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sols_wk8 - Homework 8 due Thursday December 2 before 13:30...

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Homework 8 due Thursday December 2 before 13:30 in corre- sponding group folder on door of office CM110 1. Convince yourself that the polynomial of degree n which agrees with a function f at distinct nodes x 0 , x 1 , . . . , x n may be written as p n ( x ) = f ( x n ) + ( x x n ) f [ x n , x n - 1 ] + ( x x n )( x x n - 1 ) f [ x n , x n - 1 , x n - 2 ] + . . . + ( x x n )( x x n - 1 ) . . . ( x x 1 ) f [ x n , x n - 1 , . . . , x 0 ] For equally spaced nodes x k = x 0 + kh , ( k = 0 , 1 , . . . , n ), backward differences are f n = f ( x n ) f ( x n - 1 ), and for k 0 k +1 f n = k f n − ∇ k f n - 1 . From the divided-difference formula above derive Newton’s backward difference for- mula, in the form p n ( x n + th )= f ( x n ) + t f n + t ( t + 1) 2 2 f n + . . . + t ( t + 1) . . . ( t + n 1) n ! n f n . [ Note that for interpolation, rather than extrapolation, t is negative .] ——————————————————————————————— It has been shown in lectures that the polynomial of degree n interpolating f at the nodes { x i } n i =0 is given by p n ( x ) = n i =0 f ( x i ) L i ( x ) and this is unique. But there is more than one way of writing this polynomial, of
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