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hw6solns - 151A HW 6 Solutions Will Feldman 1 Let f C n 1[a...

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151A HW 6 Solutions Will Feldman 1. Let f C n +1 ([ a, b ]) and p be the interpolating polynomial at the n + 1 equally spaced node points a = x 0 , ..., x n = b with ( b - a ) /n = h = x j +1 - x j . Recall that we have the error bound | f ( x ) - p ( x ) | ≤ | f ( n +1) ( ξ ) | ( n + 1)! Π n j =0 | x - x j | . (a) Now for x [ a, b ] arbitrary there is some j such that x j x x j +1 . We an upper bound for g ( x ) = | ( x - x j )( x - x j +1 ) | = ( x - x j )( x j +1 - x ). Just use calculus to find 0 = g 0 ( x ) = x j + x j +1 - 2 x so that the maximum value of g occurs when x = ( x j + x j +1 ) / 2. Plugging in we get that | ( x - x j )( x - x j +1 ) | ≤ ( h/ 2) 2 . (b) Now for i < j use the bound ( x - x i ) ( j - i + 1) h so that Π i<j ( x - x i ) h j Π i<j ( j - i + 1) = ( j + 1)! h j . Similarly for i > j + 1 we have that ( x i - x ) ( i - j ) h so that Π i>j +1 ( x i - x ) h n - j - 1 Π i>j +1 ( i - j ) = ( n - j )! h n - j - 1 . Combining the two above bounds with the one from (a) gives the desired result. (c) Note that ( n - j )! = ( n - j ) · ( n - j - 1) · · · 2 has n - j - 1 terms in the product and n ! ( j +1)! = n · ( n - 1) · · · ( j + 2) also has n - j - 1 terms and n - j - i n - i for i = 0 , ..., n - j - 2 so ( n - j )! = Π n - j - 2 i =0 ( n - j - i ) Π n - j - 2 i =0 (
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