Curve Fitting, Interpolation, and Extrapolation

In engineering applications, data collected from the field are usually discrete and the physical meanings of the data are not always well known. To estimate the outcomes and, eventually, to have a better understanding of the physical phenomenon, a more analytically controllable function that fits the field data is desirable. The process of finding the coefficients for the fitting function is called curve fitting; the process of estimating the outcomes in between sampled data points is called interpolation; whereas the process of estimating the outcomes beyond the range covered by the existing data is called extrapolation.

The scope of this section is limited to discussing some common interpolation methods including:

• Polynomial Interpolation

• Rational Function Interpolation

• Cubic Spline Interpolation

The bottom line is, no matter how smooth the interpolation is and how close it is to the raw data, the problem is not completely solved unless the physical meaning behind the theme has been captured.

Polynomial Interpolation

Lagrange's classical formula of polynomial interpolation is given by

The method of least squares is most commonly used to calculate the coefficients of the interpolating polynomial.

Rational Function Interpolation

Although the polynomial interpolation is probably the most widely used interpolating method, the rational function interpolation stands out when the data or function changes rapidly in some local regions, e.g., poles.

Cubic Spline Interpolation

The cubic spline interpolation uses third degree polynomials to connect the data points which often results in strikingly smooth curve fits.