3.2 When Do We Use Interpolation?

We will use the word interpolation to denote the process of determining a function whose graph passes through all of the given points. Sometimes, the word interpolation is used to mean finding values for an unknown or complicated function at points other than the known data points, especially at points lying between known data points. Interpolation is appropriate when the given data points are known to be accurate.

The most straightforward type of interpolation involves determining a polynomial that matches the given data points. This is the type of function that Professor Tapia used when solving the problem in Section 3.1. Polynomials are convenient because there is always a polynomial as close as we want to whatever function we are interested in approximating. (We say that “polynomials are dense.”) It is also easy to evaluate a polynomial at a point and to perform mathematical operations using polynomials. Unfortunately, higher-order polynomials tend to be “wiggly”, so they exhibit behavior that is often not reasonable for the situation being modeled. This means that we often choose to fit data with a function that may not pass exactly through all the data points, but which has a reasonable shape, rather than interpolate the data with a high-order polynomial that displays unreasonable behavior between the data points.

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If the data are known to be subject to measurement errors, then we don't want to require our function to pass exactly through these data points. After all, some point slightly removed from our data point might represent the true function value. Even if we feel confident that our data is good, we might not be happy with the shape of the graph of the interpolating function. We will discuss in the next section a way to fit the data without requiring that the function pass through all the data points.