线性常微分方程组解的特征数的估计

复数域上线性系统ｘ＝Ａ（ｔ）ｘ，当Ａ（ｔ）＝（ａｉｊ（ｔ））ｎ×ｎ具有（ｎ，Ｎ，ｒ） 差异性质且ｒｎ时，解的特征数ｊ有估计λｊ－ｌｉｍｔ→∞１ｔ∫ｔｔ０Ｒｅａｊ（τ）ｄτｎ－１ｒ＋１－ｎｌｉｍｔ→∞１ｔ∫ｔｔ０Ａ（τ）ｄτ，ｊ＝１，２，…，ｎ，其中Ａ（ｔ）＝ｍａｘ｛｜ａｉｊ（ｔ）｜：ｉ，ｊ＝１，２，…，ｎ，ｉ≠ｊ．｝     

关于一种循环类预条件方程组的快速求解

１引言考虑下列Ｎ阶线性方程组其中Ｃ１＝,Ｃ２＝０≤ｉ,ｊ≤Ｎ－１,是Ｎ阶循环矩阵,Ｊ１＝（Ｊ）是Ｎ阶置换矩阵,其元素分别满足１９９３年,Ｔ,Ｋ．Ｋｕ,Ｃ．Ｃ．Ｊ．Ｋｕｏ在［１］中取Ｃ１,Ｃ２为实对称循环矩阵,而Ｃ１＋Ｊ１Ｃ２作为预条件矩阵来求解在数字信号处理中有一定应用的Ｔｏｅｐｌｉｔｚ加Ｈａｎｋｅｌ线性方程组［２］,得到了一种高效的预处理其轭梯度算法．当Ｔｏｅｅｌｉｔｚ与Ｈａｎｋｅｌ矩阵之和为正定矩阵且条件数适中时,所需运算量可达到０（Ｎｌｏｇ２Ｎ）,比原有算法［２,３,４］的运算量０（Ｎ２）…     

贪婪t－相交有限序列族

设Ｆ为有限序列族，对ａ＝（ａ１，ａ２，…，ａｎ）∈Ｆ，ａｉ为整数且０≤ａｉ≤ｓｉ（整数），记ｓ（ａ）＝｛ｊ｜１≤ｊ≤ｎ，ａｊ＞０｝，ｓ（Ｆ）＝｛ｓ（ａ）｜ａ∈Ｆ｝，及Ａ｛１，２，…，ｎ｝时Ｗ（Ａ）＝Пｉ∈Ａｓｉ．称Ｆ为贪婪ｔ－相交，如对任何ａ，ｂ∈Ｆ，至少有ｔ个ａｉ，ｂｉ＞０，且Ｗ（Ａ）≥Ｗ（（｛１，２，…，ｎ｝－Ａ）＋Ｂ）对任何Ａ∈Ｓ（Ｆ）及ＢＡ（｜Ｂ｜＝ｔ－１）成立．本文得到当ｓ１＞ｓ２＞…＞ｓｎ时的最大贪婪ｔ－相交有限序列族．     

矩阵特征值的简化Brauer定理

§１　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｔｏｐｉｃｏｆｉｎｃｌｕｓｉｏｎｒｅｇｉｏｎｓｏｆｍａｔｒｉｘｅｉｇｅｎｖａｌｕｅｓｉｓｗｏｒｔｈｙｉｎｖｅｓｔｉｇａｔｉｎｇｉｎｐｒａｃｔｉｃｅａｓｗｅｌｌａｓｉｎｔｈｅｏｒｙ．Ｆｏｒｉｎｓｔａｎｃｅ，ｉｎ［１］ａｎｄ［２］ｃｈａｐｔｅｒｓｏｎｔｈｉｓｔｏｐｉｃａｒｅｐａｒｔｉｃｕｌａｒｌｙｄｅｓｉｇｎｅｄ．ＬｅｔＡ＝（ａｉｊ）∈Ｃｎ×ｎａｎｄｒｉ＝∑ｊ≠ｉ｜ａｉｊ｜（ｉ＝１，２，．．．，ｎ）．Ｂｒａｕｅｒ＇ｓｔｈｅｏｒｅｍｐｒｏｖｉｄｅｓｅｉｇｅｎｖａｌｕ…     

一个部份自回归模型的渐近有效性

本文考虑部分自回归模型Ｘｔ＝ｘｔ－１β＋ｇ（Ｕｔ）＋εｔ，ｔ≥１。这里ｇ是一未知函数，β是和待估参数，εｊ是具有０均值和方差σ＾２的ｉ．ｉ．ｄ．误差，Ｕｔｉ．ｉ．ｄ．服从［０，１］上均匀分布。     

一类Toeplitz矩阵的群逆

Ｔｏｅｐｌｉｔｚ矩阵Ｔｎ＝（ｔｉ－ｊ）ｎｉ，ｊ＝０在信号处理、系统理论、逼近论、正交多项式、积分方程数值解等许多领域常常遇到．易知，Ｔｏｅｐｌｉｔｚ矩阵Ｔｎ的逆矩阵一般不再是Ｔｏｅｐｌｉｔｚ矩阵．１９７２年，Ｇｏｈｂｅｒｇ和Ｓｅｍｅｎｃｕｌ［１］给出...     

关于不可约复矩阵的可逆性

本文推广了不可约复矩阵可逆性的一个经典结果──Ｂｅｔｔｅｒ推论，证明了如下定理：一个ｎ×ｎ不可约复矩阵Ａ是可逆的，如果它满足下列条件之一：某α∈［０，１］，且对至少一个。成立严格不等式，其中Ｎ＝｛１，２，…，ｎ｝，Ｒｉ∑ｊ∈Ｎ－（ｉ）｜αｉｊ｜，Ｃｉ＝∑ｊ∈（ｉ）｜αｊｉ｜，（ｉｉ）｜αｉｉαｊｊ｜≥ＲｉＲｊ，且对至少一对（ｉ，ｊ）成立严格不等式，同时Ａ有两行其非零非对角元的个数不小于２。     

几类凝聚图的轮廓

设Ｇ是个图，｜Ｖ（Ｇ）＝ｎ｜对Ｇ上的任一个标号ｆ：Ｖ（Ｇ）→｛１，…，ｎ｝记，且当ｊ≠ｉ时，Ｇ中有边以ｆ（－１）（ｊ）及ｆ（－１）（ｉ）为两端点｝）．称Ｐ（Ｇ）＝ｍｉｎ｛Ｐ（ｆ）：ｆ是Ｇ上的标号｝为图Ｇ的轮廓．对以Ｗ表示Ｇ中Ｗ的边界．本文证明：ｉ）若Ｇ是凝聚图，ｆ及ｆ是Ｇ上一对互逆标号，则Ｐ（Ｇ）＝Ｐ（ｆ）的充要条件是ｆ为凝聚标号，且此时若Ｇ，Ｈ均是凝聚图，则存在阶梯标号。使得路、回、完全留之间的下列乘积图也是凝聚图，且其轮廓为     

Remark on Some Classes of P—valent Functions

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＡ（ｎ,ｐ）ｂｅｔｈｅｃｌａｓｓｏｆｐ－ｖａｌｅｎｔｆｕｎｃｔｉｏｎｓｏｆｔｈｅｆｏｒｍｆ（ｚ）＝ｚｐ＋∑∞ｊ＝ｐ＋ｎａｊｚｊ,　　（ｎ,ｐ∈Ｎ）ｗｈｉｃｈａｒｅａｎａｌｙｔｉｃｉｎｔｈｅｕｎｉｔｄｉｓｃＵ＝｛ｚ;｜ｚ｜＜１｝．Ｓｕｐｐｏｓｅｆ（ｚ）ａｎｄｇ（ｚ）ａｒｅａｎａｌｙｔｉｃｉｎＵ．Ｉｆｔｈｅｒｅｅｘｉｓｔｓａｆｕｎｃｔｉｏｎｗ（ｚ）,ｗｈｉｃｈｉｓａｎａｌｙｔ－ｉｃｉｎＵａｎｄｓａｔｉｓｆｉｅｓｔｈｅｃｏｎｄｉｔｉｏｎ｜ｗ（ｚ）｜｜ｚ｜,ｓｕｃ…     

子集的交叉t－相交定理

设Ｘ为一个ｎ元集合，Ｃｎｋ为Ｘ的所有ｋ元子集全体，若Ａ∈Ａ，Ｂ∈Ｂ有｜Ａ∩Ｂ｜≥ｔ，则称（Ａ，Ｂ）为一个交叉ｔ－相交子集族．本文得到最大交叉ｔ－相交子集族和最大非空交叉２－相交子集族．证明如下两个结论．（１）若（Ａ，Ｂ）为一个交叉ｔ－相交子集族，且ａ≤ｂ及ａ＋ｂ≤ｎ＋ｔ－１，则｜Ａ＋Ｂ｜≤ｍａｘ｛（ｂｎ），（ａｎ）｝，且当（Ａ；Ｂ）＝（φ，Ｃｎｂ）或（Ｃｎａ，φ）时达到上界．（２）若（Ａ，Ｂ）为一个交叉２－相交子集族，且ａ＜ｂ，ａ＋ｂ≤ｎ－１及（ｎ，ａ，ｂ）≠（２ｉ，ｉ－１，ｉ）（ｉ为任意正整数），又Ａ，Ｂ均非空，则｜Ａ＋Ｂ｜≤１＋（ｂｎ）－（ｂ（ｎ－ａ））－ａ（（ｂ－１）（ｎ－ａ））且当（Ａ，Ｂ）＝（｛Ａ｝，Ｃｎｂ－｛Ｂ｜｜Ｂ｜＝ｂ，｜Ａ∩Ｂ｜≤１｝）时达到上界．     

SINE TRANSFORM MATRIX FOR SOLVING TOEPLITZ MATRIX PROBLEMS

1. IntroductionStrang[1] first studied the use of circulallt matrices C for solving systems of linear eqllationsTi x = b witha symmetric positive definite Toeplitz matrix.Numerous authors such as T.Chan[2],R.Chan,etc.[3],[4],[5], Tyrtyshnikov[6], Huckle[7] and T.Ku and C.Kuo[8] proposed differentfamilies of circulallt / skew- circulant precondit ioners.Appling the preconditioned conjugate gradient algorithm(PCGA) to solve the systems Ti x -b, we must find a preconditioner P such that P…     

广义块Toeplitz特征值问题的基于sine变换的预处理子

在求块Toeplitz矩阵束（Amn，Bmn）特征值的Lanczos过程中，通过对移位块Toepltz矩阵Amn-ρBmn进行基于sine变换的块预处理，从而改进了位移块Toeplitz矩阵的谱分布，加速了Lanczos过程的收敛速度．该块预处理方法能通过快速算法有效快速执行．本文证明了预处理后Lanczos过程收敛迅速，并通过实验证明该算法求解大规模矩阵问题尤其有效．     

THE SINE TRANSFORM OPERATOR IN THE BANACH SPACE OF SYMMETRIC MATRICES AND ITS APPLICATION IN IMAGE RESTORATION

In this paper, we study an operator s which maps every n-by-n symmetric matrix A, to a matrix s(A_n) that minimizes || B_n-A_n || F over the set of all matrices B_n, that can be diagonalized by the sine transform. The matrix s(A_n), called the optimal sine transform preconditioner, is defined for any n-by-n symmetric matrices A_n. The cost of constructing s(A_n) is the same as that of optimal circulant preconditioner c(A_n) which is defined in [8], The s(A_n) has been proved in [6] to be a good preconditioner in solving symmetric Toeplitz systems with the preconditioned conjugate gradient (PCG) method. In this paper, we discuss the algebraic and geometric properties of the operator s, and compute its operator norms in Banach spaces of symmetric matrices. Some numerical tests and an application in image restoration are also given.     

A FAST SINE TRANSFORM ALGORITHM FOR TOEPLITZ MATRICES AND ITS APPLICATIONS

In this paper, a fast algorithm for the discrete sine transform(DST) of a Toeplitz matrix of order N is derived. Only O(N log N) O(M) time is needed for the computation of M elements. The auxiliary storage requirement is O(N). An application of the new fast algorithm is also discussed.     

对称Toeplitz系统的快速W变换基预条件子

１．引言考虑下列Ｎ阶线性方程组其中T_N=(t_i,j)　是Ｎ×Ｎ阶实对称正定（ＳＰＤ）Ｔｏｅｐｌｉｔｚ矩阵,即０,１,…,Ｎ－１）且Ｔ_Ｎ的所有特征值为正数．Ｔｏｅｐｌｉｔｚ系统已广泛应用于数字信号处理,时间序列分析（参见［１］）以及微分方程的数值解（参见［２１］等领域．八十年代以前,考虑到Ｔｏｅｐｌｉｔｚ矩阵的特殊性,人们主要用Ｌｅｖｉｎｓｏｎ递推技术及其变形或者分而治之思想直接求解方程组（１．１）,计算复杂性为Ｏ（Ｎ~（２））或Ｏ（ＮｌｏｇＮ~（２））（参见［３］）;比Ｇａｕｓｓ法运算量级Ｏ（Ｎ~（３）…     

一类Toeplitz线性代数方程组的预处理GMRES方法

本文研究一类来源于分数阶特征值问题的Toeplitz线性代数方程组的求解.构造Strang循环矩阵作为预处理矩阵来求解该Toeplitz线性代数方程组,分析了预处理后系数矩阵的特征值性质.提出求解该线性代数方程组的预处理广义极小残量法（PGMRES）,并给出该算法的计算量.数值算例表明了该方法的有效性.     

Preconditioned Lanczos method for generalized Toeplitz eigenvalue problems

We employ the sine transform-based preconditioner to precondition the shifted Toeplitz matrix _An−ρ_Bn$A_n-\rho B_n$ involved in the Lanczos method to compute the minimum eigenvalue of the generalized symmetric Toeplitz eigenvalue problem _Anx=λ_Bnx$A_{n}x=\lambda B_{n}x$, where _An$A_n$ and _Bn$B_n$ are given matrices of suitable sizes. The sine transform-based preconditioner can improve the spectral distribution of the shifted Toeplitz matrix and, hence, can speed up the convergence rate of the preconditioned Lanczos method. The sine transform-based preconditioner can be implemented efficiently by the fast transform algorithm. A convergence analysis shows that the preconditioned Lanczos method converges sufficiently fast, and numerical results show that this method is highly effective for a large matrix.     

Toeplitz approximate inverse preconditioner for banded Toeplitz matrices

In this paper we introduce a new preconditioner for banded Toeplitz matrices, whose inverse is itself a Toeplitz matrix. Given a banded Hermitian positive definite Toeplitz matrixT, we construct a Toepliz matrixM such that the spectrum ofMT is clustered around one; specifically, if the bandwidth ofT is , all but eigenvalues ofMT are exactly one. Thus the preconditioned conjugate gradient method converges in +1 steps which is about half the iterations as required by other preconditioners for Toepliz systems that have been suggested in the literature. This idea has a natural extension to non-banded and non-Hermitian Toeplitz matrices, and to block Toeplitz matrices with Toeplitz blocks which arise in many two dimensional applications in signal processing. Convergence results are given for each scheme, as well as numerical experiments illustrating the good convergence properties of the new preconditioner.Partly supported by a travel fund from the Deutsche Forschungsgemeinschaft.Research supported in part by Oak Ridge Associated Universities grant no. 009707.     

Preconditioners for block Toeplitz systems based on circulant preconditioners

We study the numerical solution of a block system T _m,n x=b by preconditioned conjugate gradient methods where T _m,n is an m×m block Toeplitz matrix with n×n Toeplitz blocks. These systems occur in a variety of applications, such as two-dimensional image processing and the discretization of two-dimensional partial differential equations. In this paper, we propose new preconditioners for block systems based on circulant preconditioners. From level-1 circulant preconditioner we construct our first preconditioner q ₁(T _m,n ) which is the sum of a block Toeplitz matrix with Toeplitz blocks and a sparse matrix with Toeplitz blocks. By setting selected entries of the inverse of level-2 circulant preconditioner to zero, we get our preconditioner q ₂(T _m,n ) which is a (band) block Toeplitz matrix with (band) Toeplitz blocks. Numerical results show that our preconditioners are more efficient than circulant preconditioners.     

高维加权Bergman空间上的Toeplitz算子

本文研究了高维加权Bergman空间Ap(Bn,dVpφ)(1＜p＜∞).上的Toeplitz算子.利用Toeplitz算子的Berezin变换,获得了Ap(Bn,dVp)(1＜p＜∞)上具有L∞(Bn)符号的Toeplitz算子的有限乘积的有限和是紧算子的一些等价刻画,推广了加权Bergman空间Ap(D,dmpφ)上的Toeplitz算子的有限乘积的有限和是紧的当且仅当它的Berezin变换在单位圆盘的边界消失为0的结论