【作者】 刘晓娜；

【导师】 唐烁；

【作者基本信息】 合肥工业大学 ， 计算数学， 2010， 硕士

【摘要】 序列变换是加速序列收敛的一种途径,当某些序列收敛速度较慢时,通过变换可以得到一种新的序列,新序列具有较原始序列更为优良的性质,如收敛或加速收敛性质等。序列变换是依据外推法的原理,通过序列的核来插值序列的某些项,最后求解线性方程组得到的。本文首先对序列变换的相关背景做了详细的介绍,通过研究E-变换的收敛及加速收敛性质,阐明了序列变换的优点以及研究序列变换的原因。第二部分描述了三种特殊的序列变换:合成序列变换,拟线性序列变换与收缩序列变换,并研究了几种常见序列变换在加速极限周期连分式中的应用。第三部分利用合成序列变换的构造方法,将标量的合成序列变换推广到向量形式,分别构造了秩为2和k( k≥2)的向量合成序列变换,并给出了相应的收敛和加速收敛性质。第四部分主要讨论了形如K ( an/1)的极限周期连分式的加速收敛情形,将序列变换与连分式加速收敛结合起来,由著名的θ?算法构造了一个序列变换,然后分别从序列变换和加速收敛因子角度研究了此变换在加速极限周期连分式中的应用,最后实现了此变换与常见的加速收敛方法之间的分析比较。它的重要意义在于序列变换加速极限周期连分式收敛可以通过加速收敛因子来实现,这样为研究极限周期连分式的加速收敛创造了便利条件。更多还原

【Abstract】 Sequence transformation is used to accelerate the convergence of sequence. When the convergence of certain sequences is slow, a new sequence has some more excellent properties than the original sequence by transformation, such as the convergence and accelerate convergence. Sequence transformation is based on the principle of extrapolation, interpolating certain items by the kernel of the sequence, and finally is obtained by solving linear equations.In this thesis, the relevant background on the sequence of transformations was made a detailed introduction. We introduce the advantage and the reason of studying sequence transformation by E-transformation. In section 2 we introduce the concept of composite sequence transformation , quasi-linear sequence transfor- mations, and contractive sequence transformations. We also study some sequence transformation in accelerating the convergence of continued fraction. In section 3, we extend the composite sequence transformation to the vector condition by constructing the vector sequence transformation of rank 2 and k, and obtain the corresponding conclusions of convergence and accelerate convergence. In section 4, we discuss the accelerate convergence of continued fraction such as K ( an /1), combing the continued fraction with the sequence transformation. Then we construct a new sequence transformation byθ? algorithm, studying its application in continued fraction from two different angles of sequence transformation and ACFs, and realize the comparison between the sequence transformation and some common accelerate methods. The importance of it is to realize the convergence by accelerating convergence of factors, which bring the convenience in continued fraction.更多还原