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101.
Several improvements are made to an algorithm of Higham and Smith for computing the matrix cosine. The original algorithm
scales the matrix by a power of 2 to bring the ∞-norm to 1 or less, evaluates the [8/8] Padé approximant, then uses the double-angle
formula cos (2A)=2cos 2A−I to recover the cosine of the original matrix. The first improvement is to phrase truncation error bounds in terms of ‖A2‖1/2 instead of the (no smaller and potentially much larger quantity) ‖A‖. The second is to choose the degree of the Padé approximant to minimize the computational cost subject to achieving a desired
truncation error. A third improvement is to use an absolute, rather than relative, error criterion in the choice of Padé approximant;
this allows the use of higher degree approximants without worsening an a priori error bound. Our theory and experiments show
that each of these modifications brings a reduction in computational cost. Moreover, because the modifications tend to reduce
the number of double-angle steps they usually result in a more accurate computed cosine in floating point arithmetic. We also
derive an algorithm for computing both cos (A) and sin (A), by adapting the ideas developed for the cosine and intertwining the cosine and sine double angle recurrences.
AMS subject classification 65F30
Numerical Analysis Report 461, Manchester Centre for Computational Mathematics, February 2005.
Gareth I. Hargreaves: This work was supported by an Engineering and Physical Sciences Research Council Ph.D. Studentship.
Nicholas J. Higham: This work was supported by Engineering and Physical Sciences Research Council grant GR/T08739 and by a
Royal Society–Wolfson Research Merit Award. 相似文献
102.
M.?R.?CapobiancoEmail author W.?Themistoclakis 《Advances in Computational Mathematics》2005,23(4):353-374
The present paper gives a contribution of wavelet aspects to classical algebraic polynomial approximation theory. As is so often the case in classical approximation, the authors follow the pattern provided by the trigonometric polynomial case. Algebraic polynomial interpolating scaling functions and wavelets are constructed by using the interpolation properties of de la Vallée Poussin kernels with respect to the four kinds of Chebyshev weights. For the decomposition and reconstruction of a given function the structure of the involved matrices is studied in order to reduce the computational effort by means of fast discrete cosine and sine transforms.
Dedicated to Prof. Guiseppe Mastroianni on the occasion of his 65th birthday.AMS subject classification 65D05, 65T60 相似文献
103.
This paper deals with the intermediate boundary corrections required by splittings of alternating direction methods for solving three space dimensional problems involving Laplace's equation or the heat conduction equation. In addition to considering the existing splittings of Douglas and D'Yakonov, a new splitting is introduced. The extensions of the results to multidimensional problems are briefly noted. 相似文献
104.
105.
研究了利用低功率近红外光辐照In:Fe:LiNbO3晶体时写入光束的偏振方向对光致折射率变化(Δn)的影响.分别采用正常偏振(o光)和非常偏振(e光)的近红外细激光束,在In:Fe:LiNbO3晶体中进行了光折变实验.研究表明,两种偏振方向引起晶体的Δn实测曲线相似,但变化规律恰好相反,o光引起的折射率变化量约是e光的3倍左右.近红外光写入下两种偏振光束引起晶体的Δn分布规律都不同于可见光,尤其是e光辐照区域中心晶体的折射率升高.因此,通过选择不同偏振方向的近红外光可以在光折变晶体中制作不同折射率分布的非线性光学器件. 相似文献
106.
Michael M. Tung Javier Ibáñez Emilio Defez Jorge Sastre 《Mathematical Methods in the Applied Sciences》2020,43(14):8237-8252
This work deals with the simulation of a two-dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second-order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of matrix trigonometric functions. For the derivation we employ the Lagrangian formalism to determine the correct solutions, which extremize the underlying action of the system. In the numerical evaluation we develop diverse state-of-the-art algorithms which efficiently tackle equations with matrix sine and cosine functions. For this purpose, we introduce two special series related to trigonometric functions. They provide approximate solutions of the system through a suitable combination. For the final computation an algorithm based on Taylor expansion with forward and backward error analysis for computing those series had to be devised. We also implement several MATLAB programs which simulate and visualize the two-dimensional lattice and check its energy conservation. 相似文献
107.
In this paper, we are concerned with a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. First, we study the existence of mild solutions for a class of second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces on an interval [0,a]. Later, we study a couple of cases where we can establish the existence of global solutions for a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. We apply our theory to study the existence of solutions for impulsive partial differential equations. 相似文献
108.
In the context of augmented Lagrangian approaches for solving semidefinite programming problems, we investigate the possibility of eliminating the positive semidefinite constraint on the dual matrix by employing a factorization. Hints on how to deal with the resulting unconstrained maximization of the augmented Lagrangian are given. We further use the approximate maximum of the augmented Lagrangian with the aim of improving the convergence rate of alternating direction augmented Lagrangian frameworks. Numerical results are reported, showing the benefits of the approach. 相似文献
109.
110.
We study existence and uniqueness of solutions for a general class of second order abstract differential equations with state-dependent delay. Some examples related to partial differential equations with state dependent delay are presented. 相似文献