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Re: Finding Lobatto Points
Posted:
Dec 10, 1996 7:41 AM


In article <57spth$93m@ttacs7.ttu.edu>, kesinger@math.ttu.edu (Jake Kesinger) writes: > Could someone please comment on whether the following is a (theoretically) > valid method of computing the Lobatto points of degree n+1? > > 1. The Lobatto points of degree 3 are {1,0,1}. > 2. If x and y are consecutive Lobatto points of degree n, then > there is exactly one Lobatto point `z' of degree n+1 in the > interval [x,y]. > 3. Each Lobatto point is a simple root of P'_n, so P'n changes sign > at z. > 4. The secant method can be used with initial endpoints [x,y] to > approximate z. > 5. The other two Lobatto points are 1 and 1. > > This method seems to work, but I have been unable to justify it. > > I've also come across an algorithm that uses Newton's method to find > each Lobatto point with initial guess of cos(j*Pi/n), j=1..n1, but > have not found justification for that, either. > > Can anybody point me towards some references regarding such justification? this comes from the theory of orthogonal polynomials lobattopoints are 1,1 and the zeroes of pn', there pn is the nth legendre polynomial, the orthogonal polynomials for weight 1 on [1,1] orthogonal polynomials have all their roots real, simple and inside their reference interval. by rolle's theorem, this implies the same behaviour for the derivatives. for a complete proof see krylow: approximate calculation of integrals (the rule is also given in davis&rabinowitz; numerical integration) hope this helps, peter



