- - - - - o ' " A STANDARD METHOD FOR SURFACE FITTING BY ORTHOGONAL POLYNOMIALS

An algorithm for least squares fitting of a polynomial to a function z(x.y) of two independent variables is described. It is assumed that the function is given by a number of "measured values". which may be arbitrarily distributed over the ~Yplane. The polynomial is constructed as a linear combination of a set of polynomials which are orthonormalized over the set of given data points by the modified Gram Schmidt procedure. Formulas for the estimation of time and space required for the calculations and for the accuracy obtained are given. It is demonstrated that the method is numerically robust, that the costs of computations are reasonable, and that the procedure is suitable as a standard tool for least squares surface fitting. The experiments employed for the practical evaluation of the procedure are discussed. Q Introduction In many applications, functions given by "measured values" are often encountered. In order to be able to manipulate such a function; it is approximated by a simple mathematical expression. This paper is concerned with approximation by polyno­ mials of functions of two independent variables. A function z (x .y) is thus given by {(xr.Yr.Zr.wr)lr=l.2 •... Np} • where zr is the value measured at (xr.Yr) and wr is the weight assigned to this measurement. It is assumed that the measured values are not accurate, and the coefficients of the approximating polynomial P(x,y) are therefore calculated by employing the criterion of least squares. Since z(x,y) may be considered as a representation of a surface, the term "surface fitting" is commonly used for this problem. If the (xr,Yr)'s happen to be the points of intersection of a rectangular grid in the xy-plane. the' algo­ rithm of Clark Kubik and Phillips [2] may be used. This method is economic in terms of computer time and storage as compared to the general procedure discussed in this paper. Another economic method by Clenshaw and Hayes [3] may be used in tile • somewhat more general case, where the (xr,Yr) are positioned on a number of parallel T ec hn io n C om pu te r Sc ie nc e D ep ar tm en t T eh ni ca l R ep or t C S0 01 2 19 71 ----------~---------~~-~-----~-----