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1.
关于矩阵奇异值分解的注记 总被引:5,自引:0,他引:5
本文首先改进"具有奇异值分解性质的代数”一文的引理1及证明,再给出其定理i的简证,最后指出"关于‘体上矩阵的广义逆'一文的注”中一段话的错误. 相似文献
2.
Summary.
In this paper we propose an algorithm based on Laguerre's iteration,
rank two divide-and-conquer technique and a hybrid strategy
for computing singular values of bidiagonal matrices.
The algorithm is fully parallel in nature and evaluates singular
values to tiny relative error if necessary. It is competitive with QR
algorithm in serial mode in speed and advantageous in computing
partial singular values. Error analysis and numerical results are
presented.
Received
March 15, 1993 / Revised version received June 7, 1994 相似文献
3.
In this paper, a singular approach to study the solutions of an impulsive differential equation from a qualitative and quantitative point of view is proposed. In the approach, a suitable singular perturbation term is introduced and a singularly perturbed system with infinite initial values is defined, in which, the reduced problem of the singularly perturbed system is exactly the impulsive differential equation under consideration. Then the boundary layer function method is applied to construct the uniformly valid asymptotic solutions to the singularly perturbed system. Based on the continuous asymptotic solution, the discontinuous solutions of the impulsive differential equation are described and approximated. An example, namely, a classical Lotka-Volterra prey-predator model with one pulse is carried out to illustrate the main results. 相似文献
4.
A class of sign‐symmetric P‐matrices including all nonsingular totally positive matrices and their inverses as well as tridiagonal nonsingular H‐matrices is presented and analyzed. These matrices present a bidiagonal decomposition that can be used to obtain algorithms to compute with high relative accuracy their singular values, eigenvalues, inverses, or their LDU factorization. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
5.
Advances in Data Analysis and Classification - Principal component regression (PCR) is a two-stage procedure: the first stage performs principal component analysis (PCA) and the second stage builds... 相似文献
6.
Updating the singular value decomposition 总被引:4,自引:0,他引:4
Summary LetA be anm×n matrix with known singular value decomposition. The computation of the singular value decomposition of a matrixà is considered, whereà is obtained by appending a row or a column toA whenmn or by deleting a row or a column fromA whenm>n. An algorithm is also presented for solving the updated least squares problemà y–b, obtained from the least squares problemAx–b by appending an equation, deleting an equation, appending an unknown, or deleting an unknown.This research was supported by NSF grants MCS 75-06510 and MCS 76-03139 相似文献
7.
8.
We consider the following problem: Given a set of m×n real (or complex) matrices A1,…,AN, find an m×m orthogonal (or unitary) matrix P and an n×n orthogonal (or unitary) matrix Q such that P*A1Q,…,P*ANQ are in a common block-diagonal form with possibly rectangular diagonal blocks. We call this the simultaneous singular value decomposition (simultaneous SVD). The name is motivated by the fact that the special case with N=1, where a single matrix is given, reduces to the ordinary SVD. With the aid of the theory of *-algebra and bimodule it is shown that a finest simultaneous SVD is uniquely determined. An algorithm is proposed for finding the finest simultaneous SVD on the basis of recent algorithms of Murota-Kanno-Kojima-Kojima and Maehara-Murota for simultaneous block-diagonalization of square matrices under orthogonal (or unitary) similarity. 相似文献
9.
In this article, a reduced optimizing finite difference scheme (FDS) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) for Burgers equation is presented. Also the error estimates between the usual finite difference solution and the POD solution of reduced optimizing FDS are analyzed. It is shown by considering the results obtained for numerical simulations of cavity flows that the error between the POD solution of reduced optimizing FDS and the solution of the usual FDS is consistent with theoretical results. Moreover, it is also shown that the reduced optimizing FDS is feasible and efficient. 相似文献
10.
An algorithm for computing the singular value decomposition of normal matrices using intermediate complex symmetric matrices is proposed. This algorithm, as most eigenvalue and singular value algorithms, consists of two steps. It is based on combining the unitarily equivalence of normal matrices to complex symmetric tridiagonal form with the symmetric singular value decomposition of complex symmetric matrices. Numerical experiments are included comparing several algorithms, with respect to speed and accuracy, for computing the symmetric singular value decomposition (also known as the Takagi factorization). Next we compare the novel approach with the classical Golub-Kahan method for computing the singular value decomposition of normal matrices: it is faster, consumes less memory, but on the other hand the results are significantly less accurate. 相似文献
11.
A differential equation approach to nonlinear programming 总被引:5,自引:0,他引:5
Hiroshi Yamashita 《Mathematical Programming》1980,18(1):155-168
A new method is presented for finding a local optimum of the equality constrained nonlinear programming problem. A nonlinear autonomous system is introduced as the base of the theory instead of usual approaches. The relation between critical points and local optima of the original optimization problem is proved. Asymptotic stability of the critical points is also proved. A numerical algorithm which is capable of finding local optima systematically at the quadratic rate of convergence is developed from a detailed analysis of the nature of trajectories and critical points. Some numerical results are given to show the efficiency of the method. 相似文献
12.
Several lower bounds have been proposed for the smallest singular value of a square matrix, such as Johnson’s bound, Brauer-type bound, Li’s bound and Ostrowski-type bound. In this paper, we focus on a bidiagonal matrix and investigate the equality conditions for these bounds. We show that the former three bounds give strict lower bounds if all the bidiagonal elements are non-zero. For the Ostrowski-type bound, we present an easily verifiable necessary and sufficient condition for the equality to hold. 相似文献
13.
For one-step difference equations, where the matrix coefficientsmay be singular, a stability analysis based on using fundamentalsolutions and their inverses does not apply. This paper showshow well-boundedness of the Green's function leads to a kindof dichotomy of the fundamental solution, including certainparasitic solutions (which arise because of thesingularity of the fundamental solutions). This then is usedto show how one can find a stable decoupling and thus a numericalalgorithm for solving a discrete boundary-value problem. Severalexamples sustain the analysis. 相似文献
14.
Masafumi Yoshino 《manuscripta mathematica》2012,137(3-4):441-455
We shall show the solvability of semilinear Fuchsian partial differential systems in a multi-sectorial domain. Our equation contains a linearizing equation of a singular vector field to its linear part when so-called small denominators occur. We will show the existence of an analytic solution in a multi-sectorial domain without assuming any Diophantine condition. 相似文献
15.
E. F. Galba 《Ukrainian Mathematical Journal》1996,48(10):1618-1622
For a rectangular real matrix, we obtain a decomposition in weighted singular numbers. On this basis, we obtain a representation of a weighted pseudoinverse matrix in terms of weighted orthogonal matrices and weighted singular numbers. 相似文献
16.
In this paper, we study an implicit version of the sweeping process. Based on methods of convex analysis, we prove the equivalence of the implicit sweeping process with a differential equation, which enables us to show the existence and uniqueness of the solution to the implicit sweeping process in a very general framework. Moreover, this equivalence allows us to give a characterization of nonsmooth Lyapunov pairs and invariance for implicit sweeping processes. The results of the paper are illustrated with two applications to quasistatic evolution variational inequalities and electrical circuits. 相似文献
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18.
This paper is concerned with a geometric study of ()‐parameter families of constrained differential systems, where . Our main results say that the dynamics of such a family close to the impasse set is equivalent to the dynamics of a multiple time scale singular perturbation problem (that is a singularly perturbed system containing several small parameters). This enables us to use a geometric theory for multiscale systems in order to describe the behaviour of such a family close to the impasse set. We think that a systematic program towards a combination between geometric singular perturbation theory and constrained systems and problems involving persistence of typical minimal sets are currently emergent. Some illustrations and applications of the main results are provided. 相似文献
19.
Stefan Kindermann Andreas Neubauer Ronny Ramlau 《Journal of Computational and Applied Mathematics》2012,236(8):2186-2199
We study the problem of reconstructing a wavefront from measurements of Shack-Hartmann-type sensors. Mathematically, this leads to the problem of reconstructing a function from a discrete set of averages of the gradient.After choosing appropriate function spaces this is an underdetermined problem for which least squares solutions and generalized inverses can be used. We explore this problem in more detail for the case of periodic functions on a quadratic aperture, where we calculate the singular value decomposition of the associated forward operator. The nonzero singular values can be estimated which shows that asymptotically, with increasing number of measurements, the reconstruction problem becomes an ill-posed problem. 相似文献
20.
In this paper, we introduce and analyze a new singular value decomposition (SVD) called weighted SVD (WSVD) using a new inner product instead of the Euclidean one. We use the WSVD to approximate the singular values and the singular functions of the Fredholm integral operators. In this case, the new inner product arises from the numerical integration used to discretize the operator. Then, the truncated WSVD (TWSVD) is used to regularize the Nyström discretization of the first‐kind Fredholm integral equations. Also, we consider the weighted LSQR (WLSQR) to approximate the solution obtained by the TWSVD method for large problems. Numerical experiments on a few problems are used to illustrate that the TWSVD can perform better than the TSVD. 相似文献