Lanczos method and Padé-Chebyshev approximation of Markov functions

Leonid Knizhnerman

Central Geophysical Expedition, Russia, 123298, Moscow, Narodnogo Opolcheniya St., house 40, building 3


The problem of rational approximation of Markov (impedance) functions on a bounded interval of the real axis arises when constructing optimal finite-difference grids for solving differential equations (see Vladimir Druskin's abstract).

Padé-Chebyshev approximation is a popular sort of rational approximation due to its simplicity, though it is not optimal in general.

We show how to calculate a [(k-1)/k] Padé-Chebyshev approximant of a Markov function by means of a Lanczos process. We formulate an error estimate rendering concrete Gonchar-Rakhmanov-Suetin asymptotical one. We also demonstrate that such an approximation on a changed interval can produce a better approximant for a given interval.

Implementation aspects are considered.

Collaborators: David Ingerman (Princeton-MIT) and Vladimir Druskin (SDR).