How do I go about deriving the coefficients (-1/24, -27/24, 27/24,1/24) for a third-order polynomial approximation of a using the central difference method.
So far I have
ah^3-bh^2-ch+d=f(-h)
ah^3+bh^2+ch+d=f(h)
f(0)=d
f(-3/2)=a(-3/2)^3-b(-3/2)^2-c(-3/2)+d
f(-1/2) = a(-1/2)^3-b(-1/2)^2-c(-1/2)+d
f(1/2) = a(1/2)^3+b(1/2)^2+c(1/2)+d
f(3/2)=a((3/2)^3+b(3/2)^2+c(3/2)+d
I know that I am supposed to solve for c, the linear term but I don't know where to go from there
With the h=(-3/2,-1/2,0,1/2,3/2)
I know that the b and d terms are supposed to cancel but I don't know how
So far I have
ah^3-bh^2-ch+d=f(-h)
ah^3+bh^2+ch+d=f(h)
f(0)=d
f(-3/2)=a(-3/2)^3-b(-3/2)^2-c(-3/2)+d
f(-1/2) = a(-1/2)^3-b(-1/2)^2-c(-1/2)+d
f(1/2) = a(1/2)^3+b(1/2)^2+c(1/2)+d
f(3/2)=a((3/2)^3+b(3/2)^2+c(3/2)+d
I know that I am supposed to solve for c, the linear term but I don't know where to go from there
With the h=(-3/2,-1/2,0,1/2,3/2)
I know that the b and d terms are supposed to cancel but I don't know how