Summary: STABILITY AND SENSITIVITY OF DARBOUX
TRANSFORMATION WITHOUT PARAMETER
M. ISABEL BUENO AND FROILŽAN M. DOPICO
Abstract. The monic Jacobi matrix is a tridiagonal matrix which contains the parameters of
the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with
respect to a measure. Darboux transformation without parameter changes a monic Jacobi matrix
associated with a measure ” into the monic Jacobi matrix associated with xd”. This transformation
has been used in several numerical problems as in the computation of Gaussian quadrature rules.
In this paper, we analyze the stability of an algorithm which implements Darboux transformation
without parameter numerically and we also study the sensitivity of the problem. The main result
of the paper is that, although the algorithm for Darboux transformation without parameter is not
backward stable, it is forward stable. This means that the forward errors are of similar magnitude to
those produced by a backward stable algorithm. Moreover, bounds for the forward errors computable
with low cost are presented. We also apply the results to some classical families of orthogonal
Key words. Darboux transformation, orthogonal polynomials, stability, sensitivity, tridiagonal
matrices, LU factorization, LR algorithm.
AMS subject classifications. 65G50, 42C05, 15A23, 65F30, 65F35.
1. INTRODUCTION. Let ” be an absolutely continuous measure on the real
line, that is, d” = (x)dx, where is a weight function. Suppose that