interpolation of unequal intervals

interpolation of unequal intervals and lagrange interpolation

1. Gandhinagar Institute of Technology Numerical & Statistical Methods For Information Technology (2140706) Active Learning Assignment “Topic : Interpolation for unequal intervals and Lagrange's interpolation” Name: Vaani Pathak (170120107131) Branch : CE Div : B3

2. Interpolation with unequal intervals If the values of x are unequally spaced then interpolation formulae for equally spaced points cannot be used. It is therefore, desirable to develop interpolation formulae for unequally spaced values of x. There are two such formulae for unequally spaced values of x. Lagrange's interpolation formula. Newton’s interpolation formula with divided difference.

3. Lagrange Lagrange’s formula is applicable to problems where the independent variable occurs at equal and unequal intervals, but preferably this formula is applied in a situation where there are unequal intervals for the given independent series.

4. Lagrange's interpolation formula f(x) = (x-x 1 ) (x-x 2 )…. (x- x n ) y + (x- x ) (x-x 2 )….(x- x n ) y 1 (x - x 1 ) (x - x 2 )….(x - x n ) (x 1 – x )(x 1 – x 2 )….(x 1 - x n ) + (x- x ) (x- x 1 )….(x- x n-1 ) y n ( x n – x ) ( x n – x 1 )….( x n – x n-1 )

5. Example : Compute f(9.2) by using Lagrange's interpolation method from the following data : Solution : By Lagrange's interpolation formula, f(x) = (x - x 1 ) (x - x 2 ) f(x ) + (x- x ) (x- x 2 ) f(x 1 ) + (x - x 1 )(x – x 2 ) (x 1 - x )(x 1 – x 2 ) (x – x ) (x – x 1 ) f(x 2 ) (x 2 – x ) (x 2 - x 1 ) x 9 9.5 11 f(x) 2.1972 2.2513 2.3973

6. f(9.2) = (9.2 – 9.5) (9.2 - 11) (2.1972) + (9 - 9.5) (9 - 11) (9.2 – 9) (9.2-11) (2.2513) + (9.5 – 9) (9.5 – 11) (9.2 – 9) (9.2 - 9.5) (2.3979) (11 – 9) (11 - 9.5) = 1.1865 + 1.0806 + 0.048 =2.2191

7. Thank you