求分块三对角矩阵和分块周期三对角矩阵逆矩阵的快速算法

给出了分块三对角矩阵逆矩阵的快速算法,并利用所给算法得到了求分块周期三对角矩阵逆矩阵的快速算法.最后通过算例表示算法的有效性.     

三对角矩阵求逆的算法

研究了一般的非奇三对角矩阵的求逆,并给出了一个求逆矩阵的简单算法.首先研究了具有Doolittle分解的三对角矩阵的求逆,得到一个求逆的算法,然后将该算法推广到一般的非奇三对角矩阵上.最后给出了该算法与其它求逆方法的比较,可以看到该算法一方面计算量低,另一方面适用于不需任何附加条件的一般的非奇三对角矩阵.     

周期三对角矩阵逆的一种新算法

给出了一类周期三对角矩阵逆的新的递归算法.新方法充分利用周期三对角矩阵的结构特点,采用递归方法将高阶周期三对角矩阵求逆转化为低阶周期三对角矩阵的求逆.并同时得到简化的计算方法,方法可以有效地减少运算量和存储量,计算精度也有明显的优势.数值实验表明此算法是有效的.     

广义周期七对角矩阵的求逆新算法

讨论了广义周期七对角矩阵的求逆问题,利用七对角矩阵的特殊结构,通过矩阵的广义LU分解,给出了一种求解广义周期七对角逆矩阵的新型算法,该算法不需要对矩阵的各阶顺序主子式做任何限制并且适用于多种计算机代数系统,如：Mathematics,Macsyma,Matlab和Maple等.最后通过算例来说明了算法的有效性。     

分块K—循环Toeplitz矩阵求逆的快速付氏变换法

1算法描述及推导 Toeplitz矩阵及Toeplitz系统的求解在谱分析、线性预测、误差控制码、自回归滤波器设计等领域内起着重要的作用~[1-3]，而分块Toeplitz矩阵在计算机的时序分析、自回归时序模型滤波中也经常出现~[4]。对一般Toeplitz矩阵求逆，其算术复杂性为O(n~2)~[5]-[6]，其中n为Toepleitz矩阵的阶，而K-循环Toeplitz矩阵的求逆，其算术复杂性可降为O(nlog_2n)，本文提供了mn附分块K-循环Toeplitz矩阵求逆的一种快速付氏变换算法，其算术复杂性为O（mnlog_2mn）.     

三对角矩阵求逆的快速算法和逆元素的表示式

本文给出了ｎ阶三对角矩阵求逆的快速算法，其四则运算的计算量只要ｎ＾２＋７ｎ－８。同时给出了逆元素的表示式，从而得到逆元素的准确估计，大大拓广和改进了［２］、［３］的结果。     

三对角矩阵计算

1 引言 在数值计算中,有许多问题最后归结为三对角矩阵的计算,因此研究它们的计算方法是有意义的。此外,有些三对角阵的计算方法可以做为带状阵计算的借鉴。 本文讨论三对角线性方程组的解耦算法,矩阵的LR~(-1)分解,求行列式,Jacobi矩阵的特征值与特征向量的关系以及三对角阵求逆等方面的问题,与现有的算法比较,本文的算法具有计算量或存贮量较少,或计算精度较高,或编程较简单等某些特点。 设A为n阶非奇实三对角阵:     

逆为三对角矩阵的逆M-矩阵

本文研究了一类特殊的逆M-矩阵.利用有向图中的性质和方法,获得了逆M-矩阵其逆为三对角矩阵的充分必要条件,推广了常见的D-型矩阵,得到了一类矩阵为逆M-矩阵的条件.     

计算周期三对角矩阵行列式和逆矩阵的新算法

给出了一种计算周期三对角矩阵行列式和逆矩阵的新递推算法,它们的运算复杂度分别为O(n)和O(n2),该算法是文献[5]和[6]中相关算法的拓广.     

r-对称循环矩阵及逆矩阵三角分解的快速算法

根据r-对称循环矩阵的特殊结构给出了求这类矩阵本身及其逆矩阵三角分解的快速算法,算法的运算量均为O(n2),一般矩阵及逆矩阵三角分解的运算量均为O(n3).     

A note on inversion of Toeplitz matrices

It is shown that the invertibility of a Toeplitz matrix can be determined through the solvability of two standard equations. The inverse matrix can be denoted as a sum of products of circulant matrices and upper triangular Toeplitz matrices. The stability of the inversion formula for a Toeplitz matrix is also considered.     

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.     

求置换因子循环矩阵的逆阵及广义逆阵的快速算法

1 引 言 循环矩阵由于其应用非常广泛而成为一类重要的特殊矩阵,如在图象处理、编码理论、自回归滤波器设计等领域中经常会遇到以这类矩阵为系数的线性系统的求解问题.而对称循环组合系统也具有广泛的实际背景,例如造纸机的横向控制系统,具有平行结     

A fast algorithm for computing the smallest eigenvalue of a symmetric positive‐definite Toeplitz matrix

Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive‐definite (SPD) Toeplitz matrix. An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016/j.laa.2007.05.008 ). The algorithm relies on the computation of the R factor of the QR factorization of the Toeplitz matrix and the inverse of R. The simultaneous computation of R and R^?1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed. Copyright © 2008 John Wiley & Sons, Ltd.     

A direct method to solve block banded block Toeplitz systems with non-banded Toeplitz blocks

A fast solution algorithm is proposed for solving block banded block Toeplitz systems with non-banded Toeplitz blocks. The algorithm constructs the circulant transformation of a given Toeplitz system and then by means of the Sherman-Morrison-Woodbury formula transforms its inverse to an inverse of the original matrix. The block circulant matrix with Toeplitz blocks is converted to a block diagonal matrix with Toeplitz blocks, and the resulting Toeplitz systems are solved by means of a fast Toeplitz solver.The computational complexity in the case one uses fast Toeplitz solvers is equal to ξ(m,n,k)=O(mn³)+O(k³n³) flops, there are m block rows and m block columns in the matrix, n is the order of blocks, 2k+1 is the bandwidth. The validity of the approach is illustrated by numerical experiments.     

The structured distance to normality of Toeplitz matrices with application to preconditioning

A formula for the distance of a Toeplitz matrix to the subspace of {e^i?}‐circulant matrices is presented, and applications of {e^i?}‐circulant matrices to preconditioning of linear systems of equations with a Toeplitz matrix are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Algorithms for Finding the Inverses of Factor Block Circulant Matrices

In this paper,algorithms for finding the inverse of a factor block circulant matrix, a factor block retrocirculant matrix and partitioned matrix with factor block circulant blocks over the complex field are presented respectively.In addition,two algorithms for the inverse of a factor block circulant matrix over the quaternion division algebra are proposed.     

包含切比雪夫多项式的循环矩阵行列式的计算

行首加r尾r右循环矩阵和行尾加r首r左循环矩阵是两种特殊类型的矩阵,这篇论文中就是利用多项式因式分解的逆变换这一重要的技巧以及这类循环矩阵漂亮的结构和切比雪夫多项式的特殊的结构,分别讨论了第一类、第二类切比雪夫多项式的关于行首加r尾r右循环矩阵和行尾加r首r左循环矩阵的行列式,从而给出了行首加r尾r右循环矩阵和行尾加r首r左循环矩阵的行列式显式表达式.这些显式表达式与切比雪夫多项式以及参数r有关.这一问题的应用背景主要在循环编码,图像处理等信息理论方面.     

Diagonal and Toeplitz splitting iteration methods for diagonal‐plus‐Toeplitz linear systems from spatial fractional diffusion equations

The finite difference discretization of the spatial fractional diffusion equations gives discretized linear systems whose coefficient matrices have a diagonal‐plus‐Toeplitz structure. For solving these diagonal‐plus‐Toeplitz linear systems, we construct a class of diagonal and Toeplitz splitting iteration methods and establish its unconditional convergence theory. In particular, we derive a sharp upper bound about its asymptotic convergence rate and deduct the optimal value of its iteration parameter. The diagonal and Toeplitz splitting iteration method naturally leads to a diagonal and circulant splitting preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1, especially when the discretization step‐size h is small. Numerical results exhibit that the diagonal and circulant splitting preconditioner can significantly improve the convergence properties of GMRES and BiCGSTAB, and these preconditioned Krylov subspace iteration methods outperform the conjugate gradient method preconditioned by the approximate inverse circulant‐plus‐diagonal preconditioner proposed recently by Ng and Pan (M.K. Ng and J.‐Y. Pan, SIAM J. Sci. Comput. 2010;32:1442‐1464). Moreover, unlike this preconditioned conjugate gradient method, the preconditioned GMRES and BiCGSTAB methods show h‐independent convergence behavior even for the spatial fractional diffusion equations of discontinuous or big‐jump coefficients.