lagrange's interpolation formula
DESCRIPTION
lTRANSCRIPT
LAGRANGE'S INTERPOLATION FORMULA
This is again an Nth degree polynomial approximation formula to the function f(x), which is known
at discrete points xi, i = 0, 1, 2 . . . Nth. The formula can be derived from the Vandermonds
determinant but a much simpler way of deriving this is from Newton's divided difference formula. If
f(x) is approximated with an Nth degree polynomial then the Nth divided difference of f(x)
constant and (N+1)th divided difference is zero. That is
f [x0, x1, . . . xn, x] = 0
From the second property of divided difference we can write
f0 +
fn fx = 0+ . . . +
(x0 - x1) . . . (x0 - xn)(x0 - x) (xn - x0) . . . (xn - xn-1)(xn - x) (x - x0) . . . (x - xn)
or
(x - x1) . . . (x - xn) (x - x0) . . . (x - xn-1)
f(x) = f0 + . . . + fn (x0 - x1) . . . (x0 - xn) (xn - x0) . . . (xn - xn-1)
n
( n
)fi
S| | x - xj
j = 0 (xi - xj)
i = 0 j ¹ 1
Since Lagrange's interpolation is also an Nth degree polynomial approximation to f(x) and the Nth
degree polynomial passing through (N+1) points is unique hence the Lagrange's and Newton'sdivided difference approximations are one and the same. However, Lagrange's formula is moreconvinent to use in computer programming and Newton's divided difference formula is more suitedfor hand calculations.
Example : Compute f(0.3) for the data
x 0 1 3 4 7
f 1 3 49 129 813
using Lagrange's interpolation formula (Analytic value is 1.831)
(x - x1) (x - x2)(x- x3)(x - x4) (x - x0)(x - x1) (x - x2)(x - x3)
f(x) = f0+ . . . + f4 (x0 - x1) (x0 - x2)(x0 - x3)(x0 - x4) (x4 - x0)(x4 - x1)(x4 - x2)(x4 - x3)
(0.3 - 1)(0.3 - 3)(0.3 - 4)(0.3 - 7) (0.3 - 0)(0.3 - 3)(0.3 - 4)(0.3 - 7) = 1+ 3 + (-1) (-3)(-4)(-7) 1 x (-2)(-3)(-6)
(0.3 - 0)(0.3 - 1)(0.3 - 4)(0.3 - 7) (0.3 - 0)(0.3 - 1)(0.3 - 3)(0.3 - 7)
49 + 129 + 3 x 2 x (-1)(-4) 4 x 3 x 1 (-3)
(0.3 - 0)(0.3 - 1)(0.3 - 3)(0.3 - 4)
8137 x 6 x 4 x 3
= 1.831
Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
Numerical Solution of Partial Differential Equations
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