求解GPC中逆矩阵的递推算法

采用矩阵分解方法，推导出一种求解广义预测控制（GPC）中的逆矩阵的递推算法，使得计算量大为减少。     

Toeplitz矩阵之逆矩阵的新分解式及快速算法

本文利用线性方程组是否有解给出了Toeplitz矩阵可逆的条件,表明Toeplitz矩阵的逆矩阵可以表示为循环矩阵与下三角Toeplitz矩阵的乘积之和,给出了其逆矩阵列的递推公式,得到了求Toeplitz矩阵之逆矩阵的快速算法,计算复杂性为O(n2),一般n阶矩阵求逆的计算复杂性为O(n3).     

广义逆张量Padé逼近的连分式递推算法

张量指数函数已经广泛应用于控制论、图像处理和各个工程领域.鉴于此,在矩阵广义逆的基础上,首次在张量内积空间上定义一种有效的张量广义逆,从而构造张量Padé逼近的一种连分式算法.利用张量t-积成功计算张量的幂,由此递推地给出张量指数函数的幂级数展开式.在前面两个工作的基础上,利用设计的连分式算法逼近张量指数函数,其特点在于,该算法可以编程实现递推计算,而且在计算过程中不必计算张量的乘积,也不必计算张量的逆.给出的两个张量指数函数的数值实验表明,将连分式算法与目前通常使用的截断法进行比较,在不降低逼近阶的条件下,所提出算法是有效的.如果张量的维数较大,基于张量广义逆的连分式算法仍然具有一定优势.     

行对称矩阵广义逆的快速求解公式

对行对称矩阵的QR分解进行了研究,在此基础上给出了求行对称矩阵广义逆的快速求解公式,并给出了证明。将QR分解方法应用于该类行对称矩阵的广义逆的求解过程,既利用了QR分解保证足够的精度,又可大大降低求解一类具有该结构矩阵的广义逆的计算量和存储量。     

广义Nekrasov矩阵的新迭代判别法

广义Nekrasov矩阵作为一类特殊的广义严格对角占优矩阵在科学和工程实际中有着广泛的应用,因此研究这类矩阵的判定问题是非常重要的.给出了判定一个矩阵是否为广义Nekrasov矩阵的两种新的迭代算法,并用数值算例说明了算法的有效性.由于证明了广义Nekrasov矩阵就是广义严格对角占优矩阵,从而也就得到了两种新的判定广义严格对角占优矩阵的迭代算法.     

基于广义逆矩阵的B样条曲线节点消去算法研究

为了能运用广义逆矩阵理论来研究B样条曲线的节点消去问题，以解决在B样条曲线曲面拟合过程中产生的冗余节点数据，提出了一种基于广义逆矩阵的B样条曲线节点消去算法，该算法首先利用广义逆矩阵在处理奇异性问题上的独特作用来获得B样条曲线的节点可以消去的充要条件；然后在此基础上，又提出了消去多个节点的算法，算法对每个可以消去的节点都可计算相应的广义逆矩阵，而且仅进行一次矩阵的相乘即可得到由消去这个节点而产生的新的控制顶点和节点。实验表明，该算法的精度优于或近似于现有的Tiller算法，而时间效率则同于或近似于Tiller的算法。由于通过调整算法中的误差阈值，可以有效地控制消去节点后的曲线与原来曲线的误差，因此算法可以用于工程实践。     

基于广义逆矩阵的Bézier曲线降阶逼近

研究了Bézier曲线的降多阶逼近问题.利用Bézier曲线本身的升阶性质,并结合广义逆矩阵的最小二乘理论,给出了一种新的降阶逼近方法.此方法克服了一般降阶方法中每次只能降阶一次的弱点,并且得到了很好的逼近效果.     

基于广义逆矩阵的Bezier曲线降价逼近

研究了Bezier曲线的降多阶逼近问题。利用Bezier曲线本身的升阶性质,并结合广义逆矩阵的最小二乘理论,给出了一种新的降阶逼近方法。此方法克服一一般降价方法中每次只能降价一次的弱点,并且得到了很好的逼近效果。     

行满秩Toeplitz型矩阵Moore-Penrose逆的快速算法

通过构造对称分块矩阵给出了秩为m的m×n阶Toeplitz型矩阵Moore-Penrose逆的快速算法。该算法计算复杂度为O（mn）+O（m²）,而由T^T（TT^T）^-1直接求解所需运算量为O（m²n）+O（m3）。数值算例表明了该快速算法的有效性。     

小议矩阵的逆

本文通过对矩阵的逆以及逆矩阵的探讨,归纳出几种求逆矩阵的方法。     

快速递推广义预测控制

本文利用反向递推方法对广义预测控制的准则函数进行变换，得出一种递推的快速ＧＰＳ算法。     

A parallel Broyden approach to the Toeplitz inverse eigenproblem

In this work we show a portable sequential and a portable parallel algorithm for solving the inverse eigenproblem for real symmetric Toeplitz matrices. Both algorithms are based on Broyden's method for solving nonlinear systems. We reduced the computational cost for some problem sizes, and furthermore we managed to reduce spatial cost considerably, compared in both cases with parallel algorithms proposed by other authors and by us, although sometimes quasi‐Newton methods (as Broyden) do not reach convergence in all the test cases. We have implemented the parallel algorithm using the parallel numerical linear algebra library SCALAPACK based on the MPI environment. Experimental results have been obtained using two different architectures: a shared memory multiprocessor, the SGI PowerChallenge, and a cluster of Pentium II PCs connected through a myrinet network. The algorithms obtained are scalable in all the cases. Copyright © 2004 John Wiley & Sons, Ltd.     

Recursive Algorithms of Parallel Matrix Computations

Consideration was given to the recursive approach to the block algorithms of linear algebra. The problem of LL _T-decomposition (quadratic root) was used by way of example. Computational complexity was estimated both in terms of arithmetic floating-point operations and data-transfer operations required to generate recursive structures. The main area of application of the algorithms is solution of large-scale problems on parallel and distributed computer systems.     

基于粒子群优化算法的广义逆控制分配方法

为了获取具有最高分配效率的加权矩阵,将粒子群算法应用于广义逆控制分配法设计,通过计算不同广义逆阵所能有效分配的转矩可达集体积,来选择具有最高分配效率的加权矩阵。以某飞机的控制效率矩阵进行基于广义逆分配法分配器设计,通过粒子群算法对加权矩阵进行寻优,有效地选取了具有最优分配效率的广义逆阵,显著地提高了分配器的分配效率。     

递推广义预测自校正控制器

本文给出了广义预测自校正控制器的递推算法,用递推平方根法代替最小二乘法估计参数,改善了计算精度,比牛津大学克拉克等人的算法节省了计算时间.仿真实验表明,该控制器对系统时延与阶次有很好的鲁棒性,性能优于PID调节器、广义最小方差自校正器与极点配置自校正器,有较强的抑制干扰能力.     

基于广义轮换矩阵的伪随机广义二进制轮换矩阵设计

压缩感知中,测量矩阵在信号的获取和重构过程中起着重要的作用.传统的随机测量矩阵在采样率较高的情况下,能够获得比较好的重构效果,但在低采样率下的重构效果不够理想.确定性测量矩阵自身存在一些限制因素,与随机测量矩阵相比,重构效果有所降低.基于广义轮换矩阵(GR),提出了两种结构随机矩阵:广义二进制轮换矩阵(GBR)和伪随机广义二进制轮换矩阵(PGBR).仿真结果表明,相对于传统的测量矩阵,新的测量矩阵在二维图像重建方面效果较好,所需重构时间相差不大,在较低的采样率下能够获得更加精确的重建.     

Scientific applications of iterative Toeplitz solvers

Recent research on using the preconditioned conjugate gradient method as an iterative method for solving Toeplitz systems has brought much attention. One of the main important results of this methodology is that the complexity of solving a large class of Toeplitz systems can be reduced toO (n logn) operations as compared to theO(n log² n) operations required by fast direct Toeplitz solvers, provided that a suitable preconditioner is chosen under certain conditions on the Toeplitz operator. In this paper, we survery some applications of iterative Toeplitz solvers to Toeplitz-related problems arising from scientific applications. These applications include partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis. Research supported by the Cooperative Research Centre for Advanced Computational Systems. Research supported in part by HKRGC grants no. CUHK 316/94E.     

Inverse Toeplitz eigenproblem on personal computer networks

In this paper we present a parallel algorithm for solving the inverse Toeplitz Eigenvalue Problem. The algorithm has been implemented by using a cluster of personal computers, interconnected by a high‐performance Myrinet network. We have utilized standard public domain parallel environments for implementing the calculation part as well as the communications, thus producing portable software. The results obtained allow us to confirm the scalability and efficiency of the algorithm. Moreover, we have checked that by using the theoretical cost model provided by the ScaLAPACK we can predict the behaviour of the experimental results. Copyright © 2000 John Wiley & Sons, Ltd.     

基于状态空间模型广义预测控制的并行算法

本文首先基于脉动阵列经，提出了一种实时参数辨识的并行算法，然后推导出基于状态空间模型广义预测控制的两种新算法，这两种算法都可以通过阵列结构并行实现。     

Convergence Results of the Biconjugate Residual Algorithm for Solving Generalized Sylvester Matrix Equation

In this article, we investigate a variant of the biconjugate residual (BCR) algorithm to solve the generalized Sylvester matrix equation which includes the well‐known Lyapunov, Stein and Sylvester matrix equations. We show that the BCR algorithm with any (special) initial matrix pair can smoothly compute the (least Frobenius norm) solution pair of the generalized Sylvester matrix equation within a finite number of iterations in the absence of round‐off errors. Finally the accuracy and effectiveness of the BCR algorithm in comparison to some existing algorithms are demonstrated by two numerical examples.