 / 113
79 Errors: f (x) = 1 1+x. 2 : 11-deg. approx 79 / 113
80 Errors: f (x) = 1 1+x. 2 : 12-deg. approx 80 / 113
81 Errors: f (x) = 1 1+x. 2 : 13-deg. approx 81 / 113
82 Interpolation: 2-d 1. Pick fcn f (x, y) deg. n, bounds a,b and c,d for interpolation. 2. Find Chebychev nodes for f (x, y) in z [1] and z [2] space. ( ) ( ) z [1] 2j 1j j = cos π, z [2] 2j 1j j = cos π, j = 1,..., n 2n 2n 3. Convert nodes into x space: (b a)(z[1] j + 1) (d c)(z[2] j + 1) x j = a +, y j = c Calculate ξ i,j = f (x i, y j ) for all points. 5. Given y j, calculate the Chebychev coefficients c j as: n n k=1 l=1 c i,j = ξ i,jt i (z [1] k )T j(z [2] l ) n k=1 T i(z [1] k )T j(z [2] l ) 6. Approximation: n n ( ˆf (x, y) c ij T i 2 x a ) ( b a 1 T j 2 y c ) d b 1 i=1 j=1 82 / 113
83 Monom Fit: f (x) = 1 1+x. 2 : 1-deg. approx 83 / 113
84 Monom Fit: f (x) = 1 1+x. 2 : 2-deg. approx 84 / 113
85 Monom Fit: f (x) = 1 1+x. 2 : 3-deg. approx 85 / 113
86 Monom Fit: f (x) = 1 1+x. 2 : 4-deg. approx 86 / 113
87 Monom Fit: f (x) = 1 1+x. 2 : 5-deg. approx 87 / 113
88 Monom Errors: f (x) = 1 1+x. 2 : 1-deg. approx 88 / 113
89 Monom Errors: f (x) = 1 1+x. 2 : 2-deg. approx 89 / 113
90 Monom Errors: f (x) = 1 1+x. 2 : 3-deg. approx 90 / 113
91 Monom Errors: f (x) = 1 1+x. 2 : 4-deg. approx 91 / 113
92 Monom Errors: f (x) = 1 1+x. 2 : 5-deg. approx 92 / 113
93 Cheby Fit: f (x) = 1 1+x. 2 : 1-deg. approx 93 / 113
94 Cheby Fit: f (x) = 1 1+x. 2 : 2-deg. approx 94 / 113
95 Cheby Fit: f (x) = 1 1+x. 2 : 3-deg. approx 95 / 113
96 Cheby Fit: f (x) = 1 1+x. 2 : 4-deg. approx 96 / 113
97 Cheby Fit: f (x) = 1 1+x. 2 : 5-deg. approx 97 / 113
98 Cheby Fit: f (x) = 1 1+x. 2 : 6-deg. approx 98 / 113
99 Cheby Fit: f (x) = 1 1+x. 2 : 7-deg. approx 99 / 113
100 Cheby Fit: f (x) = 1 1+x. 2 : 8-deg. approx 100 / 113
101 Cheby Fit: f (x) = 1 1+x. 2 : 9-deg. approx 101 / 113
102 Cheby Fit: f (x) = 1 1+x. 2 : 10-deg. approx 102 / 113
103 Cheby Errors: f (x) = 1 1+x. 2 : 1-deg. approx 103 / 113
104 Cheby Errors: f (x) = 1 1+x. 2 : 2-deg. approx 104 / 113
105 Cheby Errors: f (x) = 1 1+x. 2 : 3-deg. approx 105 / 113
106 Cheby Errors: f (x) = 1 1+x. 2 : 4-deg. approx 106 / 113
107 Cheby Errors: f (x) = 1 1+x. 2 : 5-deg. approx 107 / 113
108 Cheby Errors: f (x) = 1 1+x. 2 : 6-deg. approx 108 / 113
109 Cheby Errors: f (x) = 1 1+x. 2 : 7-deg. approx 109 / 113
110 Cheby Errors: f (x) = 1 1+x. 2 : 8-deg. approx 110 / 113
111 Cheby Errors: f (x) = 1 1+x. 2 : 9-deg. approx 111 / 113
112 Cheby Errors: f (x) = 1 1+x. 2 : 10-deg. approx 112 / 113
113 Taste of some technical points If we have some function f (x) and some class of approximating functions, Ω We want to find the function ω(x) Ω such that: ω = arg min f g 2 ω Ω Interpolating at Chebychev nodes allows us to obtain a lower bound for the error: f (x) ˆf n 1 (x) 1 2 n 1 max n! f (n) (ξ) ξ [ 1,1] For more see Judd Chapter 6: many ways to approximate functions. 113 / 113
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