Numerical Analysis

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

(03.05.97) \underline{Piecwise Polynomial Interpolation: SPLINES} Piecewise Linear Interpolation Instead of using one interpolating polynomial of high degree, there is also the idea of using a lot of polynomials of small degree to do interpolation. For example, with the example of producing cheap and dirty sines, we said that we would use a number of linear interpolations between consecutive nodes. We say that these are piecewise polynomials because the “pieces” (the individual linear interpolating polynomials) are tied together at the “knots” (also known as the nodes). These pieces are also known as splines. Figure Here is a figure of piecewise linear interpolation

Given x_{k}, y_{k} for k = 0ton come up with the equation for a typical piecwise linear interpolant s_{k}(x)

What is the relationship between s_{k}(x) on the kth interval and s_{k+1} on the (k+1)th interval?

What are some drawbacks of doing piecewise linear interpolation instead of doing piecewise polynomial interpolation with a higher degree polynomial?

The most common piecewise polynomial interplation is done with polynomials of degree, thus leading to the famous cubic spline problem.

Cubic Splines Suppose we want to have piecewise interpolating polynomials which are continuous and smooth at all the nodes. What kinds of conditions on the s_{k}(x) and s_{k+1}(x) would this entail?

Figure Here is a figure illustrating what piecwise cubic interpolation looks like:

Given that s_{k}(x) = a_{k} + b_{k}(x − x_{k}) + c_{k}(x − x_{k})^{2} + d_{k}(x − x_{k})^{3} and that we have N + 1 data points (x_{k}, y_{k}) how many cubic splines will we use to interpolate the entire interval from x_{0} to x_{N}? How many unknown constants do we have to compute (i.e find) in order to define all of these s_{k}(x)?

How many equations do we have to solve for to find these unknown constants?

What is a natural spline? What is a clamped spline?

NOTE: Wed & Fri Classes Are Held In Fowler 127