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.

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