An extended rational interpolation method

To interpolate function, f(x), a ⩽ x ⩽ b, when we have some information about the values of f(x) and their derivatives in separate points on {x0, x1, … , xn} ⊂ [a, b], the Hermit interpolation method is usually used. Here, to solve this kind of problems, extended rational interpolation method is presented and it is shown that the suggested method is more efficient and suitable than the Hermit interpolation method, especially when the function f(x) has singular points in interval [a, b]. Also for implementing the extended rational interpolation method, the direct method and the inverse differences method are presented, and with some examples these arguments are examined numerically.