Polynomial Regression

The method of LEAST SQUARES is used in the program to fit a multivariable polynomial to a tabulated function. The values of tabulated functions must be taken for values of the Independent variables belonging to a rectangular Grid which may be unevenly spaced, that is, a function of n variables must be tabulated at the nodes of an n-dimensional rectangular grid. For example, if x and y are the independent variables, we may, for instance, have 6 values of x and 5 values of y and 30 values of the function, one for each (x,y) Combination, and the grid may look like: To Illustrate, suppose we have a tabulated function with 2 independent variables and with N data sets and we wish to fit to it a polynomial of first degree in x and second degree in y. The polynomial to be fitted is then z = a0+a1x+b0y+b1xy+c1y2+c1xy2 = (a0+a1x)+(b0+b1x)y+(c0+c1x)y2 where a0, a1, b0, b1, c0, c1 are the coefficients to be determined. To minimize the residuals by the least squares method we have to solve the following set of normal equations: