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

0351359 - ÚI 2011 RIV IN eng J - Journal Article
Knížek, J. - Tichý, Petr - Beránek, L. - Šindelář, Jan - Vojtěšek, B. - Bouchal, P. - Nenutil, R. - Dedík, O.
Note on Generating Orthogonal Polynomials and Their Application in Solving Complicated Polynomial Regression Tasks.
International Journal of Mathematics and Computation. Roč. 7, č. 10 (2010), s. 48-60. ISSN 0974-5718
Grant - others:GA MZd(CZ) NS9812; GA ČR(CZ) GAP304/10/0868
Institutional research plan: CEZ:AV0Z10300504; CEZ:AV0Z10750506
Keywords : polynomial regression * orthogonalization * numerical methods * markers * biomarkers
Subject RIV: BA - General Mathematics

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.
Permanent Link: http://hdl.handle.net/11104/0191129