Interpolation using rate of change

Question

I have a set of data with missing points, which I estimated using spline interpolation. I've now been given the rates of change at each data point. How will this change/improve my current interpolation scheme?

Answer 1

The change will probably be quite significant.

In standard cubic spline interpolation, the basic idea is to insert a cubic polynomial segment between each two consecutive data points. On a given segment, the cubic polynomial is defined by points and slopes at its extremities. You have the point values, from the original data, but the slopes must be fabricated somehow. Often this done by solving a system of linear equations that enforce $C_2$ continuity, but there are other methods, too. The problem, of course, is that these methods of fabricating slope data are all somewhat arbitrary. Unless we have some reason to believe that our data came from a $C_2$ function, how do justify using this constraint to compute slopes??

If you have the slope at each data point, then you already have enough information to construct cubic polynomial segments; no fabrication is required. The interpolation process is much simpler (no computation of slopes), and the results are likely to be quite different from those produced by any slope fabrication process.

Answer 2

In order to do that, you must use quintic splines since derivative data introduce two more continuity equations.

Searching, I found some material which could be of interest to you:

• http://fr.mathworks.com/help/curvefit/spapi.html (which means that you have all required functionalities if you are a Matlab user)
• http://fr.mathworks.com/help/curvefit/spapi.html
• http://biomch-l.isbweb.org/threads/3532-cubic-vs-quintic-spline
• http://www.pstat.ucsb.edu/faculty/yuedong/ASSIST/ssr.html
• http://en.wikipedia.org/wiki/Hermite_interpolation