0351359 20240103194317.620101209235959.9 13 s. cav_un_epca*03513580974-5718710 (2010)48-60 polynomial regression orthogonalization numerical methods markers biomarkers cav_un_auth*0050983 Knížek J. CZ G cav_un_auth*0100847 Tichý Petr Oddělení výpočetních metod Department of Computational Methods UIVT-O Department of Computational Mathematics Ústav informatiky AV ČR, v. v. i. cav_un_auth*0245237 Beránek L. CZ cav_un_auth*0101205 Šindelář Jan Stochastická informatika Department of Stochastic Informatics SI SI UTIA-B Ústav teorie informace a automatizace AV ČR, v. v. i. cav_un_auth*0019922 Vojtěšek B. CZ cav_un_auth*0266747 Bouchal P. CZ cav_un_auth*0020284 Nenutil R. CZ cav_un_auth*0266748 Dedík O. CZ NS9812 GA MZd CZ cav_un_auth*0266749 GAP304/10/0868 GA ČR CZ cav_un_auth*0266750 CEZ:AV0Z10300504 CEZ:AV0Z10750506 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. 2011 BA http://hdl.handle.net/11104/0191129 2010 cav_un_epca*0351358 International Journal of Mathematics and Computation 0974-5718 Roč. 7 č. 10 2010 48 60