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1.
Summary. Stabilized methods (also called Chebyshev methods) are explicit Runge-Kutta methods with extended stability domains along
the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs.
The aim of this paper is to show that with the use of orthogonal polynomials, we can construct nearly optimal stability polynomials
of second order with a three-term recurrence relation. These polynomials can be used to construct a new numerical method,
which is implemented in a code called ROCK2. This new numerical method can be seen as a combination of van der Houwen-Sommeijer-type
methods and Lebedev-type methods.
Received January 14, 2000 / Revised version received November 3, 2000 / Published online May 4, 2001 相似文献
2.
3.
I. I. Sharapudinov 《Mathematical Notes》2000,67(3):389-397
Let
N+2m
={−m, −m+1, …, −1, 0, 1, …,N−1,N, …,N−1+m}. The present paper is devoted to the approximation of discrete functions of the formf :
N+2m
→ ℝ by algebraic polynomials on the grid Ω
N
={0, 1, …,N−1}. On the basis of two systems of Chebyshev polynomials orthogonal on the sets Ω
N+m
and Ω
N
, respectively, we construct a linear operatorY
n+2m, N
=Y
n+2m, N
(f), acting in the space of discrete functions as an algebraic polynomial of degree at mostn+2m for which the following estimate holds (x ε Ω
N
):
whereE
n+m[g,l
2(Ω
N+m
)] is the best approximation of the function
by algebraic polynomials of degree at mostn+m in the spacel
2 (Ω
N+m
) and the function Θ
N, α
(x) depends only on the weighted estimate for the Chebyshev polynomialsτ
k
α,α
(x, N).
Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 460–470, March, 2000. 相似文献
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4.
In this paper, we present explicit formulas for discrete orthogonal polynomials over the so-called Gauss–Lobatto Chebyshev points. In particular, this allows us to compute the coefficient in the three-terms recurrence relation and the explicit formulas for the discrete inner product. The paper also contains numerical examples related to the least-squares problems. 相似文献
5.
We consider the problem of constructing polynomials, orthogonal in the Sobolev sense on the finite uniform mesh and associated with classical Chebyshev polynomials of discrete variable. We have found an explicit expression of these polynomials by classicalChebyshev polynomials. Also we have obtained an expansion of new polynomials by generalized powers ofNewton type. We obtain expressions for the deviation of a discrete function and its finite differences from respectively partial sums of its Fourier series on the new system of polynomials and their finite differences. 相似文献
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7.
V. A. Yudin 《Proceedings of the Steklov Institute of Mathematics》2009,266(Z1):227-245
On different compact sets from ℝ
n
, new multidimensional analogs of algebraic polynomials least deviating from zero (Chebyshev polynomials) are constructed.
A brief review of the analogs constructed earlier is given. Estimates of values of the best approximation obtained by using
extremal signatures, lattices, and finite groups are presented. 相似文献
8.
V. A. Yudin 《Proceedings of the Steklov Institute of Mathematics》2009,266(1):227-245
On different compact sets from ? n , new multidimensional analogs of algebraic polynomials least deviating from zero (Chebyshev polynomials) are constructed. A brief review of the analogs constructed earlier is given. Estimates of values of the best approximation obtained by using extremal signatures, lattices, and finite groups are presented. 相似文献
9.
Igor Rivin 《Proceedings of the American Mathematical Society》2005,133(5):1299-1305
We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative. As a corollary we find that and are positive definite functions. We further show that a Central Limit Theorem holds for the coefficients of our polynomials.
10.
In this paper we describe polynomials orthogonal to all powers of a Chebyshev polynomial on a segment. 相似文献
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12.
主要研究勒让德多项式与契贝谢夫多项式之间的关系的性质,利用生成函数和函数级数展开的方法,得出了勒让德多项式与契贝谢夫多项式之间的一个重要关系,这对勒让德多项式与契贝谢夫多项式的研究有一定的推动作用. 相似文献
13.
14.
Kareem T. Elgindy 《Journal of Applied Mathematics and Computing》2018,56(1-2):317-349
This paper presents for the first time a robust exact line-search method based on a full pseudospectral (PS) numerical scheme employing orthogonal polynomials. The proposed method takes on an adaptive search procedure and combines the superior accuracy of Chebyshev PS approximations with the high-order approximations obtained through Chebyshev PS differentiation matrices. In addition, the method exhibits quadratic convergence rate by enforcing an adaptive Newton search iterative scheme. A rigorous error analysis of the proposed method is presented along with a detailed set of pseudocodes for the established computational algorithms. Several numerical experiments are conducted on one- and multi-dimensional optimization test problems to illustrate the advantages of the proposed strategy. 相似文献
15.
I. E. Shparlinskii 《Siberian Mathematical Journal》1990,31(1):183-185
Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 1, pp. 217–218, January–February, 1990. 相似文献
16.
Saralees Nadarajah 《Journal of Mathematical Analysis and Applications》2007,334(2):1492-1494
Two elegant representations are derived for the modified Chebyshev polynomials discussed by Witula and Slota [R. Witula, D. Slota, On modified Chebyshev polynomials, J. Math. Anal. Appl. 324 (2006) 321-343]. 相似文献
17.
Ernst Albrecht 《Archiv der Mathematik》2009,92(5):399-404
The problem of uniqueness of the Chebyshev polynomials for bounded linear operators on normed linear spaces is investigated.
Herrn Professor Dr. Dr. h.c. Heinz K?nig zu seinem achtzigsten Geburtstag gewidmet 相似文献
18.
Roman Witu?a 《Journal of Mathematical Analysis and Applications》2006,324(1):321-343
In this paper some new properties and applications of modified Chebyshev polynomials and Morgan-Voyce polynomials will be presented. The aim of the paper is to complete the knowledge about all of these types of polynomials. 相似文献
19.
F. Luquin 《Lithuanian Mathematical Journal》1993,33(1):41-43
Supported by the University of the Basque Country. 相似文献
20.
Toufik Mansour 《Discrete Mathematics》2006,306(12):1161-1176
We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. 相似文献