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1.
Let W
n be an n × n random symmetric sparse matrix with independent identically distributed entries such that the values 1 and 0 are taken with probabilities p/n and 1-p/n, respectively; here
is independent of n. We show that the limit of the expected spectral distribution functions of W
n has a discrete part. Moreover, the set of positive probability points is dense in (- +). In particular, the points
, and 0 belong to this set. 相似文献
2.
Two recent methods (Shanno, 1978; Toint, 1980) for revisingestimates of sparse second derivative matrices in quasi-Newtonoptimization algorithms reduce to variable metric formulae whenthere are no sparsity conditions. It is proved that these methodsare equivalent. Further, some examples are given to show thatthe procedure may make the second derivative approximationsworse when the objective function is quadratic. Therefore theconvergence properties of the procedure are sometimes less goodthan the convergence properties of other published methods forrevising sparse second derivative approximations. 相似文献
3.
Sanja Singer 《BIT Numerical Mathematics》2006,46(1):141-161
Let A be a Hermitian positive definite matrix given by its rectangular factor G such that A=G*G. It is well known that the Cholesky factorization of A is equivalent to the QR factorization of G. In this paper, an analogue of the QR factorization for Hermitian indefinite matrices is constructed. This problem has been
considered by many authors, but the problem of zero diagonal elements has not been solved so far. Here we show how to overcome
this difficulty.
AMS subject classification (2000) 65F25, 46C20, 65F15 相似文献
4.
In this paper, we study the use of an incomplete Cholesky factorization (ICF) as a preconditioner for solving dense symmetric positive definite linear systems. This method is suitable for situations where matrices cannot be explicitly stored but each column can be easily computed. Analysis and implementation of this preconditioner are discussed. We test the proposed ICF on randomly generated systems and large matrices from two practical applications: semidefinite programming and support vector machines. Numerical comparison with the diagonal preconditioner is also presented. 相似文献
5.
In this paper, we propose a new factorization method for block tridiagonal symmetric indefinite matrices. We also discuss
the stability of the factorization method. As a measurement of stability, an effective condition number is derived by using
backward error analysis and perturbation analysis. It shows that under some suitable assumptions, the solution obtained by
this factorization method is acceptable. Numerical results demonstrate that the factorization is stable if its condition number
is not too large.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
6.
DUFF I. S.; REID J. K.; MUNKSGAARD N.; NIELSEN H. B. 《IMA Journal of Applied Mathematics》1979,23(2):235-250
We consider the use of 1?1 and 2?2 pivots for direct solutionof sets of linear equations whose matrix is sparse and symmetric.Inclusion of 2?2 pivots permits a stable decomposition to beobtained in the indefinite case and we demonstrate that in practicethere is little loss of speed even in positive definite cases.A pivotal strategy suitable for the sparse case is proposedand compared experimentally with alternatives. We present ananalysis of error, explain how the stability may be monitoredcheaply, discuss automatic scaling and consider implementationdetails. 相似文献
7.
8.
Pascal矩阵的一种显式分解 总被引:2,自引:0,他引:2
本文引入了两种广义Pascal矩阵凡,Pn,k,Qn,k以及两种广义Pascal函数矩阵On,k[x,y],Qn,k[x,y],证明了Pascal矩阵能够表示成(0,1)-Jordan矩阵的乘积而且Pascal函数矩阵能分解成双对角矩阵的乘积. 相似文献
9.
Tu Boxun 《数学年刊B辑(英文版)》1982,3(2):249-259
Let \Omega be a field, and let F denote the Frobenius matrix:
$[F = \left( {\begin{array}{*{20}{c}}
0&{ - {\alpha _n}}\{{E_{n - 1}}}&\alpha
\end{array}} \right)\]$
where \alpha is an n-1 dimentional vector over Q, and E_n- 1 is identity matrix over \Omega.
Theorem 1. There hold two elementary decompositions of Frobenius matrix:
(i) F=SJB,
where S, J are two symmetric matrices, and B is an involutory matrix;
(ii) F=CQD,
where O is an involutory matrix, Q is an orthogonal matrix over \Omega, and D is a
diagonal matrix.
We use the decomposition (i) to deduce the following two theorems:
Theorem 2. Every square matrix over \Omega is a product of twe symmetric matrices
and one involutory matrix.
Theorem 3. Every square matrix over \Omega is a product of not more than four
symmetric matrices.
By using the decomposition (ii), we easily verify the following
Theorem 4(Wonenburger-Djokovic') . The necessary and sufficient condition
that a square matrix A may be decomposed as a product of two involutory matrices is
that A is nonsingular and similar to its inverse A^-1 over Q (See [2, 3]).
We also use the decomosition (ii) to obtain
Theorem 5. Every unimodular matrix is similar to the matrix CQB, where
C, B are two involutory matrices, and Q is an orthogonal matrix over Q.
As a consequence of Theorem 5. we deduce immediately the following
Theorem 6 (Gustafson-Halmos-Radjavi). Every unimodular matrix may be
decomposed as a product of not more than four involutory matrices (See [1] ).
Finally, we use the decomposition (ii) to derive the following
Thoerem 7. If the unimodular matrix A possesses one invariant factor which
is not constant polynomial, or the determinant of the unimodular matrix A is I and
A possesses two invariant factors with the same degree (>0), then A may be
decomposed as a product of three involutory matrices.
All of the proofs of the above theorems are constructive. 相似文献
10.
It is demonstrated that an eigenvalue of a symmetric tridiagonalmatrix as calculated by the Sturm sequence bisection procedure,is determined in general by a leading principal partition ofthe whole matrix. The order of this leading principal partitionis called the significant order of the matrix. The significantorder depends upon the specific eigenvalue being calculated,and upon the working precision of the computer. This numericalphenomenon may be employed to accelerate the bisection procedure.In applications it either indicates the adequacy or otherwiseof the working precision for finite matrices, or in the caseof theoretically infinite matrices, the effective truncationorder for the precision employed. 相似文献
11.
设P为一给定的对称正交矩阵, 记SAR\+n\-P={A∈R\+\{n×n\}|A\+T=A,(PA)\+T=-PA}. 该文考虑下列问题问题Ⅰ〓给定X∈R\+\{n×m, Λ=diag(λ\-1,λ\-2,…, λ\-m)∈R\+\{m×m\}, 求A∈SAR\+n\-P使AX=XΛ,问题Ⅱ〓给定X,B∈R\+\{n×m, 求A∈SAR\+n\-P使
‖AX-B‖=min.问题Ⅲ设[AKA~]∈R\+\{n×n\},求A\+*∈S\-E使 ‖[AKA~]-A\+*‖=inf[DD(X]A∈S\-E[DD)]‖[AKA~]-A‖, 其中S\-E为问题Ⅱ的解集合, ‖·‖表示Frobenius范数.该文得到了问题Ⅰ有解的充要条件及解集合的表达式, 给出了解集合S\-E的通式和逼近解A\+*的具体表达式. 相似文献
12.
一个n阶本原矩阵A的k-点指数是A的最小幂指数,使得在这个幂中,存在着k个全1行.最近我们得到了n阶双对称本原矩阵的k-点指数的上确界.本文将在此基础上,以伴随图的形式给出其极矩阵的完全刻划. 相似文献
13.
EDWARDS J. T.; LlCCIARDELLO D. C.; THOULESS D. J. 《IMA Journal of Applied Mathematics》1979,23(3):277-283
A way of using the Lanczos method to find all the eigenvaluesof a large sparse symmetric matrix is described, and some empiricalobservations on the manner in which the method works in practiceare given. The method seems to be accurate, fast, and not verydemanding on storage. A formula for the number of iterationsnecessary to get the eigenvalues is proposed. 相似文献
14.
15.
在本文中,我们证明了对一个反Krylov矩阵作QR分解后,利用得到的正交矩阵可以将一个具有互异特征值的对称矩阵转化为一个半可分矩阵的形式,这个结果表明了反Krylov矩阵与半可分矩阵之间的联系.另外,我们还证明了这类对称半可分矩阵在QR达代下矩阵结构保持不变性. 相似文献
16.
A weighing matrix of order n and weight m2 is a square matrix M of order n with entries from {-1,0,+1} such that MMT=m2I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.MT=M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2pr and weight p2 where p is a prime and p≥ 5.Communicated by: K.T. Arasu 相似文献
17.
18.
19.
This paper first establishes a distance inequality of the associated diagraph of a central symmetric primitive matrix, then characters the exponent set of central symmetric primitive matrices, and proves that the exponent set of central symmetric primitive matrices of order n is {1, 2,… ,n-1}. There is no gap in it. 相似文献
20.
We consider the extremal problem to determine the maximal number
of columns of a 0-1 matrix with
rows and at most
ones in each column such that each
columns are linearly independent modulo
. For fixed integers
and
, we shall prove the probabilistic lower bound
=
; for
a power of
, we prove the upper bound
which matches the lower bound for infinitely many values of
. We give some explicit constructions. 相似文献