1. MA4229 Lectures 15, 16 Week 13 Nov 1,2 2010 Chapter 18 Interpolation by piecewise polynomials

2. Interlacing Roots of Orthog. Poly. http://en.wikipedia.org/wiki/Orthogonal_polynomials http://en.wikipedia.org/wiki/Orthogonal_polynomials#Interlacing_of_roots

3. Disadvantages of Polynomial Approx. Question Discuss other types of approximation. Question Discuss how polynomial approximation by interpolation works. Question What are some of its disadvantages? Question What are some of their disadvantages?

4. Spaces of Piecewise Polynomials Example 1.Piecewise linear is a piecewise polynomial ofA function if the restrictions degree n with respect to data points and are determined by the values

5. Spaces of Piecewise Polynomials Example 2. Piecewise cubic for determined from values at four consecutive data points foruse data points foruse data points foruse data points

6. Spaces of Piecewise Polynomials Example 3. Piecewise cubic for by the values of determined at five consecutive points near Question What is the smoothness of the functions in each of these three spaces of piecewise polynomials ? For each the value is determined whereinterpolates Then are

7. Interpolation by Piecewise Polynomials in any of these spaces of piecewise polynomials is of a function by piecewise polynomials obtained by setting This provides three interpolation methods that are Question What is the accuracy of each method ? described in detail in Section18.1 in Jackson’s book.

8. Cardinal Bases Each of these three spaces admits a basis consisting of functionsthat satisfy in one of these spaces satisfiesAny function Figure 18.1 shows cardinal functions for each space. Question How does this compare with the Lagrange basis for described by equation (4.3) on page 33 ?

9. Interpolation Using Cardinal Bases of a function easy. The function at the points Question Derive an upper bound for the operator norm is the unique function in the space that interpolates of the interpolation operator (with respect to the norm ) using these cardinal functions. Suggestion Review Section 4.4.

10. Spline Spaces For a set of knots Read Section 4.4 Piecewise poly. Approx., pp 28-31 and integerSet and define the spline space of spline functions of degreewith knots

11. Spline Spaces Question Show that Define where the coefficients Suggestion : show that Question What is

12. Cubic Spline Interpolation Let so it has the form Question Show that interpolate

13. Cubic Spline Interpolation Question Why does Question Use these equations to derive equation 18.11 on page 216 of Jackson. Question Show that if the knot spacings are equal, then This is the three term recursion for cubic splines.

14. Cardinal Cubic Splines Now we consider an infinite number of uniformly cubic splinesthat satisfy Then Question Derive equation 18.18 on page 217 spaced knotsand construct cardinal Question Why mustif is bounded ? Question Why doesimply decays decays exponentially.

15. Tutorial 8 Due Tuesday 9 November Answer all questions in these vufoils and write your Read Section 18.3 and prepare to explain the material answers up in sufficient detail to present them to the class during the tutorial session. during the tutorial session.