Do quadratic splines really use all the data points?


There are two reasons that linear splines are seldom used

  1. Each spline uses information only from two consecutive data points
  2. The slope of the splines is discontinuous at the interior data points

The answer to resolving the above concerns are to use higher order splines such as quadratic splines. Read the quadratic spline textbook notes before you go any further. You do want what I have to say to make sense to you.

In quadratic splines, a quadratic polynomial is assumed to go thru consecutive data points. So you cannot just find the three constants of each quadratic polynomial spline by using the information that the spline goes thru two consecutive points (that sets up two equations and three unknowns). Hence, we incorporate that the splines have a continuous slope at the interior points.

So does all this incorporation make the splines to depend on the values of all given data points. It does not seem so.

For example, in quadratic splines you have to assume that the first or last spline is linear. For argument sake, let that be the first spline. If the first spline is linear, then we can find the constants of the linear spline just by the knowledge of the value of the first two data points. So now we know that we can set up three equations for the three unknown constants of the second spline as follows

  1. the slope of the first spline at the 2nd data point and the slope of the second spline at the 2nd point are the same,
  2. the second spline goes thru the 2nd data point
  3. the second spline goes thru the 3rd data point

That itself is enough information to find the three constants of the second spline. We can keep using the same argument for all the other splines.

So the mth spline constants are dependent on data from the data points 1, 2, .., m, m+1 but not beyond that.

Can you now make the call on the kind of dependence or independence you have in the constants of the quadratic splines?

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu

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Published by

Autar Kaw

Autar Kaw (http://autarkaw.com) is a Professor of Mechanical Engineering at the University of South Florida. He has been at USF since 1987, the same year in which he received his Ph. D. in Engineering Mechanics from Clemson University. He is a recipient of the 2012 U.S. Professor of the Year Award. With major funding from NSF, he is the principal and managing contributor in developing the multiple award-winning online open courseware for an undergraduate course in Numerical Methods. The OpenCourseWare (nm.MathForCollege.com) annually receives 1,000,000+ page views, 1,000,000+ views of the YouTube audiovisual lectures, and 150,000+ page views at the NumericalMethodsGuy blog. His current research interests include engineering education research methods, adaptive learning, open courseware, massive open online courses, flipped classrooms, and learning strategies. He has written four textbooks and 80 refereed technical papers, and his opinion editorials have appeared in the St. Petersburg Times and Tampa Tribune.

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