Please see Directions for use for more information.
Learning Project Summary edit
Content summary edit
This learning project aims to provide as an introduction to the specialized spline functions known as B (or Basis) splines. B splines have varying applications, including numerical analysis.
- Define B Spline
- Differentiate types of B Splines
- Approximation using B Splines
Lesson 0: Prerequisite edit
B splines derive from Splines, therefore an understanding of Splines in general is beneficial to completing this lesson. In short, a Spline function approximates another function by defining a set of polynomials
where each of these polynomials defines a specific piece of the resulting Spline. might exist on the interval , might exist on the interval , and so on. The result will be a piecewise approximation to some other exact function.
Lesson 1: Definition edit
A definition of B splines assumes:
- an infinite set of knots are defined at points along the x-axis (can be spaced uniformally or not), that is,
Degree 0 (or constant) edit
With that in mind, we can now move on to the simplest of B splines, those of degree 0, which are defined as
In other words, a degree 0 B spline is equal to 0 at all points except on the interval .
It should now be easy to see that a degree 0 Spline can be formed as a weighted linear combination of degree 0 B splines so that,
Degree 1 (or linear) edit
Logically, the next B spline are those of degree 1, defined as
This might seem difficult to visualize at first glance, but its actually quite easy. Just like , it is 0 at quite nearly all points. However, we now have the two intervals, and , at which .
On the first interval it is easy to see that, substituting and give 0 and 1, respectively. Thus, this function yields an upward sloping line, with a maximum height of 1. Similarly, the second interval yields a downward sloping line, starting from the point that the first interval terminates.
Again, similarly to , it should now be easy to see that a degree 1 Spline can be formed as a weighted linear combination of degree 1 B splines so that,
Degree k (or quadratic and above) edit
The higher degree B splines, and actually including , are defined as
Lesson 2: Approximation edit
We have seen how B splines can be used to construct general Spline. Now we will discuss a process for approximating a generic function by using B splines.
Schoenberg's Approximation edit
This specific approximation utilizes (or quadratic B splines) to approximate a function with (or a quadratic Spline). The approximation is defined as
- , or the average of the next two knots
In real life, we would only approximate the function over a specific interval .
Active participants edit
Active participants in this Learning Group
Inactive participants edit
- Jlietz 18:04, 12 December 2006 (UTC)