sjut/mat210/interpolation/lagrangian 2013-14s2
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St. John's University of Tanzania
MAT210 NUMERICAL ANALYSIS2013/14 Semester II
INTERPOLATIONLagrangian Interpolation
Kaw, Chapter 5.04
MAT210 2013/14 Sem II 2 of 11
● Interpolation of functions● Approximating a complex or unknown function
with a simpler function– simpler function usually a polynomial
● Direct Method & Newton Divided Difference● Two approaches for finding the same nth order
polynomial fit for all points in an data set● Lagrangian Inteprolation
● A 3rd way, but based on a weighted average of the function values for points in the data set
Introduction
MAT210 2013/14 Sem II 3 of 11
General Form
f n(x)=∑i=0
i=n
Li(x ) f (x i)
Li (x)= ∏j=0, j≠i
i=n x−x j
x i−x j
The Lagrange elementary polynomials (Li) are the weights
MAT210 2013/14 Sem II 4 of 11
Example: Velocity
MAT210 2013/14 Sem II 5 of 11
Linear
MAT210 2013/14 Sem II 6 of 11
The Formula
f 1(x )=L0(x) f (x0)+L1(x) f (x1)
L0(x)=x−x1x0−x1
L1(x)=x−x0x1−x0
Notice the sign in the denominator
MAT210 2013/14 Sem II 7 of 11
Find v(16)
● Two closest points t=15 and t=20
Same as Direct
MAT210 2013/14 Sem II 8 of 11
Quadratic
MAT210 2013/14 Sem II 9 of 11
Quadratic v(16) ● Use t=10,15,20 as before
Same as Direct
MAT210 2013/14 Sem II 10 of 11
The Error is the Error● The equations are the same, just found in
different ways and kept in different forms
|ϵa|=| vquadratic−v linear
vquadratic|
Same as Direct
MAT210 2013/14 Sem II 11 of 11
In Conclusion● The methods so far are Polynomial
Interpolation Methods and create one nth degree polynomial for all n points.● Direct: Curve fitting by solving n+1
simultaneous equations● NDD: Can be linked to Taylor Series and is
recursve● Lagrangian: Something a weighted average of
function values based on relative distance from the point of interest
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