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* Present address: School of Computer Science, McGill University, Montreal 101, Quebec, Canada. The Lanczos process is theoretically ideal for finding severalextreme eigenvalues of large sparse symmetric matrices, butbecause of loss of orthogonality in practice it has largelybeen ignored, or used with re-orthogonalization. Here it isshown that it can still be an extremely efficient and accuratealgorithm if used in an iterative manner. An error analysisis given showing the most accurate algorithms, and the conclusionsare supported by computational results. 相似文献
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矩阵方程AXB+CYD=E对称最小范数最小二乘解的极小残差法 总被引:1,自引:0,他引:1
<正>1引言本文用R~(n×m)表示全体n×m实矩阵集合,用SR~(n×n)表示全体n×n实对称矩阵集合,OR~(n×n)表示全体n×n实正交矩阵集合.用I_n表示n阶单位矩阵,用A*B表示矩阵A与B的Hadamard乘积.对任意矩阵A,B∈R~(n×m),定义内积〈A,B〉=tr(B~T A),其中 相似文献
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实对称矩阵的特征值问题,无论是低阶稠密矩阵的全部特征值问题,或高阶稀疏矩阵的部分特征值问题,都已有许多有效的计算方法,迄今最重要的一些成果已总结在[5]中。本文利用规范矩阵的一些重要性质将对于Hermite矩阵(特别是对弥矩阵)特征值问题的一些有效算法推广到规范矩阵的特征值问题,由于对复规范阵的推广是简单的,而且实际上常遇到的是实矩阵(这时常要求只用实运算),因此我们着重讨论实规范矩阵的特征值问题。 相似文献
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Gene H. Golub Ahmed H. Sameh Vivek Sarin 《Numerical Linear Algebra with Applications》2001,8(5):297-316
A parallel algorithm is proposed for the solution of narrow banded non‐symmetric linear systems. The linear system is partitioned into blocks of rows with a small number of unknowns common to multiple blocks. Our technique yields a reduced system defined only on these common unknowns which can then be solved by a direct or iterative method. A projection based extension to this approach is also proposed for computing the reduced system implicitly, which gives rise to an inner–outer iteration method. In addition, the product of a vector with the reduced system matrix can be computed efficiently on a multiprocessor by concurrent projections onto subspaces of block rows. Scalable implementations of the algorithm can be devized for hierarchical parallel architectures by exploiting the two‐level parallelism inherent in the method. Our experiments indicate that the proposed algorithm is a robust and competitive alternative to existing methods, particularly for difficult problems with strong indefinite symmetric part. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
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正交平衡区组设计(或广义正交表)的数据分析类似于正交拉丁方(或正交表)的数据分析.利用类似于正交表数据分析中的投影矩阵的正交分解技术,研究正交平衡区组设计的统计分析模型,给出了方差分析中的二次型以及各因子的二次型的分布性质,从而给出正交平衡区组设计统计模型中的方差分析方法. 相似文献
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The convex feasibility problem asks to find a point in the intersection of finitely many closed convex sets in Euclidean space. This problem is of fundamental importance in the mathematical and physical sciences, and it can be solved algorithmically by the classical method of cyclic projections.In this paper, the case where one of the constraints is an obtuse cone is considered. Because the nonnegative orthant as well as the set of positive-semidefinite symmetric matrices form obtuse cones, we cover a large and substantial class of feasibility problems. Motivated by numerical experiments, the method of reflection-projection is proposed: it modifies the method of cyclic projections in that it replaces the projection onto the obtuse cone by the corresponding reflection.This new method is not covered by the standard frameworks of projection algorithms because of the reflection. The main result states that the method does converge to a solution whenever the underlying convex feasibility problem is consistent. As prototypical applications, we discuss in detail the implementation of two-set feasibility problems aiming to find a nonnegative [resp. positive semidefinite] solution to linear constraints in n [resp. in
, the space of symmetric n×n matrices] and we report on numerical experiments. The behavior of the method for two inconsistent constraints is analyzed as well. 相似文献
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Yongxin Yuan 《Applied mathematics and computation》2011,218(7):3753-3762
An efficient iterative method for updating the mass, gyroscopic and stiffness matrices simultaneously using a few of complex measured modal data is developed. By using the proposed iterative method, the unique symmetric solution can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrices. Numerical results show that the presented method can be used to update finite element models to get better agreement between analytical and experimental modal parameters. 相似文献
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An iterative method is proposed to solve generalized coupled Sylvester matrix equations, based on a matrix form of the least-squares QR-factorization (LSQR) algorithm. By this iterative method on the selection of special initial matrices, we can obtain the minimum Frobenius norm solutions or the minimum Frobenius norm least-squares solutions over some constrained matrices, such as symmetric, generalized bisymmetric and (R, S)-symmetric matrices. Meanwhile, the optimal approximate solutions to the given matrices can be derived by solving the corresponding new generalized coupled Sylvester matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the present method. 相似文献
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We present an algorithm for the approximation of the dominant singular values and corresponding right and left singular vectors of a complex symmetric matrix. The method is based on two short-term recurrences first proposed by Saunders, Simon and Yip [24] for a non-Hermitian linear system solver. With symmetric matrices, the recurrence can be modified so as to generate a tridiagonal symmetric matrix from which the original triplets can be approximated. The recurrence formally resembles the Lanczos method, in spite of substantial differences which make usual convergence results inapplicable. Implementation aspects are discussed, such as re-orthogonalization and the use of alternative representation matrices. The method is very efficient over existing approaches which do not exploit the symmetry of the problem. Numerical experiments on application problems validate the analysis, while showing satisfactory results, especially on dense matrices. © 1997 by John Wiley & Sons, Ltd. 相似文献
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Sergey I. Solov’ëv 《Linear algebra and its applications》2006,415(1):210-229
This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to nonlinear eigenvalue problems with very large sparse ill-conditioned matrices monotonically depending on the spectral parameter. To compute the smallest eigenvalue of large-scale matrix nonlinear eigenvalue problems, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors, and inner products of vectors. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem. 相似文献
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A non-interior point algorithm based on projection for second-order cone programming problems is proposed and analyzed. The
main idea of the algorithm is that we cast the complementary equation in the primal-dual optimality conditions as a projection
equation. By using this reformulation, we only need to solve a system of linear equations with the same coefficient matrix
and compute two simple projections at each iteration, without performing any line search. This algorithm can start from an
arbitrary point, and does not require the row vectors of A to be linearly independent. We prove that our algorithm is globally convergent under weak conditions. Preliminary numerical
results demonstrate the effectiveness of our algorithm. 相似文献
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The preconditioned iterative solvers for solving Sylvester tensor equations are considered in this paper.By fully exploiting the structure of the tensor equation,we propose a projection method based on the tensor format,which needs less flops and storage than the standard projection method.The structure of the coefficient matrices of the tensor equation is used to design the nearest Kronecker product(NKP) preconditioner,which is easy to construct and is able to accelerate the convergence of the iterative solver.Numerical experiments are presented to show good performance of the approaches. 相似文献
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An inverse problem for the determination of an unknown spacewise-dependent coefficient in a parabolic equation is considered. The problem is reformulated as a nonclassical parabolic equation along with the initial and boundary conditions. The iterative fixed point projection method is applied to solve the reformulated problem. The comparison analysis of proposed method with a least square method and some numerical examples are presented. 相似文献
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We present an error analysis of the symmetric Lanczos algorithm in finite precision arithmetic. The loss of orthogonality among the computed Lanczos vectors is explained with the help of a recurrence formula. A backward error analysis shows that semiorthogonality among the Lanczos vectors is enough to guarantee the accuracy of the computed quantities up to machine precision. The results of this analysis are then extended to the more general case of the Lanczos algorithm with a semiorthogonalization strategy. Based on the recurrence formula, a new reorthogonalization method called partial reorthogonalization is introduced. We show that both partial reorthogonalization and selective orthogonalization as introduced by Parlett and Scott [15] are semiorthogonalization strategies. Finally we discuss the application of our results to the solution of linear systems of equations and to the eigenvalue problem. 相似文献
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在共轭梯度思想的启发下,结合线性投影算子,给出迭代算法求解了线性矩阵方程AXB+CYD=E的M对称解[X,Y]及其最佳逼近.当矩阵方程AXB+CYD=E有M对称解时,应用迭代算法,在有限的误差范围内,对任意初始M对称矩阵对[X_,Y_1],经过有限步迭代可得到矩阵方程的M对称解;选取合适的初始迭代矩阵,还可得到极小范数M对称解.而且,对任意给定的矩阵对[X,Y],矩阵方程AXB+CYD=E的最佳逼近可以通过迭代求解新的矩阵方程AXB+CYD=E的极小范数M对称解得到.文中的数值例子证实了该算法的有效性. 相似文献
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An iterative scheme for variational inequalities 总被引:1,自引:0,他引:1
Stella Dafermos 《Mathematical Programming》1983,26(1):40-47
In this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational
inequalities. This iterative scheme not only contains, as special cases the projection, linear approximation and relaxation
methods but also induces new algorithms. Then, we show that under appropriate assumptions the proposed iterative scheme converges
by establishing contraction estimates involving a sequence of norms in En induced by symmetric positive definite matrices Gm. Thus, in contrast to the above mentioned methods, this technique allows the possibility of adjusting the norm at each step
of the algorithm. This flexibility will generally yield convergence under weaker assumptions. 相似文献