 Using Sampled Data to Estimate Derivatives, Integrals, and Interpolated Values

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In applied mathematics, a function is often only available as a set of sampled points. Even so, much can be inferred about the unknown function. This Demonstration shows how to approximate a linear operation (selected by dropdown menu) on .

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Sampled points are represented here by locator positions. The results are calculated from a polynomial of selected order that best fits these points (using least-squared error).

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Contributed by: Robert L. Brown (March 2011)
Open content licensed under CC BY-NC-SA

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More interesting than the numerical result is the formula used to calculate it. This tool demonstrates an astonishing fact:

The least squares estimator of any linear (Wolfram MathWorld) operator (Wolfram MathWorld) applied to a continuous (Wolfram MathWorld) real analytic (Wolfram MathWorld)function will always be a linear combination (Wolfram MathWorld) of the sampled data. Moreover, the coefficients will be rational if the values of the samples are rational.

These formulas are often very simple. A look at some of the bookmarked examples shows that many famous rules in numerical analysis are a result of this principle. These rules are perfect for real-time data sampling and analysis.

The ability to generate rules for over-sampled and over-constrained data is important for estimating derivatives. Derivatives are very sensitive to errors or noise in the data.

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Robert L. Brown

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