Note on Generating Orthogonal Polynomials and Their Application in Solving Complicated Polynomial Regression Tasks
Abstract
The paper deals with efficient numerical solving the proposed statistical model using modern algorithms of numerical linear algebra. In particular, the main ingredients are: Numerically stable generation of vectors of values of orthogonal polynomials (the "design" matrix Psí) based on the MGS Arnoldi algorithm with reorthogonalization, algebraic derivation of the inversion of the matrix Psí'Omega(-1)Psí and, consequently, algebraic derivation of the solution of the system of normal equations, and, finally, efficient computation of testing quantities based on the Cholesky decomposition of relatively small matrices.
The techniques presented in this paper represent economized way of solving the problem. From the point of view of practical computing, we save the computer memory as well as the time requirements.
We manipulate only with small matrices, we do not compute inversions of large matrices and we do not even need to solve linear algebraic systems with large matrices.