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A more accurate algorithm for computing the Christoffel transformation

Summary: A more accurate algorithm for computing the
Christoffel transformation
MarŽia I. Bueno and FroilŽan M. Dopico a,b
aDepartment of Mathematics, The College of William and Mary
P.O. Box 8795, Williamsburg, VA 23187-8795, USA. mbueno@math.uc3m.es
bDepartamento de MatemŽaticas, Universidad Carlos III de Madrid,
Avda. de la Universidad, 30. 28911 LeganŽes, Spain. dopico@math.uc3m.es
A 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. The basic Christoffel transformation with
shift transforms the monic Jacobi matrix associated with a measure d” into the
monic Jacobi matrix associated with (x - )d”. This transformation is known for
its numerous applications to quantum mechanics, integrable systems, and other
areas of mathematics and mathematical physics. From a numerical point of view,
the Christoffel transformation is essentially computed by performing one step of
the LR algorithm with shift, but this algorithm is not stable. We propose a more
accurate algorithm, estimate its forward errors, and prove that it is forward stable,
i.e., that the obtained forward errors are of similar magnitude to those produced by
a backward stable algorithm. This means that the magnitude of the errors is the best


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics