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Re: Finding Lobatto Points
Posted:
Dec 10, 1996 7:41 AM
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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..n-1, 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
lobatto-points are -1,1 and the zeroes of pn', there pn is the n-th
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
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