Polynomial interpolation is a method of estimating values between known data points. When graphical data contains a gap, but data is available on either side of the gap or at a few specific points within the gap, an estimate of values within the gap can be made by interpolation.
The simplest method of interpolation is to draw straight lines between the known data points and consider the function as the combination of those straight lines. This method, called linear interpolation, usually introduces considerable error. A more precise approach uses a polynomial function to connect the points. A polynomial is a mathematical expression comprising a sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient. The simplest polynomials have one variable. Polynomials can exist in factored form or written out in full. For example:
(x - 4) (x + 2) (x + 10)
x2 + 2x + 1
3y3 - 8y2 + 4y - 2
The value of the largest exponent is called the degree of the polynomial.
If a set of data contains n known points, then there exists exactly one polynomial of degree n-1 or smaller that passes through all of those points. The polynomial's graph can be thought of as "filling in the curve" to account for data between the known points. This methodology, known as polynomial interpolation, often (but not always) provides more accurate results than linear interpolation.
The main problem with polynomial interpolation arises from the fact that even when a certain polynomial function passes through all known data points, the resulting graph might not reflect the actual state of affairs. It is possible that a polynomial function, although accurate at specific points, will differ wildly from the true values at some regions between those points. This problem most often arises when "spikes" or "dips" occur in a graph, reflecting unusual or unexpected events in a real-world situation. Such anomalies are not reflected in the simple polynomial function which, even though it might make perfect mathematical sense, cannot take into account the chaotic nature of events in the physical universe.