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Colloquium, Spring 2006

Day: Monday, March 13th Time: 4:00pm Place: BH 428 Title: Darboux transformations and orthogonal polynomials: Matrix interpretation and numerical aspects. Speaker: Maribel Bueno Abstract: Some algorithms for computing Jacobi matrices called Darboux transformations have important applications in different areas of mathematics and mathematical physics. A monic Jacobi matrix is a tridiagonal matrix which contains the coefficients of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure. The Darboux transformation with shift and without parameter, also known as Christoffel transformation with shift, transforms the monic Jacobi matrix associated with a measure into the monic Jacobi matrix associated with a polynomial perturbation of this measure. However the existing algorithms to compute Christoffel transformation are not stable. A similar situation occurs with Darboux transformation with shift and parameter, also known as Geronimus transformation with shift. It computes the monic Jacobi matrix associated with a rational modification of a measure. In this talk, I present new algorithms to compute these standard transformations and prove that they are forward stable. This means that the obtained forward errors are of similar magnitude to those produced by a backward stable algorithm. I also provide condition numbers that allow us to estimate forward errors in O(n) flops. Moreover, the new algorithms yield smaller forward errors than the previous ones and for values of the shift large enough, both become stable. At the end of the talk, I will present some open problems related with the topic of this talk.