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Some variants of the numerical Picard iterations method are presented to solve an IVP for an ordinary differential system. The term numerical emphasizes that a numerical solution is computed. The method consists in replacing the right hand…
We design nearly-linear time numerical algorithms for the problem of multivariate multipoint evaluation over the fields of rational, real and complex numbers. We consider both \emph{exact} and \emph{approximate} versions of the algorithm.…
The cubic spline interpolation method, the Runge--Kutta method, and the Newton-Raphson method are extended to dual versions (developed in the context of dual numbers). This extension allows the calculation of the derivatives of complicated…
A procedure for the construction and the classification of multilattices in arbitrary dimension is proposed. The algorithm allows to determine explicitly the location of the points of a multilattice given its space group, and to determine…
Quantum computing requires a universal set of gate operations; regarding gates as rotations, any rotation angle must be possible. However a real device may only be capable of $B$ bits of resolution, i.e. it might support only $2^B$ possible…
For $m,n \in \mathbb{N}$, $m\geq 1$ and a given function $f : \mathbb{R}^m\longrightarrow \mathbb{R}$, the polynomial interpolation problem (PIP) is to determine a unisolvent node set $P_{m,n} \subseteq \mathbb{R}^m$ of…
In this paper, we prove a discrete Rubio de Francia extrapolation theorem via factorization of discrete Muckenhoupt weights and discrete iterated Rubio de Francia algorithm and its duality.
A new approach for integration of the initial value problem for ordinary differential equations is suggested. The algorithm is based on approximation of the solution by a system of functions that contains orthogonal exponential polynomials.
We introduce an extension of interpolation theory to more than two spaces by employing a functional parameter, while retaining a fully functorial and systematic framework. This approach allows for the construction of generalized…
We address the problem of approximating an unknown function from its discrete samples given at arbitrarily scattered sites. This problem is essential in numerical sciences, where modern applications also highlight the need for a solution to…
We consider two non-linear generalizations of fractal interpolating functions generated from iterated function systems. The first corresponds to fitting data using a Kth-order polynomial, while the second relates to the freedom of adding…
We present a new formula for divided difference and few new schemes of divided difference tables in this paper. Through this, we derive new interpolation, numerical differentiation and numerical integration formulas with arbitrary order of…
We propose an iterative algorithm to interpolate graph signals from only a partial set of samples. Our method is derived from the well known Papoulis-Gerchberg algorithm by considering the optimal value of a constant involved in the…
In this paper we present a new multilevel quasi-interpolation algorithm for smooth periodic functions using scaled Gaussians as basis functions. Recent research in this area has focussed upon implementations using basis function with finite…
This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice -- a setting that, as pointed out by Zeng, Leung, Hickernell (MCQMC2004, 2006) and Zeng,…
This research is concerned with finding the roots of a function in an interval using Chebyshev Interpolation. Numerical results of Chebyshev Interpolation are presented to show that this is a powerful way to simultaneously calculate all the…
In this paper, we first describe a matricial Newton-type algorithm designed to solve the multivariable spectrum approximation problem. We then prove its global convergence. Finally, we apply this approximation procedure to multivariate…
For the last almost three decades, since the famous Buchberger-M\"oller(BM) algorithm emerged, there has been wide interest in vanishing ideals of points and associated interpolation polynomials. Our paradigm is based on the theory of…
We propose a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the…
Singular and oscillatory functions feature in numerous applications. The high-accuracy approximation of such functions shall greatly help us develop high-order methods for solving applied mathematics problems. This paper demonstrates that…