Fast and accurate interpolation of large scattered data sets on the sphere

In this paper a new efficient algorithm for spherical interpolation of large scattered data sets is presented. The solution method is local and involves a modified spherical Shepard’s interpolant, which uses zonal basis functions as local approximants. The associated algorithm is implemented and optimized by applying a nearest neighbour searching procedure on the sphere. Specifically, this technique is mainly based on the partition of the sphere in a suitable number of spherical zones, the construction of spherical caps as local neighbourhoods for each node, and finally the employment of a spherical zone searching procedure. Computational cost and storage requirements of the spherical algorithm are analyzed. Moreover, several numerical results show the good accuracy of the method and the high efficiency of the proposed algorithm.     

Algorithms for cardinal interpolation using box splines and radial basis functions

Summary We describe an algorithm for (bivariate) cardinal interpolation which can be applied to translates of basis functions which include box splines or radial basis functions. The algorithm is based on a representation of the Fourier transform of the fundamental interpolant, hence Fast Fourier Transform methods are available. In numerical tests the 4-directional box spline (transformed to the characteristical submodule of ²), the thin plate spline, and the multiquadric case give comparably equal and good results.     

A cutting plane method for solving minimax problems in the complex plane

It is shown how the combined discretization and cutting plane method for general convex semi-infinite programming problems recently presented in [40] can be effectively implemented for the solution of minimax problems in the complex plane. In contrast to other approaches, the minimax problem does not have to be linearized. The performance of the algorithm is demonstrated by a number of highly accurate numerical examples.     

Numerical factorization of a polynomial by rational Hermite interpolation

We derive a class of iterative formulae to find numerically a factor of arbitrary degree of a polynomialf(x) based on the rational Hermite interpolation. The iterative formula generates the sequence of polynomials which converge to a factor off(x). It has a high convergence order even for a factor which includes multiple zeros. Some numerical examples are also included.     

Numerical stability of nonequispaced fast Fourier transforms

This paper presents some new results on numerical stability for multivariate fast Fourier transform of nonequispaced data (NFFT). In contrast to fast Fourier transform (of equispaced data), the NFFT is an approximate algorithm. In a worst case study, we show that both approximation error and roundoff error have a strong influence on the numerical stability of NFFT. Numerical tests confirm the theoretical estimates of numerical stability.     

Pseudospectral methods for solving an equation of hypergeometric type with a perturbation

Almost all, regular or singular, Sturm-Liouville eigenvalue problems in the Schrödinger form

     

Obtaining the Quantum Fourier Transform from the classical FFT with QR decomposition

We present the detailed process of converting the classical Fourier Transform algorithm into the quantum one by using QR decomposition. This provides an example of a technique for building quantum algorithms using classical ones. The Quantum Fourier Transform is one of the most important quantum subroutines known at present, used in most algorithms that have exponential speed-up compared to the classical ones. We briefly review Fast Fourier Transform and then make explicit all the steps that led to the quantum formulation of the algorithm, generalizing Coppersmith’s work.     

Thinning algorithms for scattered data interpolation

Multistep interpolation of scattered data by compactly supported radial basis functions requires hierarchical subsets of the data. This paper analyzes thinning algorithms for generating evenly distributed subsets of scattered data in a given domain in ℝ ^d .     

Algorithms for inversion of diagonal plus semiseparable operator matrices

LetH be a Hilbert space andRHH be a bounded linear operator represented by an operator matrix which is a sum of a diagonal and of a semiseparable type of order one operator matrices. We consider three methods for solution of the operator equationRx=y. The obtained results yields new algorithms for solution of integral equations and for inversion of matrices.     

A new type of Hermite matrix polynomial series

Conventional Hermite polynomials emerge in a great diversity of applications in mathematical physics, engineering, and related fields. However, in physical systems with higher degrees of freedom it will be of practical interest to extend the scalar Hermite functions to their matrix analogue. This work introduces various new generating functions for Hermite matrix polynomials and examines existence and convergence of their associated series expansion by using Mehler’s formula for the general matrix case. Moreover, we derive interesting new relations for even- and odd-power summation in the generating-function expansion containing Hermite matrix polynomials. Some new results for the scalar case are also presented.     

Two interpolation operators on irregularly distributed data in inner product spaces

Two interpolation operators in inner product spaces for irregularly distributed data are compared. The first is a well-known polynomial operator, which in a certain sense generalizes the classical Lagrange interpolation polynomial. The second can be obtained by modifying the first so as to get a partition-of-unity interpolant. Numerical tests and considerations on errors show that the two operators have very different approximation performances, and that by suitable modifications both can provide acceptable results, working in particular from R^m to Rⁿ and from C[−π,π] to R.     

Curve and surface construction using Hermite subdivision schemes

In this paper we present a very efficient Hermite subdivision scheme, based on rational functions, and outline its potential applications, with special emphasis on the construction of cubic-like B-splines — well suited for the design of constrained curves and surfaces.     

Frequency-dependent interpolation rules using first derivatives for oscillatory functions

We construct frequency-dependent rules to interpolate oscillatory functions y(x)$y(x)$ with frequency ω$\omega$ of the form,

y(x)=_f1(x)cos(ωx)+_f2(x)sin(ωx),$y(x)=f_1(x)\cos (\omega x)+f_2(x)\sin (\omega x)\text{,}$

A nonmonotone smoothing-type algorithm for solving a system of equalities and inequalities

In this paper, we investigate a smoothing-type algorithm with a nonmonotone line search for solving a system of equalities and inequalities. We prove that the nonmonotone algorithm is globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are reported.     

A reliable modification of the cross rule for rational Hermite interpolation

Claessens' cross rule [8] enables simple computation of the values of the rational interpolation table if the table is normal, i.e. if the denominators in the cross rule are non-zero. In the exceptional case of a vanishing denominator a singular block is detected having certain structural properties so that some values are known without further computations. Nevertheless there remain entries which cannot be determined using only the cross rule.In this note we introduce a simple recursive algorithm for computation of the values of neighbours of the singular block. This allows to compute entries in the rational interpolation table along antidiagonals even in the presence of singular blocks. Moreover, in the case of non-square singular blocks, we discuss a facility to monitor the stability.Dedicated to Professor G. Mühlbach on the occasion of his 50th birthday     

Solving systems of fractional differential equations using differential transform method

This paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of systems of fractional differential equations. The solutions of our model equations are calculated in the form of convergent series with easily computable components. Some examples are solved as illustrations, using symbolic computation. The numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations.     

Homotopy method for solving ball-constrained variational inequalities

In this paper, we present a smoothing homotopy method for solving ball-constrained variational inequalities by utilizing a similar Chen-Harker-Kanzow-Smale function to smooth Robinson’s normal equation. Without any monotonicity condition on the defining map F, for the starting point chosen almost everywhere in Rⁿ, the existence and convergence of the homotopy pathway are proven. Numerical experiments illustrate that the method is feasible and effective.     

Biorthogonal systems for solving Volterra integral equation systems of the second kind

With the aid of biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a system of nonlinear Volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically.     

Multiplier Theorems for the Short-Time Fourier Transform

So-called short-time Fourier transform multipliers (also called Anti-Wick operators in the literature) arise by applying a pointwise multiplication operator to the STFT before applying the inverse STFT. Boundedness results are investigated for such operators on modulation spaces and on L _p -spaces. Because the proofs apply naturally to Wiener amalgam spaces the results are formulated in this context. Furthermore, a version of the Hardy-Littlewood inequality for the STFT is derived. This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No K67642.