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1.
A symmetrizer of the matrix A is a symmetric solution X that satisfies the matrix equation XA=AX. An exact matrix symmetrizer is computed by obtaining a general algorithm and superimposing a modified multiple modulus residue arithmetic on this algorithm. A procedure based on computing a symmetrizer to obtain a symmetric matrix, called here an equivalent symmetric matrix, whose eigenvalues are the same as those of a given real nonsymmetric matrix is presented.Supported by CSIR. 相似文献
2.
《Journal of Computational and Applied Mathematics》1988,21(1):27-40
A symmetric solution X satisfying the matrix equation XA = AtX is called a symmetrizer of the matrix A. A general algorithm to compute a matrix symmetrizer is obtained. A new multiple-modulus residue arithmetic called floating-point modular arithmetic is described and implemented on the algorithm to compute an error-free matrix symmetrizer. 相似文献
3.
A two-way chasing algorithm to reduce a diagonal plus a symmetric semi-separable matrix to a symmetric tridiagonal one and an algorithm to reduce a diagonal plus an unsymmetric semi-separable matrix to a bidiagonal one are considered. Both algorithms are fast and stable, requiring a computational cost of N
2, where N is the order of the considered matrix. 相似文献
4.
A. M. Abdel‐Rehim Andreas Stathopoulos Kostas Orginos 《Numerical Linear Algebra with Applications》2014,21(4):473-493
The technique that was used to build the eigCG algorithm for sparse symmetric linear systems is extended to the nonsymmetric case using the BiCG algorithm. We show that, similar to the symmetric case, we can build an algorithm that is capable of computing a few smallest magnitude eigenvalues and their corresponding left and right eigenvectors of a nonsymmetric matrix using only a small window of the BiCG residuals while simultaneously solving a linear system with that matrix. For a system with multiple right‐hand sides, we give an algorithm that computes incrementally more eigenvalues while solving the first few systems and then uses the computed eigenvectors to deflate BiCGStab for the remaining systems. Our experiments on various test problems, including Lattice QCD, show the remarkable ability of eigBiCG to compute spectral approximations with accuracy comparable with that of the unrestarted, nonsymmetric Lanczos. Furthermore, our incremental eigBiCG followed by appropriately restarted and deflated BiCGStab provides a competitive method for systems with multiple right‐hand sides. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
5.
An n×n real matrix P is said to be a symmetric orthogonal matrix if P = P?1 = PT. An n × n real matrix Y is called a generalized centro‐symmetric with respect to P, if Y = PYP. It is obvious that every matrix is also a generalized centro‐symmetric matrix with respect to I. In this work by extending the conjugate gradient approach, two iterative methods are proposed for solving the linear matrix equation and the minimum Frobenius norm residual problem over the generalized centro‐symmetric Y, respectively. By the first (second) algorithm for any initial generalized centro‐symmetric matrix, a generalized centro‐symmetric solution (least squares generalized centro‐symmetric solution) can be obtained within a finite number of iterations in the absence of round‐off errors, and the least Frobenius norm generalized centro‐symmetric solution (the minimal Frobenius norm least squares generalized centro‐symmetric solution) can be derived by choosing a special kind of initial generalized centro‐symmetric matrices. We also obtain the optimal approximation generalized centro‐symmetric solution to a given generalized centro‐symmetric matrix Y0 in the solution set of the matrix equation (minimum Frobenius norm residual problem). Finally, some numerical examples are presented to support the theoretical results of this paper. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
6.
Raf Vandebril Marc Van Barel Nicola Mastronardi 《Numerical Linear Algebra with Applications》2005,12(7):625-658
The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n × n matrix, this algorithm requires O(n3) operations per iteration step. To reduce this complexity for a symmetric matrix to O(n), the original matrix is first reduced to tridiagonal form using orthogonal similarity transformations. In the report (Report TW360, May 2003) a reduction from a symmetric matrix into a similar semiseparable one is described. In this paper a QR algorithm to compute the eigenvalues of semiseparable matrices is designed where each iteration step requires O(n) operations. Hence, combined with the reduction to semiseparable form, the eigenvalues of symmetric matrices can be computed via intermediate semiseparable matrices, instead of tridiagonal ones. The eigenvectors of the intermediate semiseparable matrix will be computed by applying inverse iteration to this matrix. This will be achieved by using an O(n) system solver, for semiseparable matrices. A combination of the previous steps leads to an algorithm for computing the eigenvalue decompositions of semiseparable matrices. Combined with the reduction of a symmetric matrix towards semiseparable form, this algorithm can also be used to calculate the eigenvalue decomposition of symmetric matrices. The presented algorithm has the same order of complexity as the tridiagonal approach, but has larger lower order terms. Numerical experiments illustrate the complexity and the numerical accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
7.
Charles J. Colbourn Sosina S. Martirosyan Tran Van Trung Robert A. Walker II 《Designs, Codes and Cryptography》2006,41(1):33-57
A covering array CA(N;t,k,v) is an N × k array such that every N × t sub-array contains all t-tuples from v symbols at least once, where t is the strength of the array. Covering arrays are used to generate software test suites to cover all t-sets of component interactions. Recursive constructions for covering arrays of strengths 3 and 4 are developed, generalizing
many “Roux-type” constructions. A numerical comparison with current construction techniques is given through existence tables
for covering arrays.
相似文献
8.
9.
The rank-one modification algorithm of theLDM
t factorization was given by Bennett [1]. His method, however, could break down even when the matrix is nonsingular and well-conditioned. We introduce a pivoting strategy for avoiding possible break-down as well as for suppressing error growth in the modification process. The method is based on a symbolic formula of the rank-one modification of the factorization of a possibly singular nonsymmetric matrix. A new symbolic formula is also obtained for the inverses of the factor matrices. Repeated application of our method produces theLDM
t-like product form factorization of a matrix. A numerical example is given to illustrate our pivoting method. An incomplete factorization algorithm is also introduced for updating positive definite matrix useful in quasi-Newton methods, in which the Fletcher and Powell algorithm [2] and the Gill, Murray and Saunders algorithm [4] are usually used.This paper is presented at the Japan SIAM Annual Meeting held at University of Tokyo, Japan, October 7–9, 1991. 相似文献
10.
A graph G is minimal harmoniously colorable if it has a proper vertex coloring in which each pair of colors occurs exactly once on an edge. In particular, if D is a 2-design we consider the graph G whose vertices are the points and blocks of D and where two vertices of G are adjacent if and only if the corresponding elements of D are incident. It will be shown that if D is symmetric then G is minimal harmoniously colorable if and only if D is a Hadamard design with corresponding Hadamard matrix of a certain form. We obtain some results if D is nonsymmetric, and construct two classes of nonsymmetric minimal harmoniously colorable designs. © 1994 John Wiley & Sons, Inc. 相似文献
11.
Some examples of orthogonal matrix polynomials satisfying odd order differential equations 总被引:2,自引:1,他引:1
It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form
W(t)=tαe-teAttBtB*eA*t,