A Problem of Mixed Numerical Derivatives
Michael Monagan, CECM@SFU
Given f(x), a function of x, the Maple diff command can often compute a formula for f'(x). E.g. for the Bessel function J_v(x), we have > diff(BesselJ(v,x),x); v BesselJ(v, x) -BesselJ(v + 1, x) + --------------- x But Maple cannot compute the derivative of J_v(x) wrt the parameter v. A consequence of this is that Maple cannot graph this derivative. In such cases numerical derivatives are necessary. I'll describe how to compute f'(x) and f''(x) accurate to n digits of precision based on the formulae f'(x) = (f(x+h)-f(x-h))/2/h + O(h^2) f''(x) = (f(x+h) - 2 f(x) + f(x-h))/h^2 + O(h^2). The problem, which I do not know how to solve, is how to do this for mixed derivatives. E.g. how to estimate the partial derivative diff(f(x,y),x,y) at a point x=a,y=b. I.e. D[1,2](f)(a,b) in Maple. This work was motivated by professor Yury Brychkov's work where he gave formulae for derivatives of special functions w.r.t. parameters but restricted to special cases of the parameter.