求实对称矩阵部分特征值的并行算法

提出了并行求解实对称稠密矩阵部分特征值的反幂法的预处理方法.该方法基于带状矩阵特征问题反幂法的信息传递复杂度低的特点,采用Householder变换并行算法约化大型实对称稠密矩阵为一定带宽的带状矩阵,针对带状矩阵用反幂法求解矩阵的在某一点的近似特征值;其中针对反幂法迭代中遇到的线性方程组,采用文献中的并行预处理共轭梯度算法求解.最后在Lenovo深腾1800集群上进行数值实验,并与预处理前反幂法的计算结果进行了比较,实验结果表明,经过预处理后的并行性远高于直接采用反幂法的并行性.     

基于LDLT分解求实对称矩阵特征值的递归算法

基于线性代数与矩阵理论,给出利用LDLT分解计算实对称矩阵特征值的递归算法。该算法可求出实对称矩阵在给定区间内的特征值的个数,并可计算满足精度要求的特征值。理论分析和实际测试证明该算法是有效的。     

The determination of the eigenvalues of large sparse symmetric matrices

A recursive algorithm for the implicit derivation of the determinant of a large sparse symmetric matrix derived from the finite difference discretisation of the Biharmonic equation is developed in terms of its leading principal minors. The algorithm is shown to yield a sequence of polynomials from which the eigenvalues can be obtained by use of a bisection process.     

Computation of eigenvalues of a real matrix

This paper presents a computational procedure for finding eigenvalues of a real matrix based on Alternate Quadrant Interlocking Factorization, a parallel direct method developed by Rao in 1994 for the solution of the general linear system Ax=b. The computational procedure is similar to LR algorithm as studied by Rutishauser in 1958 for finding eigenvalues of a general matrix. After a series of transformations the eigenvalues are obtained from simple 2×2 matrices derived from the main and cross diagonals of the limit matrix. A sufficient condition for the convergence of the computational procedure is proved. Numerical examples are given to demonstrate the method.     

Alternating sequential-parallel calculation of eigenvalues for symmetric matrices

An “alternating sequential-parallel” system (ASP) is introduced, and its advantages over other models are discussed. Algorithms of Jacobi, Givens and Householder are modified for execution on this system. Efficiencies are computed for all methods and prove that the proposed methods achieve considerable speedups.     

A note on the eigenvalues of a real matrix

Necessary and sufficient conditions are found for the eigenvalues of a real matrix to lie within a certain region of the complex plane.     

Time-optimal control of certain plants with positive real eigenvalues

This note considers the time-optimal control problem for plants with positive real eigenvalues and a single saturable control input. It is demonstrated that a previous analysis (Fuller 1973, 1974 a) of plants with negative real eigenvalues in simple ratio is also applicable, with certain restrictions, to plants with positive real eigenvalues in the same simple ratio. For a second-order plant a region of controllability in the state plane is algebraically determined. For third and higher-order plants, a region of controllability is partially defined.     

Parallel block tridiagonalization of real symmetric matrices

Two parallel block tridiagonalization algorithms and implementations for dense real symmetric matrices are presented. Block tridiagonalization is a critical pre-processing step for the block tridiagonal divide-and-conquer algorithm for computing eigensystems and is useful for many algorithms desiring the efficiencies of block structure in matrices. For an “effectively” sparse matrix, which frequently results from applications with strong locality properties, a heuristic parallel algorithm is used to transform it into a block tridiagonal matrix such that the eigenvalue errors remain bounded by some prescribed accuracy tolerance. For a dense matrix without any usable structure, orthogonal transformations are used to reduce it to block tridiagonal form using mostly level 3 BLAS operations. Numerical experiments show that block tridiagonal structure obtained from this algorithm directly affects the computational complexity of the parallel block tridiagonal divide-and-conquer eigensolver. Reduction to block tridiagonal form provides significantly lower execution times, as well as memory traffic and communication cost, over the traditional reduction to tridiagonal form for eigensystem computations.     

On real factorizations of symmetric circulant sparse matrices

For the matricesA mentioned in the headline we determine the limit points up to which there is possible a real factorization of the formA=QQ ^T . HereQ=(q _ij ) is a circulant matrix, where from the elementsq _ij andq _ji withi≠j always one element is vanishing.     

Parallel computation of the eigenvalues of symmetric Toeplitz matrices through iterative methods

This paper presents a new procedure to compute many or all of the eigenvalues and eigenvectors of symmetric Toeplitz matrices. The key to this algorithm is the use of the “Shift–and–Invert” technique applied with iterative methods, which allows the computation of the eigenvalues close to a given real number (the “shift”). Given an interval containing all the desired eigenvalues, this large interval can be divided in small intervals. Then, the “Shift–and–Invert” version of an iterative method (Lanczos method, in this paper) can be applied to each subinterval. Since the extraction of the eigenvalues of each subinterval is independent from the other subintervals, this method is highly suitable for implementation in parallel computers. This technique has been adapted to symmetric Toeplitz problems, using the symmetry exploiting Lanczos process proposed by Voss [H. Voss, A symmetry exploiting Lanczos method for symmetric Toeplitz matrices, Numerical Algorithms 25 (2000) 377–385] and using fast solvers for the Toeplitz linear systems that must be solved in each Lanczos iteration. The method compares favourably with ScaLAPACK routines, specially when not all the spectrum must be computed.     

Stable and efficient algorithm for selected eigenvalues and eigenvectors of the general symmetric eigenproblem

A stable and efficient algorithm for the general symmetric matrix eigenproblem is given which allows the calculation of selected eigenvalues and eigenvectors in large Hylleraas-CIs. The method is based on a new version of the inverse iteration. The algorithm was successfully applied to a Hylleraas-CI for the Li-atom with dimensions of the basis up to 1589.     

An efficient method for computing eigenvalues of a real normal matrix

Jacobi-based algorithms have attracted attention as they have a high degree of potential parallelism and may be more accurate than QR-based algorithms. In this paper we discuss how to design efficient Jacobi-like algorithms for eigenvalue decomposition of a real normal matrix. We introduce a block Jacobi-like method. This method uses only real arithmetic and orthogonal similarity transformations and achieves ultimate quadratic convergence. A theoretical analysis is conducted and some experimental results are presented.     

Computation of extreme eigenvalues in higher dimensions using block tensor train format

We consider approximate computation of several minimal eigenpairs of large Hermitian matrices which come from high-dimensional problems. We use the tensor train (TT) format for vectors and matrices to overcome the curse of dimensionality and make storage and computational cost feasible. We approximate several low-lying eigenvectors simultaneously in the block version of the TT format. The computation is done by the alternating minimization of the block Rayleigh quotient sequentially for all TT cores. The proposed method combines the advances of the density matrix renormalization group (DMRG) and the variational numerical renormalization group (vNRG) methods. We compare the performance of the proposed method with several versions of the DMRG codes, and show that it may be preferable for systems with large dimension and/or mode size, or when a large number of eigenstates is sought.     

Hankel norm approximation of real symmetric rational matrix functions

In this paper the recent results for Hankel norm approximation of rational matrix valued functions are studied for the special case of real symmetric rational matrix functions. It is shown how to obtain all real symmetric Hankel norm approximants of a given real symmetric function.     

Computing the fine structure of real reductive symmetric spaces

Much of the structure of Lie groups has been implemented in several computer algebra packages, including LiE, GAP4, Chevie, Magma and Maple. The structure of reductive symmetric spaces is very similar to that of the underlying Lie group and a computer algebra package for computations related to symmetric spaces would be an important tool for researchers in many areas of mathematics. Until recently only very few algorithms existed for computations in symmetric spaces due to the fact that their structure is much more complicated than that of the underlying group.