It is known that if A is a Toeplitz matrix，then A enjoys a circulant and skew circulant splitting(de—
noted by CSCS)，i．e．，A=C+ S with C a circulant matrix and S a skew circulant matrix． Based on the CSCS iteration，we give m-step preconditioners P for certain classes of Toeplitz matrices in this paper．We
show that if both C and S are positive definite，then the spectrum of the preconditioned matrix(PA)^* PA are
clustered around one for some moderate size ． Experimental results show that the proposed preconditioners
perform slightly better than T．Chan’S preconditioners  for some moderate size m．info:eu-repo/semantics/publishedVersio

Liu, Zhongyun

Yu, Jing

Zhang, Yan

Zhang, Yulin

Universidade do Minho: RepositoriUM

第 36卷 第 1期 2016年 3月 数学理论与应用 MATHEMATICAL THE0RY AND APPLICATIoNS Vo1．36 No．1 Mar． 2O16 m- step Prec0ndi ti0ners for Nonhermitian Positive Definite Toeplitz Systems Zhongyun Liu Jing Yu Yah Zhang Yulin Zhang。 (1．School of Mathematics and Statistics，Changsha University of Science and Technology，Changsha 410004，China； 2．Centro de M atemdtica，Universidade do Minho，Braga 4710—057，Portuga1) Abstract It is known that if A is a Toeplitz matrix，then A enjoys a circulant and skew circulant splitting(de— noted by CSCS)，i．e．，A — C+ S with C a circulant matrix and S a skew circulant matrix．Based on the CSCS iteration[73，we give m—step preconditioners P for certain classes of Toeplitz matrices in this paper．We show that if both C and S are positive definite，then the spectrum of the preconditioned matrix(P A) P A are clustered around one for some moderate size ． Experimental results show that the proposed preconditi0ners perform slightly better than T．Chan’S preconditioners in[-33 for some moderate size m． Key words Irculant— skew circulant splitting m — step polynomial preconditioners Conjugate gradient method Toeplitz matrix 非埃米特正定 Toeplitz矩阵的 m一步预处理子 刘仲云 于静 张艳 张育林。 (1．长沙理工大学数 学与统计学院，长沙，410004； 2．Minho大学数学中心，4710—057，葡萄牙) 摘 要 众所周知，如果A是 Toeplitz矩阵，那么矩阵A有一循环与反循环分裂(记为 CSCS)E7]，可写为A — C4-S，其中 C为循环矩阵，s为反循环矩阵．本文针对某类 Toeplitz矩阵，提 出了一个 m步的预处理子 P ，这个预 处理子 P 是 基于 CSCS迭代方 法构 建的．本 文 中证 明 当C和 S都是 正定矩阵时 ，对于适 当的 m ， 预处理矩阵 (P *A) *(P *A)的谱半径聚集于 1．实验结果表明，对于适 当的 m，本文提 出的预处理子 优 于 T— Otan预处理子i 3 I． 关键词 循环与反循环分裂 m一步多项式预处理子 共轭梯度方法 Toeplitz矩阵 1 Introduction Consider the solution to a large linear system of equations 湖南省研 究生创新项 目(CX2015B374) 收稿 日期 ：2016年 02月 24日 26 数学理论与应用 Ax — b。 (1) by Preconditioned COnjugate gradient(PEG)method。where A is an × nonhermitian posi tive definite Toeplitz matrix where A is constant along its diagonals． Toeplitz linear systems arise in a variety of applications in mathematics，scientific corn— puting and engineering，see for instance[3]and references therein．There are two main types of methods for solving Toeplitz systems：direct methods and iterative methods．The di— rect methods are based on the idea of solving Toeplitz systems recursively．The operational cost of these direct methods is O(n )E 5 3．The second type of method is iterative methods． Conj ugate Gradient(CG)method is a popular method for solving Toeplitz systems．An im— portant property of a Toeplitz matrix is that it can be embedded into 2n× 2n circulant matrix． Thus the operational cost for a Toeplitz matrix— vector multiplication is O(nlogn)by using the Fast Fourier Transforms(FFT)．One of the main important results of this methodology is that the complexity of solving a large class of Toeplitz systems can be reduced to O(nlogn) operations as compared to the O(n )operations required by fast direct Toeplitz solvers，pro～ vided that a suitable preconditioner is chosen under certain conditions on Toeplitz solvers．In the context of the preconditioners for Toeplitz matrices，various preconditioners proposed u— sualy correspond to different classes of Toeplitz matrices with certain structures and proper— ties． That is why the theory and algorithms of preconditioners for Toeplitz systems have been intrigued the researchers for decades，see for instance[3]and references therein． Here we consider the z—step preconditioners(proposed in[1]and denoted by Pm)for such classes of Toeplitz matrices of which each consists of the sum of a positive definite cireu— lant matrix and a positive definite skew circulant matrix． Thepreconditioners are constructed based on CSCS iteration proposed in E7]．Note that the system(1)have the same solution as the following preconditioned system ： (P A ) (P A )z 一 (P A) P b． (2) For such classes of Toeplitz matrices，we’11 show that the eigenvalues of the coefficient matrix in (2)is clustered around one for some moderate large sizem ． when the COnl ugate gradient method is applied to solve system (2)，we therefore expect fast convergence． In this the sequel， 2 m — step preconditioners based on CSCS iteration section，we first review some basic definitions，notation then introduce the CSCS iteration for Toeplitz system and (1) preliminaries used in and finally construct 非埃米特正定 Toeplitz矩阵的 m一步预处理子 27 them — step preconditioners based on CSCS． 2．1 Preliminaries Recall that a matrixA ∈ C ”is said to be positive definite if oZ" Ax> 0 for all nonzero z ∈ C“(z denotes the COnjugate transpose of a vector z)，and that the expression A — M — N is called，respectively。a splitting of A if M  is nonsingular and a convergent splitting A if the spectral radius of N is less than one，i．e．，p(M- N)< 1 The following two lemmas are classical results in matrix is invertible if there is a matrix norm I I·l I such that satisfied，then(I—H) 一 ∑ 。H analysis[4]． H I I< 1． A matriXI— H If this condition ∈ ● lS Let A ∈ ，and￡> 0 be given．There 1’S a matrix norm I1·f l such that p(A)≤ 1 1 A ≤ iD(A)+￡． The folowing conclusion is very often used in iterative methods[6]．Given a matrix A and H such that(I— H)～ exists，there exists such that H — MdNH and A — MH—NH，where MH i It is said that A ： M H— NH is an induced splitting 2．2 The CSCS iteration a unique pair of matrices S nonsingular． of A by H ． nonsingular M H ，N H ， Recal that any Toeplitz matrixA enjoys a circulant and skew—circulant splitting[7-1． A —C+S is a CSCS ofA，then the CSCS iteration for solving(1)in[7]can be described follows． If aS Algorithm ． (CSCS iteration)Given an initial guess 。，for k一 0，1，2，⋯ until conver ges，compute f(口f+C)x抖÷一(aI—S)x +6 ， l(口J+ S)x外 一 (aI—C)x抖专+b where口is positive constants． (3) It is shown in[7]that the CSCS iteration is convergent unconditionaly if bothC and S are positive definite，and that the parameter a has an optimal choice． 2．3 m - step preconditioners For a general n 。——step approximate where H = N Before giving onsingular matrixA 。ifA — M — inverse preconditioner of A can be P 一 (I+ H + H。+ ⋯ + see E1-1． N is a convergent splitting，then an m defined as H 一 )M- ． (4) the preconditioners of this paper，we first establish the following theorem． 28 数学理论与应用 Let A — C+ S be the circulant and skew— circulant splitting of A ．If both C and S are positive definite．then the CSCS iteration induces a convergent splitting A — M ——N with M 一 ± and M N一(口J+s)～(aI—c)(口J+c)一-(口J—s)． 厶a Proof In fact，one CSCS iteration (3)consists of two half step iterations，which can be thought of as the following single step iteration z = Hx + Gb， (5) where H 一 (aI+ S)_。(aI— C)(aI+ C)～ (aI— S)and G 一 2a(aI十 S)一 (aI+ C)_。．From the assumption，口> 0，which implies that G is invertible．By Lemma 2．1，we then have the following induced splitting by H ， A — G — G一 H ． By settingM — G and N — G一 H ，we thus complete the proof． (6) For system (1)，if A — C+ S is a CSCS with positive definite C and S ，by Theorem 4， we then have that A — G一 一 H is a convergent splitting，where G and H are defined as in (6)．Thus the m — step preconditioner P in (4)for A can be defined． 3 Analysis of the convergence The following lemma concerns an error bound which describes the convergence rate of the preconditioned system．([2])Let lz be the kth iteration of the CG method applied to the symmetric positive definite system Bx— b and z be the exact solution of the system．If the ei genvalues of B are ordered such that 0< a≤ 1≤ ⋯ ≤ ≤ 卢，where a and卢are two con— stants,then⋯ SC-Xk ll 1~2(7r十- 11) ⋯ X-X0⋯ ，where 1 l l is the energy norm an by f f I l f l ==vTBv and y= ( a)寺≥ 1． This Lemma tell US that the more clustered the eigenvalues are，the faster the conver— gence rate will be． The following theorem is the main result of this paper，in which the bounds of eigenvalues of the preconditioned matrix in (2)is given． Let A — C+ S be the circulant and skew— circulant splitting of A ，let the preconditioner P be defined as in(4)，and the eigenvalues of the preconditioned matrix A 一 (P A) (P A ) in(2)be ordered such that ≤ ，广1≤ ⋯ ≤ 1．If both C and S are positive definite，then we have the following bounds (1一l l H l l ) ≤ (A)≤ (1+1 1 H l l )。， (7) 非埃米特正定 Toeplitz矩阵的 m一步预处理子 29 where H ： H ，A一 (P A) (P A)，l【．1 l is the matrix norm in Lemma Table 1 No．of iters．with P and Cr in Example 4．1—4．2 Proof Now we consider the preconditioned linear system (2)，PmA — I— H ， since i0(H )< 1，we get P(H )< 1．By lemma 2．1，for any sufficient small e> 0，there exists a norm I1·l1 >0，such thatlD(H)≤I j H l f ≤10(H)+e< 1．It suffices to show that (A) ≥ (1一l l H l I ) and 1(A)≤ (1+l l H )。．We first prove the left inequality(7)：By lemma 2．1，we get ≥ (A)一 p(A一 ) 1 ≥ (I+H+H +⋯ I l ((I— H) ) (I—H) l1 ≥ ≥ —————。——— L—。———一  (1+lI H I I +I l H I l + ⋯)。 For the right inequality of(7)，we have (I—H) I1 ： ( 一ll H 『f )。． (A)≤』D((j—H) )(J—H))≤l l((I—H) )(I—H))l1 ≤ (1+f I H l1 ) ． Thus we complete the proof． 4 Numerical examples All the numerical tests were done on a Founder desktop PC with Pentium dual。——core E6700 CPU 3．20 GHz with M atlab 7．4．0．287 (R2007a)．W hen CG method is applied to the preconditi0ned system (2)，the initial guessx。is chosen to be zero vector．The stopping cri— teria iS r一 I l。 II ≤ lO-。，where is the residual vector at kth iteration． To verify the effectiveness of our preconditioners，three kinds of generating functions were tested and listed as follows 30 数学理论与应用 Example 4．1 (1，1，⋯ ，1) ． a0— 5，a1一一 1，a一1— 1，a一2一 一 2，and“j一 0，elsewhere；b— e— Example 4．2 ao一 10，a1一一 1— 2*i，a2一一 1— 3*i，a—l： 2*i，n一2= 3*i，b 0e ． Example 4．3 a 一 (0．1+I 1)_。， ≥ 0；and a ： i(O．1+} I)一， < 0，b—Ae． Table 2 No．of iters with P and Cr in Example 4．3 For comparison，we also test T．Chan’S circulant preconditioner CT in E 33．The numer— ical results are illustrated in Tables 1— 2，where 7z is the order of coefficient matrix A ，Cr is Chan’S preconditioner and P ，7 一 0，1，2 are our new preconditioners． In the numerical test of Example，the scalar is taken to be 0．9，1．0 and 1．1，respectively．Numerical experi— ments show that our preconditioners P ，m一 1，2，perform slightly better than Chan’S one． References 1-13 L．Adams，M —step preconditioned conjugate gradient methods，SIAM J．Sci．Statist．Comput．6 [2] [33 [43 [5] (1985)，452—462． 0．Axelsson，Iterative Solution Methods，Cambridge University Press，Cambridge，1996． R．Chan，M．Ng，Conjugate gradient methods for Toeplitz systems，SlAM Rev．38(1 996)，427—482． R．A．Horn and C．R．Johnson，Matrix Analysis，Cambridge University Press，Cambridge，UK，1985 G．H．Golub and C．F．Van Loan，Matrix Computations，Johns Hopkins University Press，Baltimore and London，3rd edition，1996． [63 Z．Y．Liu，H．B．Wu，L．Lin，The two—stage iterative methods for symmetric positive definite matri— ces。Applied Math．and Computation 114(2000)，1— 12 E7]M．K．Ng，Circulant and skew—circulant splitting methods for Toeplitz systems，Journal of Computa tional and Applied Math．159(2003)，101— 108 

m-step preconditioners for nonhermitian positive definite Toeplitz systems

http://repositorium.sdum.uminho.pt/bitstream/1822/50409/1/%e9%9d%9e%e5%9f%83%e7%b1%b3%e7%89%b9%e6%ad%a3%e5%ae%9aToeplitz%e7%9f%a9%e9%98%b5%e7%9a%84m-%e6%ad%a5%e9%a2%84%e5%a4%84%e7%90%86%e5%ad%902016.pdf

m-step preconditioners for nonhermitian positive definite Toeplitz systems

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