It is well known that collocation methods are equivalent to Runge-Kutta methods, with the entries of the matrix aij in the Butcher's tableau given by
aij = \int _{t_{m-1}}^{t_m+c_i} \ell_{j}(t) dt,
where \ell _j is the j-th lagrangian polynomial basis for the set c_j, j=1, ... , k. I want to know if there is an alternative formula for the entries of the above matrix. That is, I want to know if one can find an analytical formula to solve the above integral. I am particularly interested in the case when c_j is a Gauss-Legendre quadrature point scaled to [t_{m-1}, t_m].
In case of single variable, the integral of Lagrange polynomial over an interval I = [a,b] when we partition I evenly, we get the Newton's formula for approximating the integral on I. This can be: Trapezoidal rule, Simpson's rules.... generally referred to as Newton Cotes formula for integration.
Thanks!
Tekle Gemechu Thank you, this is actually a good idea, at least for the lowest order Lagrange polynomials. I guess that will do, although I would like to know of a general formula for general order.
Never mind. I have just realized that there exists a general formula and that the error term is also known in general. Thank you Tekle Gemechu , your answer helped solving the problem
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