Science > Physics > closed form solution from Lagrange Interpolating Poly
| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
28 Jul 2005 03:33:12 PM |
| Object: |
closed form solution from Lagrange Interpolating Poly |
Let i and j be integers (indices) that range from {0,...,H}.
Let M be an (H+1)x(H+1) Vandermonde matrix with the following matrix
elements:
M(i,j) = g_i^j = "g_i to the power of j"
where {g_0, g_1, ..., g_H} are H+1 positive real numbers inside the
range (0,R).
Let v be an (H+1) vector with matrix elements
v(i) = g_i / (R-g_i)
Let z be the (H+1) vector determined by solving the matrix equation:
M z = v
Taking advantage of the special form of M and v,
find a closed form solution for the H+1 elements of z for arbitrary H.
I know one can identify z to the coefficients of the Lagrange
Interpolating
Polynomial. However, because the vector v has a particularly simple
form in this
case, one can actually go a step FURTHER and obtain direct closed-form
expressions
for all the elements of z. (I've done it for H=1,2,3,4,5 using
mathematica).
Can anyone write down the general formula for z? It falls into some
obvious patterns...
K. Onyee
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| User: "" |
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| Title: Re: closed form solution from Lagrange Interpolating Poly |
29 Jul 2005 01:14:20 AM |
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Suppose that g_0,g_1,...,g_H are distinct and denote
w(x)=(x-g_0)(x-g_1)...(x-g_H)
L_k(x)= w(x)/((x-g_k)w'(g_k))
A(i,j)= L_j^{(i)}(0)/i! .
=======================
Because L_k(x)=SUM_{i=0 to i=H}A(i,k)x^i , by choosing x=g_j ,
j in {0,1,...,H}, we obtain
SUM_{i=0 to i=H}A(i,k)g_k^i = L_k(g_j)= 0 when k=/=j or =1 (when
k=j).
This means that H:=||A(i,j)|| is the inverse of the Vandermonde matrix
V:= ||g_i^j|| . Therefore , if V:=(v_0,v_1,...,v_H)^T then
the solution Z:=(z_0,z_1,...,z_H)^T of the matrix equation
H*Z= V , is Z=H*V . In conclusion
z_i = SUM_{j=0 to j=H}A(i,j)v_j , i=0,1,...,H
In your case
V_i= SUM_{j=0 to j=H}L_j^{(i)}(0)g_j/((R-g_j)i!).
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| User: "double d" |
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| Title: Re: closed form solution from Lagrange Interpolating Poly |
01 Aug 2005 04:28:28 AM |
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You might be able to do this sum to achieve true simplification.
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| User: "double d" |
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| Title: Re: closed form solution from Lagrange Interpolating Poly |
28 Jul 2005 11:43:22 PM |
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for H=0, the solution is z = g0/ (R-g0).
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