Orthogonal Rational Functions and Structured Matrices Orthogonal Rational Functions and Structured Matrices

Marc VAN BAREL1
Department of Computer Science
Katholieke Universiteit Leuven
Belgium
Email: Marc.VanBarel@cs.kuleuven.ac.be

Given the complex numbers y1,y2,¼,yn all different from each other. Let us consider the vector space Rn of all proper rational functions having possible poles in y1,y2,¼,yn:

Rn : = span{1, 1
x-y1
 , 1
x-y2
 ,¼, 1
x-yn
 }.
Given the complex numbers x0,x1,¼,xn which together with the numbers yi are all different from each other, and the weights 0 ¹ wi, i = 0,1,¼,n, we define the following bilinear form
áf,yñ: = n
å
 i = 0 
 wi2 f(xi)
y(xi)
 
 .
This bilinear form defines an inner product in the space Rn. Let us consider an orthonormal basis
®
a
 
n 
 : = [ a0,a1,¼,an ]
for Rn satisfying the following properties

aj Î Rj \Rj-1
       (R0 : = Æ)
áai, aj ñ = di,j
      (Kronecker delta)
for i,j = 0,1,2,¼,n.

In this talk we will show that the coefficients of a recurrence relation for these orthonormal rational functions aj are the elements of a structured matrix S+Dy. Here Dy denotes the diagonal matrix whose diagonal elements are y0,y1,¼,yn where y0 can be chosen arbitrarily. The lower triangular part of S is the lower triangular part of a rank 1 matrix. When the numbers xi are all on the real line or all on the unit circle, also the upper triangular part has this property.

The matrix S can be constructed as the matrix having the structure as indicated above such that S+Dy has eigenvalues xi and such that the first component of the normalised eigenvector corresponding to xi equals wi. We will give an algorithm to compute the matrix S in this way. This algorithm requires O(n3) flops in the general case and O(n2) flops when the numbers xi are all real or all on the unit circle.

Footnotes:

1 Joint work with Nicola MASTRONARDI, Dario FASINO and Luca GEMIGNANI.

File translated from TEX by TTH, version 1.94.
On 16 Apr 2002, 17:56.