需要切比雪夫多项式的高阶导数

Question

I need to evaluate high order (up to 4) derivatives of Chebyshev polynomials at points of the so called Chebyshev grid,

x(j)=cos(πj/N), j=0,...,N

Anyone know how to do that? I tried iterative methods but they are too clumsy. I remember seeing something like that in an old paper but now it's nowhere to be found.

Any help appreciated.

Answer 1

One way to do this (though this might be the iterative method you reject) is to use the recurrence:

T[n+1]'/(n+1) - T[n-1]'/(n-1) = 2T[n]   n>=2

This requires one to be able to compute the derivatives of the first 3 polynomials by hand, but since

T[0](x) = 1
T[1](x) = x
T[2](x) = 2*x*x-1

this is straightforward.

The coefficients in the recurrence are independent of x, so that if T[j,k] is the k'th derivative of the j'th Chebyshev poly, we can easily differentiate it, getting

T[n+1, k]/(n+1) - T[n-1,k]/(n-1) = 2T[n,k-1]    n>=2

So the code could be:

compute the T[n,0] (ie the polynomials) at the point, for n=0..deg
initialise T[j,d] for j=0,1,2 and the required degrees
for j=1..deg
   use the recurrence to compute the remaining polynomials