Question

(b) Derive a numerical differentiation formula of order O(h4) by applying Richardson ertrapolation to 2 MI 2h 120

Answer 1

f^1(x0 = 1/2h [f9x + h) -f(x-h)]-h^2/6 f^111(x) -h^4/120 f^(5) (x) + rightarrow q (1) substitute h/2 by h then the we get f^1 (x0 = 1/2(h/2) [f(x + h/2) -f(x -h/2) - h^2/2^2 times 6 f^111(x)- h^4/2^4 times 120 f^(5) (x) rightarrow (2) multiply with (*0 4 to (20 then we get (5) 4 f^1 (x) = 4/h [f(x + h/2) -f(x-h/2) -h^2/6 f^111(x) -h^4/480 f(x)----(3) (3) -(1) rightarrow 4f^1(x) - f^1(x) = 4/h [f(x + h/2) -f(x-h/2)]-1/2h[f(x + h)-f(x-h)]- h^2/6 f^111(x) + h^2/6 f^1119x) + (h^4/480 + h^4/120) f(x) +------ 3f^19x) = 1/2h [-f(x + h)-f9x-h) + 8f(x + h/2) - 8f(x-h/2)] + 3h^4/480 f(x0 +--- f^1(x0 = 1/6h [-f9x + h) -f9x-h) + 8f(x + h/2) -8f(x-h/2)] + h^4/480 f(x) +----