2,977 research outputs found
Multipoint secant and interpolation methods with nonmonotone line search for solving systems of nonlinear equations
Multipoint secant and interpolation methods are effective tools for solving
systems of nonlinear equations. They use quasi-Newton updates for approximating
the Jacobian matrix. Owing to their ability to more completely utilize the
information about the Jacobian matrix gathered at the previous iterations,
these methods are especially efficient in the case of expensive functions. They
are known to be local and superlinearly convergent. We combine these methods
with the nonmonotone line search proposed by Li and Fukushima (2000), and study
global and superlinear convergence of this combination. Results of numerical
experiments are presented. They indicate that the multipoint secant and
interpolation methods tend to be more robust and efficient than Broyden's
method globalized in the same way
The linear pencil approach to rational interpolation
It is possible to generalize the fruitful interaction between (real or
complex) Jacobi matrices, orthogonal polynomials and Pade approximants at
infinity by considering rational interpolants, (bi-)orthogonal rational
functions and linear pencils zB-A of two tridiagonal matrices A, B, following
Spiridonov and Zhedanov.
In the present paper, beside revisiting the underlying generalized Favard
theorem, we suggest a new criterion for the resolvent set of this linear pencil
in terms of the underlying associated rational functions. This enables us to
generalize several convergence results for Pade approximants in terms of
complex Jacobi matrices to the more general case of convergence of rational
interpolants in terms of the linear pencil. We also study generalizations of
the Darboux transformations and the link to biorthogonal rational functions.
Finally, for a Markov function and for pairwise conjugate interpolation points
tending to infinity, we compute explicitly the spectrum and the numerical range
of the underlying linear pencil.Comment: 22 page
Adaptive meshless centres and RBF stencils for Poisson equation
We consider adaptive meshless discretisation of the Dirichlet problem for Poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions. New meshless stencil selection and adaptive refinement algorithms are proposed in 2D. Numerical experiments show that the accuracy of the solution is comparable with, and often better than that achieved by the mesh-based adaptive finite element method
Symmetric Contours and Convergent Interpolation
The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as
applied to the multipoint Pad\'e approximants is the fact that given a germ of
an algebraic function and a sequence of rational interpolants with free poles
of the germ, if there exists a contour that is "symmetric" with respect to the
interpolation scheme, does not separate the plane, and in the complement of
which the germ has a single-valued continuation with non-identically zero jump
across the contour, then the interpolants converge to that continuation in
logarithmic capacity in the complement of the contour. The existence of such a
contour is not guaranteed. In this work we do construct a class of pairs
interpolation scheme/symmetric contour with the help of hyperelliptic Riemann
surfaces (following the ideas of Nuttall \& Singh and Baratchart \& the author.
We consider rational interpolants with free poles of Cauchy transforms of
non-vanishing complex densities on such contours under mild smoothness
assumptions on the density. We utilize -extension of the
Riemann-Hilbert technique to obtain formulae of strong asymptotics for the
error of interpolation
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