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1.
Some Identities Involving Square of Fibonacci Numbers and Lucas Numbers   总被引:8,自引:0,他引:8  
By studying the properties of Chebyshev polynomials, some specific and meaningful identities for the calculation of square of Chebyshev polynomials, Fibonacci numbers and Lucas numbers are obtained.  相似文献
2.
一些包含契贝谢夫多项式的恒等式   总被引:6,自引:2,他引:4  
讨论了著名的契贝射夫多项式的一些性质,并给出了一些有趣的恒等式。  相似文献
3.
关于正、余弦函数的一组恒等式   总被引:5,自引:0,他引:5  
:利用第二类契贝谢夫多项式的性质得到了关于正余弦函数的一组有趣的恒等式 .  相似文献
4.
On a quadrature formula of Micchelli and Rivlin   总被引:4,自引:0,他引:4  
Micchelli and Rivlin (1972) obtained a quadrature formula of highest algebraic degree of precision for the Fourier-Chebyshev coefficients An(f), which is based on the divided differences of f′ at the zeros of the Chebyshev polynomial Tn(x). We give here a simple approach to questions of this type, which applies to the coefficients in arbitrary orthogonal expansion of f. As an auxiliary result we obtain a new interpolation formula and a new representation of the Turán quadrature formula.  相似文献
5.
The convergence problem of many Krylov subspace methods,e.g., FOM, GCR, GMRES and QMR, for solving large unsymmetric (non-Hermitian) linear systems is considered in a unified way when the coefficient matrixA is defective and its spectrum lies in the open right (left) half plane. Related theoretical error bounds are established and some intrinsic relationships between the convergence speed and the spectrum ofA are exposed. It is shown that these methods are likely to converge slowly once one of the three cases occurs:A is defective, the distribution of its spectrum is not favorable, or the Jordan basis ofA is ill conditioned. In the proof, some properties on the higher order derivatives of Chebyshev polynomials in an ellipse in the complex plane are derived, one of which corrects a result that has been used extensively in the literature. Supported by the China State Major Key Project for Basic Researches, the National Natural Science Foundation of China, the Doctoral Program of the Chinese National Educational Commission, the Foundation of Returned Scholars of China and Liaoning Province Natural Science Foundation.  相似文献
6.
本文通过一个例子说明了文献[3]中定理6.9的不完善之处,并建立了:若f∈Cr[-1,1],则  相似文献
7.
一个序列的组合解释及其应用   总被引:2,自引:0,他引:2       下载免费PDF全文
该文给出了一个序列的组合解释,讨论了这个序列在研究两类Chebyshev多项式,广义Fibonacci序列和广义Lucas序列中的一些应用.  相似文献
8.
Algorithms for the integration and derivation of Chebyshev series   总被引:1,自引:0,他引:1  
General formulas for the mth integral and derivative of a Chebyshev polynomial of the first or second kind are presented. The result is expressed as a finite series of the same kind of Chebyshev polynomials. These formulas permit to accelerate the determination of such integrals or derivatives. Besides, it is presented formulas for the mth integral and derivative of finite Chebyshev series and a numerical algorithm for the direct evaluation of the mth derivative of such a series.  相似文献
9.
Convergence of CG and GMRES on a tridiagonal Toeplitz linear system   总被引:1,自引:0,他引:1  
The Conjugate Gradient method (CG), the Minimal Residual method (MINRES), or more generally, the Generalized Minimal Residual method (GMRES) are widely used to solve a linear system Ax=b. The choice of a method depends on A’s symmetry property and/or definiteness), and MINRES is really just a special case of GMRES. This paper establishes error bounds on and sometimes exact expressions for residuals of CG, MINRES, and GMRES on solving a tridiagonal Toeplitz linear system, where A is Hermitian or just normal. These expressions and bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first or second kind. AMS subject classification (2000)  65F10, 65N22  相似文献
10.
A pseudospectral method for generating optimal trajectories of linear and nonlinear constrained dynamic systems is proposed. The method consists of representing the solution of the optimal control problem by an mth degree interpolating polynomial, using Chebyshev nodes, and then discretizing the problem using a cell-averaging technique. The optimal control problem is thereby transformed into an algebraic nonlinear programming problem. Due to its dynamic nature, the proposed method avoids many of the numerical difficulties typically encountered in solving standard optimal control problems. Furthermore, for discontinuous optimal control problems, we develop and implement a Chebyshev smoothing procedure which extracts the piecewise smooth solution from the oscillatory solution near the points of discontinuities. Numerical examples are provided, which confirm the convergence of the proposed method. Moreover, a comparison is made with optimal solutions obtained by closed-form analysis and/or other numerical methods in the literature.  相似文献
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