Chapter 7: Interpolation Topics

Topics and Sections from the Text:

• What is interpolation? How does it differ from finding a function to approximate a set of data points? (Section 7.1)
• Understand the relationship between interpolation and solving a system of equations; what does this tell you about existence, uniqueness, and conditioning of the problem? (Section 7.2)
• Polynomial Interpolation: Know how to find a polynomial interpolant using a Monomial Basis, Lagrange Interpolation, and Newton Interpolation. For each, know how to set up the system of equations (the basis matrix), know the advantages and disadvantages of each method. For Newton, know how to find the interpolant incrementally (and how this technique can be beneficial when adding data points to existing points) and using divided differences. (Sections 7.3.1 - 7.3.3)
• Orthogonal Polynomials: What does it mean for two polynomials to be orthogonal? For a set of polynomials to be orthonormal? What are the Chebyshev polynomials and why are they important? (Section 7.3.4)
• Understand the problem of trying to approximate a continuous function with a polynomial of high degree. Understand the role of Chebyshev points in minimizing this problem. (Section 7.3.5)
• Piecewise Polynomial Interpolation: Know what it is; know about Hermite Cubic Interpolation (what conditions are imposed); know about splines (cubic splines in particular) (Section 7.4); be able to figure out how many equations and how many unknowns there are for given interpolation requirements; be able to set up the equations in simple cases (e.g. cubic or quadratic spline interpolating three points).

Questions and Exercises From Chapter 7

Review Questions: pages 333-335, #7.1, 7.3 - 7.5, 7.10, 7.12, 7.13, 7.15 - 7.21, 7.23, 7.24, 7.27 - 7.29

Exercises: pages 335-336, #7.1 - 7.5, 7.15

Additional Exercises

1. Do there exist a, b, c, and d so that the following function is a natural cubic spline?

                        -x                   for  x in [-10, -1]
         S(x) =      ax^3 + bx^2 + cx + d    for x in [-1, 1]
                         x                   for x in [1, 10]
   

2. List all the ways in which the following fails to be a natural cubic spline.

                       x + 1             for x in [-2, -1]
         S(x) =     x^3 - 2x + 1         for x in [-1, 1]
                       x - 1             for x in [1, 2]
   

3. Given the data points (-1, 3), (1, 5), (2, 1). Write the equations one would need to solve to interpolate these points with
   (a) A quadratic spline
   (b) A natural cubic spline