class_13

class_13 - CS205 Class 13 Covered in class: 1, 3, 5...

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CS205 – Class 13 Covered in class: 1, 3, 5 Readings: 6.7, 7.2 to 7.3.3 1. Interpolation a. polynomial of degree n 2 1 2 3 1 n n y c c x c x c x i. Monomial basis 1 1 2 2 3 3 1 1 n n y c c c c where the basis function are 1 ( ) j j x x for 1,2, , 1 j n 1. Polynomial interpolation given a set of 1 n points ( , ) i i x y , find the unique n degree polynomial 2 1 2 3 1 n n y c c x c x c x that interpolates them. 2. Solve Ax = y where A is the     1 1 n n Vandermonde matrix with rows 2 (1, , , , ) n i i i x x x for each data point ( , ) i i x y 3. Example: i.e. for (1,3), (2,4), (5,-3) for quadratic we would have 3 4 3 25 5 1 4 2 1 1 1 1 3 2 1 c c c so we get 6 / 5 2 / 7 3 / 1 3 2 1 c c c and thus the equation is 2 7 5 1 3 2 6 ( ) f x x x   which looks like: 4. But this is not an ideal basis, because as polynomials get higher, the functions have lots of overlap. Plotting 2 8 ( ) 1, , ,..., f x x x x we can see this effect: 0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 ii. Lagrange interpolation 1 1 2 2 3 3 1 1 n n y c
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class_13 - CS205 Class 13 Covered in class: 1, 3, 5...

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