泰勒公式-文献翻译.doc

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1、精品文档，仅供学习与交流，如有侵权请联系网站删除重 庆 理 工 大 学文 献 翻 译二级学院 数学与统计学院 班 级 111010101 学生姓名 学 号 11101010110 复杂矩阵用泰勒公式的解决方法一个复杂的RR矩阵A中的解决方法R（A）是与A的频谱A空交集的任何域名的解析函数。在任何给定的0A的附近著名的泰勒展开R（A）的修改考虑到R0只有第一大国（A）是线性无关。在这个框架的主要工具给出了多变量多项式查看MATHML源取决于不变V1，V2，.，VRR（A）的（m为最小多项式的程度）。这些功能被用于以代表的R0（A）的随后的权力的系数作为它们的第一米的线性组合。一 简介如在1中，预

2、解R（A）的（I-A）一种非奇异正方形矩阵A（表示单位矩阵）-1所示的希尔伯特同一性的后果是参数的一个解析函数在与A的频谱A因此，空交集的任何域D使用泰勒展开任何固定0D的附近，我们可以在1R（A）的表示公式发现使用R0的一切权力（A）。在这篇文章中，通过使用一些前面的结果回忆说，例如，在2中，我们写下使用R0（A）的权力，只有有限数量的表示公式。这似乎是因为R0的（A）是线性无关的只有第一个权力是自然的。在此框架的主要工具是由多变量多项式给出查看MATHML来源（参见2，3，4，5和6），根据不同的不变量V1，V2，.，vr中的R（A）;这里m表示极小多项式的程度。二 权力矩阵和F k,n功

3、能我们还记得在本节一定的成效上表示公式矩阵的权力（见23456和其参考文献）。为简单起见，我们指的是情况下，当基质是非贬损使得M = R。命题2.1。设A是一个矩阵，由U1表示，U2，.，UR A的不变量，并通过其特征多项式（按照惯例u01）;那么对于A的非负整数指数的权力下表示公式也是如此：功能Fk,n(u1.ur)该显示为系数（2.1）由递推关系定义和初始条件：此外，如果A是非奇异（ur0），则式（2.1）仍然保持对于n的负值，只要我们定义FK，n功能对于n的负值如下：三 泰勒展开式的解决对策我们认为解决方法矩阵R（A）定义如下：注意，有时有标志的公式的变化。 Eq.（3.1），但这个当然

4、不是必须的。这是众所周知的，该预解是一个解析（理性）的在复平面不含A的频谱的每个域D功能，此外它消失在无穷远，以便R的仅奇点（极点）（A）的是A的特征值6事实证明，不变量V1，V2，.，R（A）的VR都与A的方程联作为命题2.1，Eq（3.2）的结果，R（A）的整数幂可以如下表示。定理3.1对于每个且其中是由Eq（3.2）给出。由（A）A的谱半径表示，对于每一个，使得（A）分钟（|，|），希尔伯特的身份也是如此（见1）：因此，对于每A，我们有 总的来说所以，对于每一个0D，R（A）可以在泰勒级数扩大这是D.绝对和一致收敛定义其中v（）由等式限定。（3.2），我们可以证明下面的定理。定理3.2该

5、解决方法R（A）在任何正规点0邻里的泰勒展开式（3.7）可以写成的形式 因此，我们可以得出一个结果：Tuilun3.1 对于每0A和= 1,2，.，R的级数展开是收敛的。证明：回顾（3.3），我们可以写因此，考虑到的初始条件（2.3）我们可以写所以（3.10）成立。级数展开收敛（3.11）是初始扩张（3.7）的收敛琐碎的后果。四 结束语泰勒公式的用途很广泛，它是数学分析中的重要组成部分，它的理论方法已成为研究函数极限和估计误差等方面的不可缺少的工具，它即是微积分中值定理的推广应用，又是运用高阶导数研究函数性态的重要工具之一。特别在计算机的计算能力和解决复杂算式的能力快速发展的今天，因为计算机无

6、法解决太过复杂的函数，数学问题的解决习惯是把未知问题转换成已知问题，把复杂问题转化成多个简单问题，泰勒公式的发展泰勒级数就是这样的计算机可以很好解决的简单函数，一旦把一个函数展开成泰勒级数的形式，就可以交给计算机来解决了，所以泰勒公式在当代早已不可或缺。不单在计算机的各个领域都有着重要的应用，而且泰勒公式解决复杂问题的能力在数学领域的研究方面也有了很大的作用。值得一提的是，解决方法R（A）是用于表示矩阵A其实解析函数，用f表示（Z）的复变函数z的函数，分析含A的谱域作主题元素，并用表示A的特征值显着与多重K，拉格朗日 - 西尔维斯特公式（见4）由下式给出其中是与特征值K相关的投影仪，并通过k乔

7、丹曲线，域的Dk的边界，分离与所有其他特征值固定K，回顾里斯表示的公式，可以得出当K仅已知大约，本投影不能被利用残余定理导出。在这种情况下，有必要整合R（A）中沿k（即可能是Gershgorin圆），利用预解的已知的表示（参见3）或者通过取代R（A）与它的泰勒展开，并假设为初始点任何0K内的Dk。这是最好的公式依赖于相关的稳定性和计算成本。从理论角度，公式（3.7），（3.10）及（4.1）似乎是从稳定性来看等效，因为所有所需要的给定的矩阵A的不变量的知识然而，在我们看来，在情况考虑，式（3.10）似乎是相对于（3.7）比较便宜，因为它需要一个近似r系列的基本功能，而不是无穷级数矩阵。参考文献

8、1I。 Glazman，Y H. Liubitch达马迪安（主编），分析linaire丹斯莱ESPACES德维finies：曼努埃尔等problmes，Traduit杜齐名俄语，和平号空间站，莫斯科（1972）2M。 Bruschi，体育利玛窦Sulle potenze二UNA矩阵的计算quadrata德拉quale SIA诺托IL polinomio MINIMO Pubbl。 IST。垫。申请。 FAC。 ING。大学。梭哈。罗马，四路，13（1979），页9-183 V.N。线性代数多佛酒吧Faddeeva计算方法。 Inc。，纽约（1959）4 F.R。 Gantmacher矩阵，二卷

9、的理论。 1，2（KA赫希，跨。）切尔西出版公司，纽约（1959）5M。 Bruschi，体育利玛窦Sulle funzioni FK，正荣polinomi迪卢卡斯迪seconda物种generalizzati Pubbl。 IST。垫。申请。 FAC。 ING。大学。梭哈。罗马，四路，14（1979年），第49-586M。 Bruschi，体育利玛窦的显式F（A）SIAM J.数学。肛门。，13（1982），第162-165On Taylors formula for the resolvent of a complex matrixThe resolvent R(A) of a compl

10、ex rr matrix A is an analytic function in any domain with empty intersection with the spectrum A of A. The well known Taylor expansion of R(A) in a neighborhood of any given 0A is modified taking into account that only the first powers of R0(A) are linearly independent. The main tool in this framewo

11、rk is given by the multivariable polynomials View the MathML source depending on the invariants v1,v2,vr of R(A) (m denotes the degree of the minimal polynomial). These functions are used in order to represent the coefficients of the subsequent powers of R0(A) as a linear combination of the first m

12、of them.1. IntroductionAs a consequence of the Hilbert identity in 1, the resolvent R(A)(IA)1 of a nonsingular square matrix A (I denoting the identity matrix) is shown to be an analytic function of the parameter in any domain D with empty intersection with the spectrum A of A. Therefore, by using T

13、aylor expansion in a neighborhood of any fixed 0D, we can find in 1 a representation formula for R(A) using all powers of R0(A).In this article, by using some preceding results recalled, e.g., in 2, we write down a representation formula using only a finite number of powers of R0(A). This seems to b

14、e natural since only the first powers of R0(A) are linearly independent. The main tool in this framework is given by the multivariable polynomials View the MathML source (see 2, 3, 4, 5 and 6), depending on the invariants v1,v2,vr of R(A); here m denotes the degree of the minimal polynomial.2. Power

15、s of matrices andFk,nfunctionsWe recall in this section some results on representation formulas for powers of matrices (see e.g.2,3,4,5and6and the references therein). For simplicity we refer to the case when the matrix is non derogatory so thatm=r.Proposition2.1.Let A be an complex matrix, and deno

16、te by u1,u2,ur the invariants of A ,and byits characteristic polynomial (by convention u01); then for the powers of A with non negative integral exponents the following representation formula holds true:The functionsFk,n(u1.ur)that appear as coefficients in(2.1)are defined by the recurrence relation

17、and initial conditions:Furthermore, ifAis nonsingular(ur0), then formula(2.1)still holds for negative values ofn, provided that we define theFk,nfunction for negative values ofnas follows:3. Taylor expansion of the resolventWe consider the resolvent matrixR(A)defined as follows:Note that sometimes t

18、here is a change of sign in Eq.(3.1), but this of course is not essential.It is well known that the resolvent is an analytic (rational) function ofin every domainDof the complex plane excluding the spectrum ofA, and furthermore it is vanishing at infinity so the only singular points (poles) ofR(A)ar

19、e the eigenvalues ofA.In6it is proved that the invariantsv1,v2,vrofR(A)are linked with those ofAby the equationsAs a consequence ofProposition2.1, and Eq.(3.2), the integral powers ofR(A)can be represented as follows.Theorem3.1.For every Aand ,where the are given by Eq. (3.2).Denoting by (A) the spe

20、ctral radius of A, for every , such that (A)min(|,|), the Hilbert identity holds true (see 1):Therefore for everyA, we have and in generalso, for every0D,R(A)can be expanded in the Taylor serieswhich is absolutely and uniformly convergent inD.Definingwhere thev()are defined by Eq.(3.2), we can prove

21、 the following theorem.Theorem3.2.The Taylor expansion(3.7)of the resolvent R(A)in a neighborhood of any regular point 0 can be written in the formTherefore we can derive as a consequence:Corollary3.1.For every 0A and =1,2,r the series expansionsare convergent.Proof.Recalling(3.3), we can writeThere

22、fore, taking into account the initial conditions(2.3)we can writeso(3.10)holds true. The convergence of series expansions(3.11)is a trivial consequence of the convergence of the initial expansion(3.7).4. Concluding remarksTaylor formula uses very wide, it is the mathematical analysis of an important

23、 part of its theoretical approach has become an indispensable tool for studying the function and the estimation error limits and other aspects, it is the application that is value theorem of calculus, and It is an important tool to study the use of higher derivative function of the state. Particular

24、ly in computing power and the ability to solve complex computer formula of rapid development of today, because the computer can not be solved too complex function, mathematical problem solving habit is to convert the issue into the unknown known issues, the complex problem into multiple simple issue

25、, Taylor series development of Taylor formula is simple functions such computers can be solved, once a function of the Taylor series expansion of the form, you can give computers to solve, so Taylor formula in contemporary or had not missing. Not only in all areas of the computer they have important

26、 applications, and Taylor Formula ability to solve complex problems in mathematics research has also been a great role.It is worth noting that the resolventR(A)is a keynote element for representing analytic functions of a matrixA. In fact, denoting byf(z)a function of the complex variablez, analytic

27、 in a domain containing the spectrum ofA, and denoting bythe distinct eigenvalues ofAwith multiplicitiesk, the LagrangeSylvester formula (see4) is given bywhereis the projector associated with the eigenvaluek, andDenoting byka Jordan curve, the boundary of the domainDk, separating a fixedkfrom all o

28、ther eigenvalues, recalling the Riesz formula, it follows thatWhenkis only known approximately, this projector cannot be derived by using the residue theorem.In this case it is necessary to integrateR(A)alongk(being possibly a Gershgorin circle), by using the known representation of the resolvent (s

29、ee3)or by substitutingR(A)with its Taylor expansion, and assuming as initial point any0kinsideDk.Which is the best formula depends on the relevant stability and computational cost. From the theoretical point of view, formulas(3.7),(3.10)and(4.1)seem to be equivalent from the stability point of view,

30、 since all require knowledge of invariants of the given matrixA. However, in our opinion, in the situation considered, Eq.(3.10)seems to be less expensive with respect to(3.7), since it requires one to approximaterseries of elementary functions instead of an infinite series of matrices.References1I.

31、 Glazman, Y. Liubitch H. Damadian (Ed.), Analyse linaire dans les espaces de dimension finies: Manuel et problmes, Traduit du russe par, Mir, Moscow (1972)2M. Bruschi, P.E. Ricci Sulle potenze di una matrice quadrata della quale sia noto il polinomio minimo Pubbl. Ist. Mat. Appl. Fac. Ing. Univ. Stu

32、d. Roma, Quad., 13 (1979), pp. 9183V.N. Faddeeva Computational Methods of Linear Algebra Dover Pub. Inc., New York (1959)4F.R. Gantmacher The Theory of Matrices, Vols. 1, 2 (K.A. Hirsch, Trans.)Chelsea Publishing Co., New York (1959)5M. Bruschi, P.E. Ricci Sulle funzioniFk,ne i polinomi di Lucas di seconda specie generalizzati Pubbl. Ist. Mat. Appl. Fac. Ing. Univ. Stud. Roma, Quad., 14 (1979), pp. 49586M. Bruschi, P.E. Ricci An explicit formula forf(A) SIAM J. Math. Anal., 13 (1982), pp. 162165译 文 评 阅导师评语（应根据学校“译文要求”，对学生外文翻译的准确性、翻译数量以及译文的文字表述情况等作具体的评价）指导教师： 年 月 日【精品文档】第 9 页