I am trying to fit a polynomial through three points, where I also know the derivatives at the two endpoints. I don't need a truly general solution. My specific problem is constrained as follows:
f(0) = 1; f'(0) = 0
f(w) = 1/2, where 0 < w < 1/2; f'(w) is unconstrained
f(1) = 0; f'(1) = 0
I believe this means I have five degrees of freedom, rather than the usual four DOF of the cubic spline interpolation problem. Therefore I'm expecting to fit a quartic polynomial through my points.
I know the Taylor series approach to polynomial interpolation, where you have a point and the nth derivatives at that point. I also know the Vandermonde matrix inversion approach to polynomial interpolation, where you know n points, and the derivatives are not constrained.
I don't know of any polynomial interpolation approach which makes use of a mix of point and derivative information, other than the cubic spline interpolation. Is there a method which handles my specific problem?
Thanks for any advice.
Let $$f(x)=ax^4+bx^3+cx^2+dx+e$$ Then $$f'(x)=4ax^3+3bx^2+2cx+d$$ Your conditions are $$f(0)=e=1\\f'(0)=d=0\\f(1)=a+b+c+1=0\\f'(1)=4a+3b+2c=0\\f(w)=aw^4+bw^3+cw^2+1=1/2$$ The last three lines form a system of three linear equations with three unknowns. You can use Cramer's rule to solve it.