JACOBI MATRICES FOR SUMS OF WEIGHT-FUNCTIONS

Orthogonal polynomials are conveniently represented by the tridiagonal Jacobi matrix of coefficients of the recurrence relation which they satisfy. Let J1 and J2 be finite Jacobi matrices for the weight functions w1 and w2, resp. Is it possible to determine a Jacobi matrix J, corresponding to the weight functions w = w1 + w2 using only J1 and J2 and if so, what can be said about its dimension? Thus, it is important to clarify the connection between a finite Jacobi matrix and its corresponding weight function(s). This leads to the need for stable numerical processes that evaluate such matrices. Three new O(n2) methods are derived that "merge" Jacobi matrices directly without using any information about the corresponding weight functions. The first can be implemented using any of the updating techniques developed earlier by the authors. The second new method, based on rotations, is the most stable. The third new method is closely related to the modified Chebyshev algorithm and, although it is the most economical of the three, suffers from instability for certain kinds of data. The concepts and the methods are illustrated by small numerical examples, the algorithms are outlined and the results of numerical tests are reported.