Friday, 27 August, 2010

Facts of Lagrange Interpolation

"Lagrangian interpolation is praised for analytic utility and beauty but deplored for numerical practice." This heading, from the extended table of contents of one of the most enjoyable textbooks of numerical analysis [1],expresses a widespread view. 

[...] Given (x0, f0), (x1, f1), . . ., (xn, fn) with arbitrary spaced xj, Lagrange had the idea of multiplying each fj by a polynomial that is 1 at xj and 0 at the other n nodes and then taking the sum of these n + 1 polynomials. Clearly, this gives the unique interpolation polynomial of degree n or less. [...]
                  ( Erwin Kreyszig, Advanced Engineering Mathematics )

Figure : Lagrange Interpolation of function 1/(1+x*x)


[...] Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as Runge's phenomenon; the problem may be eliminated by choosing interpolation points at Chebyshev nodes. [...]
                  ( Wikipedia )

Although their are superior interpolation methods than lagrange interpolation method. But it is quite easy to understand and it is superior than Taylor Series Approximation of a function. It can be seen easily by following diagram.
Here I choose f(x) = e^x in the interval [0, 2]

[1]  F. S. Acton, Numerical Methods That [Usually] Work, AMS, Providence, RI, 1990.