The set of i/o difference equations (1) is called strongly row-reduced if the leading coefficient
matrix [L.sub.[mu]] has full rank over [A.sub.l].
Dividing equation (2.20) by the leading coefficient
, we have
It remains to examine the leading coefficient
, where it turns out that only the summand k = m gives a contribution, and by using equation (17b) we get
1 Compute n zeros, [mathematical expression not reproducible] of pn = [p.sub.n] (*; [[mu].sup.(a,[rho ] )]), and leading coefficient
[[gamma]n] of pn.
Rules for Leading Coefficients
of the Polynomials [p.sub.[alpha]](k).
In this case, [A.sub.[GAMMA]]([alpha], 1) is a polynomial in g of degree [alpha] - 1 and leading coefficient
We begin with a known transformation that homogenizes the leading coefficient
a2(x), and show how it can be used to generalize the stability results from the previous sections.
Summarizing, we see that the Laurent coefficients of S (or of 1/S) contain surprisingly good approximations of two main parameters of the OPUC: they match asymptotically the Verblunsky coefficients, and the partial sums of the squares of their absolute values represent (up to a normalizing constant) the leading coefficient
of the orthonormal polynomials.
Condition numbers are found from the Lagrange's representation of the polynomial p(z) where we assume that the leading coefficient
[a.sub.n] is known exactly whereas the computed values fl(p([w.sub.i])) satisfy fl(p([w.sub.i])) = p([w.sub.i])(1 + [[epsilon].sub.i]), where [absolute value of [[epsilon].sub.i]] [less than or equal to] n x eps and eps is the machine precision.
Furthermore, taking into account the polar decomposition for the leading coefficient
of ([[PHI].sub.n]), we can assume that such a matrix coefficient is a positive definite matrix, and thus, we can choose this normalization in order to have uniqueness for our sequence of matrix orthonormal polynomials [1, page 333].
The other topic is that of descriptor systems, where the leading coefficient
of the polynomial is singular.
As shown in [1, 10], we remark that the dominant asymptotic behavior of [[z.sup.n]][W.sub.l](z) and [[z.sub.n]] E]r(z) (for any power series A(z) = [summation] [a.sub.n] [z.sup.n], A(z) denotes the nth coefficient of A(z), namely, [[z.sup.n]]A(z) = [a.sub.n].) is governed by the leading coefficients
[b.sub.l] and [e.sub.r].