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Note on Generating Orthogonal Polynomials and Their Application in Solving Complicated Polynomial Regression Tasks

J. Knizek, Petr Tichy, Ladislav Beranek, Jan Sindelar, Borivoj Vojtesek, Pavel Bouchal, Rudolf Nenutil, Ondrej Dedik


In this paper, we describe efficient algorithms for computing solutions of numerically exacting parts of used complicated polynomial regression tasks. In particular, we use a numerically stable way of generating the values of normalized orthogonal polynomials on a discrete set of points; we use “the Arnoldi algorithm with reorthogonalization”, which is the key ingredient of our approach. The generated vectors can then be considered orthogonal also in finite precision arithmetic (up to a small inaccuracy proportional to machine precision). We then use the special algebraic structure of the covariance matrix to find algebraically the inversion of the matrix of the system of normal equations. Therefore, we do not need to compute numerically the inversion of the covariance matrix and we do not even need to solve the system of normal equations numerically. Some consequences of putting the algorithms mentioned into practice are discussed.


Polynomial regression; Orthogonalization; Numerical methods; Markers; Biomarkers

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