On the saturation ofL p w -approximation by (0-q′-q) type Hermite-Fejér interpolating polynomials

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

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Adaptive Mesh Iteration Method for Trajectory Optimization Based on Hermite-Pseudospectral Direct Transcription

Mathematical Problems in Engineering ◽

10.1155/2017/2184658 ◽

2017 ◽

Vol 2017 ◽

pp. 1-7

Author(s):

Humin Lei ◽

Tao Liu ◽

Deng Li ◽

Jikun Ye ◽

Lei Shao

Keyword(s):

Error Estimate ◽

Relative Error ◽

Iteration Method ◽

Trajectory Optimization ◽

Simulation Experiment ◽

Adaptive Mesh ◽

The State ◽

State Equations ◽

Direct Transcription ◽

Interpolating Polynomials

An adaptive mesh iteration method based on Hermite-Pseudospectral is described for trajectory optimization. The method uses the Legendre-Gauss-Lobatto points as interpolation points; then the state equations are approximated by Hermite interpolating polynomials. The method allows for changes in both number of mesh points and the number of mesh intervals and produces significantly smaller mesh sizes with a higher accuracy tolerance solution. The derived relative error estimate is then used to trade the number of mesh points with the number of mesh intervals. The adaptive mesh iteration method is applied successfully to the examples of trajectory optimization of Maneuverable Reentry Research Vehicle, and the simulation experiment results show that the adaptive mesh iteration method has many advantages.

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Solving multi-choice linear programming problems by interpolating polynomials

Mathematical and Computer Modelling ◽

10.1016/j.mcm.2011.04.009 ◽

2011 ◽

Vol 54 (5-6) ◽

pp. 1405-1412 ◽

Cited By ~ 23

Author(s):

M.P. Biswal ◽

S. Acharya

Keyword(s):

Linear Programming ◽

Interpolating Polynomials ◽

Linear Programming Problems

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Approximate Solution of a Singular Integral Equation with a Cauchy Kernel on the Euclidean Plane

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2017-0052 ◽

2018 ◽

Vol 18 (4) ◽

pp. 741-752

Author(s):

Dorota Pylak ◽

Paweł Karczmarek ◽

Paweł Wójcik

Keyword(s):

Integral Equation ◽

Approximate Solution ◽

Singular Integral Equation ◽

Singular Integral ◽

Euclidean Plane ◽

Singular Integral Equations ◽

Cauchy Kernel ◽

Challenging Problem ◽

One Dimensional ◽

Interpolating Polynomials

AbstractMultidimensional singular integral equations (SIEs) play a key role in many areas of applied science such as aerodynamics, fluid mechanics, etc. Solving an equation with a singular kernel can be a challenging problem. Therefore, a plethora of methods have been proposed in the theory so far. However, many of them are discussed in the simplest cases of one–dimensional equations defined on the finite intervals. In this study, a very efficient method based on trigonometric interpolating polynomials is proposed to derive an approximate solution of a SIE with a multiplicative Cauchy kernel defined on the Euclidean plane. Moreover, an estimation of the error of the approximated solution is presented and proved. This assessment and an illustrating example show the effectiveness of our proposal.

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