Least-squares orthogonal polynomial approximation in several independent variables
Robert S. Caprari · Computers in Physics · 1993
A systematic technique for generating orthonormal polynomials in two independent variables by application of the Gram–Schmidt orthogonalization procedure of linear algebra is presented. A linear least-squares approximation for experimental data or an arbitrary function is generated from these polynomials. The least-squares coefficients are computed without recourse to matrix arithmetic, which ensures both numerical stability and simplicity of implementation as a self-contained numerical algorithm. The Gram–Schmidt procedure is then utilized to generate a complete set of orthogonal polynomials of fourth degree. A general technique for the transformation of the polynomial representation from an arbitrary basis into the familiar sum of products form is presented, together with a specific implementation for fourth degree polynomials. The computational integrity of this algorithm is verified by reconstructing arbitrary fourth degree polynomials from their values at randomly chosen points in their domain.