LAGRANGE'S INTERPOLATION FORMULA
This is again an N^{th}
degree polynomial approximation formula to the function f(x), which
is known at discrete points x_{i},
i = 0, 1, 2 . . .
N^{th}.
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 N^{th}
degree polynomial then the N^{th}
divided difference of f(x) constant and (N+1)^{th}
divided difference is zero. That is
f [x_{0}, x_{1}, . . . x_{n}, x] = 0
From the second property of divided difference we can write
f_{0}

+

f_{n}


f_{x}

= 0 


+ . . . +


(x_{0 } x_{1}) . . . (x_{0 } x_{n})(x_{0
}
x)

(x_{n } x_{0}) . . . (x_{n}  x_{n1})(x_{n
}
x)


(x_{ } x_{0}) . . . (x_{ } x_{n})

or

(x_{ } x_{1}) . . . (x_{ } x_{n})


(x_{ } x_{0}) . . . (x  x_{n1})


f(x) =


f_{0 }+ . . . +


f_{n}


(x_{0 } x_{1}) . . . (x_{0 } x_{n})


(x_{n } x_{0}) . . . (x_{n}
 x_{n1})


n

(

n



)f_{i}


S

 


x  x_{j}


j = 0


(x_{i}  x_{j})

i = 0

j ¹ 1



Since Lagrange's interpolation is also an N^{th}
degree polynomial approximation to f(x) and the N^{th}
degree polynomial passing through (N+1)
points
is unique hence the Lagrange's and Newton's divided difference approximations
are one and the same. However, Lagrange's formula is more convinent to
use in computer programming and Newton's divided difference formula is
more suited for 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_{ } x_{1}) (x_{ } x_{2})(x
x_{3})(x_{ } x_{4})


(x_{ } x_{0})(x_{ } x_{1})
(x_{ } x_{2})(x_{ } x_{3})


f(x) =


f_{0}+ . . . +


f_{4}


(x_{0 } x_{1}) (x_{0 } x_{2})(x_{0
}
x_{3})(x_{0 } x_{4})


(x_{4 } x_{0})(x_{4 } x_{1})(x_{4
}
x_{2})(x_{4 } x_{3})



(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)



813

7 x 6 x
4 x 3


= 1.831
