# Interpolation Topics For Test #2

The following topics and corresponding sections of the text will be covered on test #2.

• 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)

### 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, 7.3, 7.5, 7.15