Fast and Stably Recursive Algorithm for Computing Second Derivative of Associated Legendre Functions

Based on the basic principles of elementary algebra,a fast and stably recursive algorithm for computing second derivative of associated Legendre's functions is derived.Numerical tests suggest that this new approach is exactly as precise as general ones.But the principal strength of the new approach is that it is much faster than general ones in computation speed(at least twice as fast).The method is non-singular(the relative accuracy can be achieved to 5×10-10 up to degree and order 3 600 at the poles) and simplicity of formulation and implementation(just need a few lines of code).The approach can compute the second derivative of associated Legendre functions of any latitude quickly and accurately,which is very important for gravity gradient data processing of GOCE satellite.