Lagrange Interpolating Polynomials and Kronecker Delta
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We are given the Lagrange polynomial in the form:
P_n(x)= y_0*L_0(x)+...+y_n*L_n(x) and
y_i = f(x_i) and L_i(x)= ((x-x_0)***(x-x_n)) / ((x_i-x_0)***(x_i-x_n)).
We must show that L_0(x) + ... + L_n(x) = 1 for all x and n=3.
Later we are to generalize this for all n>0.
But let's just focus on n=3 for now; I may be able to get the rest later.
We know that each L_i(x) equals the Kronecker Delta, i.e. it is either 0 or
1. It is also proven that P_n(x_i)=y_i. From this I concluded that L_i(x)=1
only when x=x_i and zero otherwise. But if I choose any x other than x_0 to
x_3, L_i(x) will be zero. Can I use P_n(x) and let every y_i = 1? Will this make
the sum of the L's equal to 1?
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Solution Summary
Lagrange interpolating polynomials and kronecker delta are investigated.
Solution Preview
Actually the term corresponds to x_i does not appear in both the Numerator and Denominator of the following equation; see the correct equation below.
L_i(x)= [ ...
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