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INTERPOLATION

INTERPOLATION

LAGRANGRIAN INTERPOLATING POLYNOMIALS
The Lagrangian form of an interpolating polynomial is a modification of the Newton form of the interpolating polynomial that has been devised to avoid the divided differences:
pn(x)  =  n⎲⎳i = 0Li(x)f(xi)        =  Lo(x)f(xo) + L1(x)f(x1) + ... + Li(x)f(xi) + ... + Ln(x)f(xn)       Lj(x)  =  n∏j = 0, j ≠ i(x − xj)/(xi − xj)      
where i = 0, n is the number of distinct points where values are known, which results in an nth-order polynomial (since there are n + 1 known points).
An example of a linear (n = 1) Lagrangian polynomial between two points is
p1(x)  =  L0(x)f(x0) + L1(x)f(x1)       L0(x)  =  (x − x1)/(x0 − x1)       L1(x)  =  (x − x0)/(x1 − x0)      
A second-order Lagrangian polynomial with three known points is
p2(x)  =  L0(x)f(x0) + L1(x)f(x1) + L2(x)f(x2)       L0(x)  =  ((x − x1))/((x0 − x1))((x − x2))/((x0 − x2))       L1(x)  =  ((x − x0))/((x1 − x0))((x − x2))/((x1 − x2))       L2(x)  =  ((x − x0))/((x2 − x0))((x − x1))/((x2 − x1))      
SPLINES
The problem with polynomial interpolation is that the increasing order of the interpolation can result in higher-order oscillations and overshoots in the computed function that can cause errors in relatively simple interpolation errors. Spline techniques are lower-order schemes that circumvent this problem. A spline ensures that the curve fit through the points passes through the known points. A spline can be of different orders, where the order of the spline indicates the kind of curve that exists between points. For example, a spline of order i or Si results in interpolations of the order
S1  =  a2x + a1       S2  =  b3x2 + b2x + b1       S3  =  c4x3 + c3x2 + c2x + c1      
Applying the first-order spline as the example, points are interpolated between each known point or knot in the following manner
p(x)  =  p(x0) + m0(x − x0)  x0 ≤ x ≤ x1       p(x)  =  p(x1) + m1(x − x1)  x1 ≤ x ≤ x2       ⋮   p(x)  =  p(xi) + mi(x − xi) xi ≤ x ≤ xi + 1       ⋮   p(x)  =  p(xn − 1) + mn − 1(x − xn − 1)  xn − 1 ≤ x ≤ xn      
where the slope, mi, is defined as
mi = ((p(xi + 1) − p(xi)))/((xi + 1 − xi))
In order to use splines, the function must have some specific mathematical properties so that the interpolating curves between each point will meet correctly at the knot points. For an nth-order spline:
  • the derivatives from 1 to n − 1 of the function must be equal at the interior knots
  • the derivatives from 1 to n − 1 of the function must be zero at the end knots
The most commonly used spline is the cubic (third-order) spline.

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