divisiondifferences

The concept of division differences is a method used in numerical analysis for the interpolation of data points. It provides a systematic way to construct a polynomial that passes through a given set of data points. The process involves calculating a series of divided differences, which are essentially generalized slopes.

Given a set of n+1 data points (x0, y0), (x1, y1), ..., (xn, yn), the divided differences are denoted by f[xi, xj, ..., xk]. The first-order divided differences are computed as the difference in the y-values divided by the difference in the x-values between two points. For example, f[xi, xj] = (yj - yi) / (xj - xi).

The second-order divided differences are then calculated using the first-order divided differences. Specifically, f[xi, xj, xk] = (f[xj, xk] - f[xi, xj]) / (xk - xi). This process is continued for higher orders.

These divided differences are then used to form the Newton form of the interpolating polynomial. The polynomial is expressed as a sum of terms, where each term is a product of a divided difference and a series of (x - xi) terms. The Newton form is advantageous because adding a new data point only requires the calculation of new divided differences and the addition of a single new term to the existing polynomial, rather than recomputing the entire polynomial. This makes it an efficient method for incremental interpolation.