the inter vai interpolation of/ l ailu second derivatives Dunve the tollowing bo
ID: 3167431 • Letter: T
Question
the inter vai interpolation of/ l ailu second derivatives Dunve the tollowing bound on the error due to linear where h = x1-30. interpolation points influence interpolation error through the polynomial 10. Tp (E ). Suppose we are interpolating the function f over the interval 10. The (-1, 1] using linear interpolation. (a) If z 0 =-1 and x| = 1, determine the maximum value of the expression (z - zo)(a - 1) for -1 1. b) If z0V2/2 and z1 V2/2, determine the maximum value of the expression |(z -xo)(x-zi)l for -1 K z 1. How does this compare to the (c) Select any two numbers from the interval [-1, 1] to serve as the interpolation zo)(-1) for -1 3 1, and compare to the maxima found in parts l. The interpolation points infuence interpolation error through the polynomial maximum found in part (a)? points zo and zi. Determine the maximum value of the expression l(x (a) and (b). Suppose we are interpolating the function f over the inter ,l using quadratic interpolation. (a) If zo = , = 0 and x2 1, determine the maximum value of the expression |(z-2)| for -13r sExplanation / Answer
We are asked to find the absolute max value of the function f=|(x-x0)(x-x1)|
a) Given: x0=-1;x1=1
=> f= |(x-(-1)1)(x-1) |= |(x+1)(x-1)|=|x2-1|
Now differentiating w.r.t x gives,
f' = 0
=> 2x=0
=>x= 0
Thus max value of f is at x=0 and its maximum value is f(0)=1
b)Given:x0=-sqrt2/2 ; x1=sqrt2/2
Doing this in the same way,we get
f= | (x+sqrt(2)/2)( x- sqrt(2)/2)|=|x2-2/4|=|x2-1/2|
f'=0
=>2x=0 gives x=0
Thus f is max at x= 0 and its max value is f(0) = 0.5
c) For this part select x0=-1/2 and x1=1/2
=> f=|(x+1/2)(x-1/2)|=|x^2-1/4|
=> f'=0
=>x=0
thus max f=f(0) =1/4
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