分块K—循环Toeplitz矩阵求逆的快速付氏变换算法

本文讨论了分块K－循环Toeplitz系统,导出分块K－循环Toeplitz矩阵求逆的一种快速付氏变换算法,其算法复杂性为O（mnlog2mn）。     

分块K

本文讨论了分块K-循环Toeplitz系统,导出分块K-循环Toeplitz矩阵求逆的一种快速付氏变换算法,其算法复杂性为O(mnlog2 mn).     

(n1,n2,…,nk)型k重(r1,r2,…,rk)-循环矩阵求逆的快速算法

本文给出了(n1，n2，…，nk)型k重(r1，r2，…，rk)-循环矩阵求逆的快速算法，其计算复杂性为O[(Ⅱ↑ki=1ni)logsⅡ↑ki=1ni]。     

求Sylvester矩阵逆矩阵的快速算法

本文利用m+n阶Sylvester矩阵的位移结构并在假设该矩阵的所有顺序主子矩阵可逆的条件下给出了求解Sylvester矩阵的逆的一种快速算法.该算法所需计算量为O(m+n)~2,而高斯-约当消去法所需计算量为O(m+n)~3.最后通过数值算例说明了算法的有效性.     

Toeplitz矩阵相乘的一种新快速算法

将Toeplitz矩阵分解为一个循环矩阵和一个下三角Toeplitz矩阵之和,以及一般卷积向循环卷积的转化,借助快速Fouier变换(FFT),导出了一种计算两个n阶Toeplitz矩阵乘积的新快速算法,其算法复杂性为2n2 63/4n log2n-15n-34次实乘运算,4n2 63/2n log2n-18n 23次实加运算,与已有的优化算法相比,在实乘次数有所降低的同时,实加次数降低了近1/3,是目前复杂性最小的一种算法.     

r-循环矩阵开平方的两个快速算法

本文利用快速富里叶变换(FFT)和矩阵分块逐次降阶的方法，给出了两种n阶r—循环矩阵开平方的快速算法，其计算复杂性均为O(nlog2n)。     

关于Vandermonde矩阵的串行与并行复杂性

为n+1阶Vandermonde矩阵,简称V阵。 本文首先给出求解相应线性代数方程组(简称V型方程组)的递推算法。算术运算总次数为O(n~2)级,接着进一步利用快速插值算法导出求V阵逆的O(n~2)算法,并分析了这两种算法的并行时间复杂性。     

Toeplitz矩阵在图像恢复中的应用

从某种意义来说,图像退化的过程可以等价为传输函数和噪声对原始图像矩阵进行线性变换,因而此像恢复过程可以等价于在了解传输函数的基础上,尝试由最小二乘给出解答。当传输函数可分离时,该问题可进一步转化为Toeplitz矩阵的求逆。通过仿真实验,验证了该方法的有效性并给出了算法的数值稳定性分析。     

GPU上循环矩阵的快速求逆算法

循环矩阵是一种特殊类型的Toeplitz矩阵,在很多专业领域尤其是图像和数字信号处理中有广泛的应用。计算其逆矩阵的快速算法由三个步骤组成:(1)使用离散傅立叶变换将矩阵的第一行元素转换到频率空间;(2)计算转换后的频谱中每个幅度的倒数;(3)在调整过的频谱上施加傅立叶反变换,获得逆矩阵的第一行元素,从而构建原始循环矩阵的逆矩阵。此算法的特点是每个数据元素的计算过程完全相同,同时独立于其它元素的计算,因而非常适合在GPU上运行。本文在GPU上实现了上述循环矩阵求逆的快速算法,将其转换为一个正方形的图形绘制。实验结果表明,该算法在GPU上的运行速度比在CPU上提高了大约10倍。     

对称Toeplitz线性方程组的基于余弦变换的最佳预优矩阵

利用第四类离散余弦变换矩阵构造出求解对称Toeplitz线性方程组的最佳预优矩阵,构造该预优矩阵所需的运算量为O(n).理论和数值实验显示,利用本文中所构造的预优矩阵求解对称Toeplitz线性方程组所需的迭代次数与现有的其它类型预优矩阵差不多,但预优矩阵的构造要更简单.     

Toeplitz矩阵相乘的快速卷积算法

本文利用Ｔｏｅｐｌｉｔｚ矩阵可分解为循环阵与斜循环阵之和的特点２，借助于卷积的ＦＦＴ算法，推导出计算两个Ｔｏｅｐｌｉｔｚ矩阵之积的一种新的快速算法，其乘法复杂性为２ｎ＾２＋Ｏ（ｎｌｏｇ２ｎ）。     

基于随机间距稀疏 Toeplitz 测量矩阵的压缩传感

选择合适的测量矩阵是压缩传感理论实用化的关键之一. 本文在Toeplitz矩阵独立元素中随机地引入零元,形成随机间距稀疏Toeplitz矩阵, 使得随机独立变元个数可以减少到原Toeplitz矩阵的1/2～1/16,甚至更少, 非零元个数同样大大减少,有利于数据传输和存储.模拟实验表明随机间距稀疏 Toeplitz矩阵在重建效果优于Gauss矩阵和原Toeplitz矩阵的同时,重建时间只有Gauss矩阵和一般Toeplitz矩阵重建时间的约15%～40%.     

Matrix structures in parallel matrix computations

We survey certain techniques such as approximation, interpolation and embedding in matrix algebra, which are particularly useful to devise fast parallel algorithms for some matrix-structured problems. The problems we consider are triangular Toeplitz matrix inversion and its algebraic counterpart polynomial division, matrix multiplication, and some computations concerning banded Toeplitz matrices.     

Fast inversion of triangular Toeplitz matrices

An algorithm requiring0(n log n)operations is proposed for the inversion of triangular Toeplitz matrices. This algorithm is faster than existing techniques, even for relatively small values ofn, and can easily be written in Fortran.     

Inversion of the generalized Fibonacci matrix by convolution

A convolution formula containing the generalized Fibonacci numbers and applications of this formula are investigated. Starting from the convolution formula, we derive combinatorial identities involving generalized and usual Fibonacci numbers, as well as the Lucas numbers. The inversion of a lower triangular matrix and the generalized inversion of strictly lower triangular Toeplitz matrix whose non-zero elements are generalized Fibonacci numbers are considered.     

Solving certain queueing problems modelled by toeplitz matrices

By using the concept of generating function associated with a Toeplitz matrix, we analyze existence conditions for the probability invariant vector π of certain stochastic semi-infinite Toeplitz-like matrices. An application to the shortest queue problem is shown. By exploiting the functional formulation given in terms of generating functions, we devise a weakly numerically stable algorithm for computing the probability invariant vector π. The algorithm is divided into three stages. At the first stage the zeros of a complex function are numerically computed by means of an extension of the Aberth method. At the second stage the first k components of π are computed by solving an interpolation problem, where k is a suitable constant associated with the matrix. Finally, at the third stage a triangular Toeplitz system is solved and its solution is refined by applying the power method or any other refinement method based on regular splittings. In the solution of the triangular Toeplitz system and at each step of the refinement method, special FFT-based techniques are applied in order to keep the arithmetic cost within the O(n log n) bound, where n is an upper bound to the number of the computed components. Numerical comparisons with the available algorithms show the effectiveness of our algorithm in a wide set of cases.     

A Toeplitz algorithm for polynomial J-spectral factorization

A block Toeplitz algorithm is proposed to perform the J-spectral factorization of a para-Hermitian polynomial matrix. The input matrix can be singular or indefinite, and it can have zeros along the imaginary axis. The key assumption is that the finite zeros of the input polynomial matrix are given as input data. The algorithm is based on numerically reliable operations only, namely computation of the null-spaces of related block Toeplitz matrices, polynomial matrix factor extraction and linear polynomial matrix equations solving.     

A new version of Kovarik’s approximate orthogonalization algorithm without matrix inversion

In 1970 Kovarik proposed approximate orthogonalization algorithms. One of them (algorithm B) has quadratic convergence but requires at each iteration the inversion of a matrix of similar dimension to the initial one. An attempt to overcome this difficulty was made by replacing the inverse with a finite Neumann series expansion involving the original matrix and its adjoint. Unfortunately, this new algorithm loses the quadratic convergence and requires a large number of terms in the Neumann series which results in a dramatic increase in the computational effort per iteration. In this paper we propose a much simpler algorithm which, by using only the first two terms in a different series expansion, gives us the desired result with linear convergence. Systematic numerical experiments for collocation and Toeplitz matrices are also described.