Lagrange Interpolating Polynomials
- Ryan Hirst
Suppose we have a function f(x), whose values are known at a set of points , and we would like to estimate f(x) at points which are not given. If we approximate f(x) by another function y(x) with the same values as f(x) at the given points, we can use y(x) to interpolate additional values.
In Analysis, it is often convenient to write L(x) as a sum of n polynomials, each of order n-1: Where is the x-value of the kth tabular point. It can be shown that satisfies the given conditions. Alternately, I set up the problem more simply. Suppose, for example, n=3. Let points . Then I have Written as a matrix multiplication, with the coefficients as the unknowns: Or, in general, , with solution The inverse matrix can be found using Gaussian elimination. Corresponding tools in GGB: Geogebra recognizes input A^(-1) as matrix inverse. L(x) is the same as Polynomial[<list of points>] __________ Given enough points ( sufficiently large n), L(x) will converge to a smooth function. If the tabular poitns themselves are not exact, caution is in order. Small changes in the given values can dramatically alter the behavior of the interpolated curve. For example, suppose L(x) above is, in fact, identically f(x), but there are errors in our data. Try perturbing the points. Use Offset to slide a new set of points along the curve; Jitter to introduce some random displacement. play to browse possible changes to L(x) caused by this displacement.