scholarly journals APPROXIMATION BY INTERPOLATING POLYNOMIALS IN SMIRNOV-ORLICZ CLASS

2006 ◽  
Vol 43 (2) ◽  
pp. 413-424 ◽  
Author(s):  
Ramazan Akgun ◽  
Daniyal M. Israfilov
Keyword(s):  
Orlicz Class ◽  
Interpolating Polynomials

scholarly journals Approximation by interpolating polynomials

1978 ◽  
Vol 22 (1) ◽  
pp. 27-32
Author(s):  
A.S.B Holland ◽  
B Kuttner ◽  
B.N Sahney
Keyword(s):  
Interpolating Polynomials

scholarly journals Bounding the Number of Common Zeros of Multivariate Polynomials and Their Consecutive Derivatives

2018 ◽  
Vol 28 (2) ◽  
pp. 253-279
Author(s):  
O. GEIL ◽  
U. MARTÍNEZ-PEÑAS
Keyword(s):  
Upper Bound ◽  
Existence And Uniqueness ◽  
Combinatorial Nullstellensatz ◽  
Finite Collection ◽  
Multivariate Polynomials ◽  
Grobner Basis ◽  
Evaluation Codes ◽  
Number Of Zeros ◽  
Interpolating Polynomials ◽  
Common Zeros

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.

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

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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