Note on Fitting of Functions of Several Independent Variables

Introduction. Thacher and Milne [1, 2] have recently described useful techniques for interpolation of functions of several independent variables. Their techniques involve fitting to the values of the function to be interpolated, given at an arbitrary set of base points in s-space, an interpolatory function which is required to be a linear combination of specified basis functions. As the number of base points is taken equal to the number of basis functions, a unique interpolatory function is obtained (except in singular cases, which are not considered). The present writer has been concerned with situations involving curvefitting rather than interpolation in the strict sense, where the given data are derived from experimental observations and the number of base points exceeds the number of basis functions. In such a case, the matrix D Qf values of the basis functions at the base points is not square.' It is the primary purpose of this note to point out that, if the least-squares criterion of fit is adopted, the principal formulas of Thacher and Milne carry over to this case, merely replacing the inverse of D by its pseudoinverse [3, 4]. Some remarks on computational methods and a numerical example are also included.