Numerical Analysis

Math 370  MWF 2:30-3:25
Spring 1997  127 Fowler

(02.26.97) \underline{Polynomial Interpolation (continued)} Weierstrass Approximation Theorem If f is defined and continuous on [a,b] and ϵ > 0 is given, then there exists a polynomial P (x), defined on [a,b] such that |f(x) − P (x)| < ϵ, for all x ∈ [a, b]. What does this mean? Write out what this theorem means, in your own words. Then share your meaning with your nearest neighbor.

\underline{Example} Recall the previous example of linear interpolation that we went through with U.S. Census Data. Consider the more general problem of having two data points (sometimes called nodes) (x_{0}, y_{0}) and (x_{1}, y_{1}). Write down a formula for the value of the interpolated value P (x) below...

Lagrange Polynomials The general form of the interpolating polynomial P (x) of degree 1 through the points (x_{0}, y_{0}) and (x_{1}, y_{1}) has the form:

P (x) = \frac{(x − x_{1})}{x_{0} − x_{1}}y_{0} + \frac{(x − x_{0})}{(x_{1} − x_{0})}y_{1}

This can also be written as P (x) = y_{0}L_{0}(x) + y_{1}L_{1}(x), where L_{k}(x) are called Lagrange Polynomials of degree 1 for x_{k}. To be precise we should write these Lagrange polynomials as L_{1,0} and L_{1,1} to show that they are of degree 1 for x_{0} and x_{1}, respectively. Let’s show that the answer you wrote down to the generic linear interpolation scheme is the same thing as the form when written using Lagrange Polynomials \underline{Example} Show that P (x_{0}) and P (x_{1}) has the required values that you want.

What does this tell you about properties of L_{k}(x_{j})? (Is there a difference whether k = j or k ≠ j)?

In general Lagrange polynomials of degree n, are given by

L_{n,k}(x) = ∏_{i≠k^{i=0}}^{n} \frac{x − x_{i}}{x_{k} − x_{i}}

Using a Lagrange Polynomial of degree n to interpolate a function value P (x) between n + 1 function values y_{k} produces

P (x) = ∑_{k=1}^{n} L_{n,k}(x)y_{k}

Error Formula for Lagrange Interpolation The error between a function f(x) and its interpolating polynomial P (x) derived from the nodes x_{0}, x_{1}, … x_{n} on [a,b] is given by

e(x) = |f(x) − P (x)| = \frac{f^{(n+1)}(ξ(x))}{(n + 1)!}(x − x_{0})(x − x_{1}) ⋯ (x − x_{n})

where ξ(x) is in (a, b) \underline{Example} Determine the second degree interpolating polynomial for the function f(x) = 1/x which uses the nodes x_{0} = 2, x_{1} = 2.5 and x_{2} = 4

(a) First write down the Lagrange polynomials of degree 2 and

(b) Combine your answer above to form P (x)

(c) Use your interpolant to approximate P (3)

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