Function interpolation, Lagrange Interpolation
1.Iterative Methods.
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones.
Definition:
2. Lagrange Interpolation formula.
Given few real values x1, x2, x3, …, xn and y1, y2, y3, …, yn and there will be a polynomial P with real coefficients satisfying the conditions P(xi) = yi, ∀i = {1, 2, 3, …, n} and degree of polynomial P must be less than the count of real values i.e., degree(P) < n. The Lagrange Interpolation formula for different orders i.e., nth order is given as,
Lagrange Interpolation Formula for nth order is-
Lagrange Interpolation Formula for 1st order polynomial is-
Similarly for 2nd Order polynomial, the Lagrange Interpolation formula is-
Proof of Lagrange Theorem.
Substitute observations xi to get Ai
Put x = x0 then we get A0
f(x0) = y0 = A0(x0 – x1)(x0 – x2)(x0 – x3)…(x0 – xn)
A0 = y0/(x0 – x1)(x0 – x2)(x0 – x3)…(x0 – xn)
By substituting x = x1 we get A1
f(x1) = y1 = A1(x1 – x0)(x1 – x2)(x1 – x3)…(x1 – xn)
A1 = y1/(x1 – x0)(x1 – x2)(x1 – x3)…(x1 – xn)
In similar way by substituting x = xn we get An
f(xn) = yn = An(xn – x0)(xn – x1)(xn – x2)…(xn – xn-1)
An = yn/(xn – x0)(xn – x1)(xn – x2)…(xn – xn-1)
3. Use Lagrange's formula find the interpolating polynomial that approximates to the function described by following table
4. Use Lagrange's formula find the interpolating polynomial that approximates to the function described by following table
5. Use Lagrange's formula find the interpolating polynomial that approximates to the function described by following table
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