共查询到20条相似文献,搜索用时 15 毫秒
1.
G.W. Stewart 《Numerische Mathematik》1994,68(1):143-147
Summary.
This note gives a new convergence proof for iterations based on
multipoint formulas. It rests on the very general assumption that if
the desired fixed point appears as an argument in the formula then
the formula returns the fixed point.
Received March 24, 1993 / Revised version received
January 1994 相似文献
2.
Summary. Graeffe iteration was the choice algorithm for solving univariate polynomials in the XIX-th and early XX-th century. In this paper, a new variation of Graeffe iteration is given, suitable to IEEE floating-point arithmetics of modern digital computers. We prove that under a certain generic assumption the proposed algorithm converges. We also estimate the error after N iterations and the running cost. The main ideas from which this algorithm is built are: classical Graeffe iteration and Newton Diagrams, changes of scale (renormalization), and replacement of a difference technique by a differentiation one. The algorithm was implemented successfully and a number of numerical experiments are displayed. Received May 29, 1998 / Revised version received September 13, 1999 / Published online April 5, 2001 相似文献
3.
Miodrag S. Petkovi 'c Carsten Carstensen Miroslav Trajkov ' i c 《Numerische Mathematik》1995,69(3):353-372
Summary.
Classical Weierstrass' formula
[29] has been often the subject of investigation of many
authors. In this paper we give some further
applications of this formula for finding the zeros of polynomials and
analytic functions. We are concerned with the problems of
localization of polynomial zeros and the construction of iterative methods for
the simultaneous approximation and inclusion of these zeros.
Conditions for the safe convergence of Weierstrass' method,
depending only on initial approximations, are given. In particular,
we study polynomials with interval coefficients. Using an interval
version of Weierstrass' method enclosures in the form of disks
for the complex-valued set containing all zeros of a
polynomial with varying coefficients are obtained. We also present
Weierstrass-like algorithm for approximating, simultaneously, all
zeros of a class of analytic functions in a given closed region.
To demonstrate the proposed algorithms, three numerical
examples are included.
Received September 13, 1993 相似文献
4.
Summary. We construct in this paper an analogous method to Bairstow's one, for
trigonometric polynomials with real coefficients.
We also present some numerical examples which illustrate this method.
Received November 11, 1992/Revised version received May 13, 1993 相似文献
5.
Christian Kanzow 《Numerische Mathematik》1998,80(4):557-577
Summary. We consider a quadratic programming-based method for nonlinear complementarity problems which allows inexact solutions of
the quadratic subproblems. The main features of this method are that all iterates stay in the feasible set and that the method
has some strong global and local convergence properties. Numerical results for all complementarity problems from the MCPLIB
test problem collection are also reported.
Received February 24, 1997 / Revised version received September 5, 1997 相似文献
6.
Krzysztof C. Kiwiel 《Numerische Mathematik》1994,68(3):325-340
Summary.
A quadratic programming method is given for minimizing a
sum of piecewise linear functions and a proximal quadratic term,
subject to simple bounds on variables. It may be used for search
direction finding in nondifferentiable optimization algorithms. An
efficient implementation is described that updates a Cholesky
factorization of active constraints and provides good accuracy via
iterative refinement. Numerical experience is reported for some
large problems.
Received March 29, 1993 / Revised version received December 18, 1993 相似文献
7.
Summary. By providing a matrix version of Koenig's theorem we reduce the problem of evaluating the coefficients of a monic factor
r(z) of degree h of a power series f(z) to that of approximating the first h entries in the first column of the inverse of an Toeplitz matrix in block Hessenberg form for sufficiently large values of n. This matrix is reduced to a band matrix if f(z) is a polynomial. We prove that the factorization problem can be also reduced to solving a matrix equation for an matrix X, where is a matrix power series whose coefficients are Toeplitz matrices. The function is reduced to a matrix polynomial of degree 2 if f(z) is a polynomial of degreeN and . These reductions allow us to devise a suitable algorithm, based on cyclic reduction and on the concept of displacement rank,
for generating a sequence of vectors that quadratically converges to the vector having as components the coefficients of the factor r(z). In the case of a polynomial f(z) of degree N, the cost of computing the entries of given is arithmetic operations, where is the cost of solving an Toeplitz-like system. In the case of analytic functions the cost depends on the numerical degree of the power series involved
in the computation. From the numerical experiments performed with several test polynomials and power series, the algorithm
has shown good numerical properties and promises to be a good candidate for implementing polynomial root-finders based on
recursive splitting strategies. Applications to solving spectral factorization problems and Markov chains are also shown.
Received September 9, 1998 / Revised version received November 14, 1999 / Published online February 5, 2001 相似文献
8.
Qingchuan Yao 《Numerische Mathematik》1999,81(4):647-677
This paper proposes some modified Halley iterations for finding the zeros of polynomials. We investigate the non-overshoot
properties of the modified Halley iterations and other important properties that play key roles in solving symmetric eigenproblems.
We also extend Halley iteration to systems of polynomial equations in several variables.
Received March 20, 1996 / Revised version received December 5, 1997 相似文献
9.
Summary. We consider a smoothing-type method for the solution of linear programs. Its main idea is to reformulate the primal-dual
optimality conditions as a nonlinear and nonsmooth system of equations, and to apply a Newton-type method to a smooth approximation
of this nonsmooth system. The method presented here is a predictor-corrector method, and is closely related to some methods
recently proposed by Burke and Xu on the one hand, and by the authors on the other hand. However, here we state stronger global
and/or local convergence properties. Moreover, we present quite promising numerical results for the whole netlib test problem
collection.
Received August 9, 2000 / Revised version received September 28, 2000 / Published online June 7, 2001 相似文献
10.
Colm Art O'Cinneide 《Numerische Mathematik》1996,73(4):507-519
Summary.
Recently the author showed that the Grassmann-Taksar-Heyman
(GTH) algorithm computes the steady-state
distribution of a finite-state
Markov chain with low relative error.
Here it is shown that the
LU decomposition
computed in the course of the
GTH algorithm also has low relative error.
The proof requires a refinement of the methods used in
the earlier paper.
Received September 2, 1994 / Revised version received
July 17, 1995 相似文献
11.
Summary. Let be a complex polynomial of
degree with and Cauchy radius 1 about the
origin. We discuss the order of magnitude
of the minimal number such that
Previous estimates of are improved to . Some other related properties of these polynomials
are also exhibited.
Received March 3, 1993 相似文献
12.
Summary. We present a new class of integration methods for differential equations on manifolds, in the framework of Lie group actions.
Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie
algebra. The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves the inversion
of its differential, a task that can be challenging. To succeed, it is necessary to consider carefully how to choose a basis
for the Lie algebra, and the ordering of the basis is important as well. For semisimple Lie algebras, one may take advantage
of the root space decomposition to provide a basis with desirable properties. The problem of ordering leads us to introduce
the concept of an admissible ordered basis (AOB). The existence of an AOB is established for some of the most important Lie
algebras. The computational cost analysis shows that the approach may lead to more efficient solvers for ODEs on manifolds
than those based on canonical coordinates of the first kind presented by Munthe-Kaas. Numerical experiments verify the derived
properties of the new methods.
Received April 2, 1999 / Revised version received January 18, 2000 / Published online August 2, 2000 相似文献
13.
Summary. We give the asymptotic formula for the error in cardinal interpolation. We generalize the Mazur Orlicz Theorem for periodic
function.
Received February 22, 1999 / Revised version received October 15, 1999 / Published online March 20, 2001 相似文献
14.
A.M. Cohen 《Numerische Mathematik》1994,68(2):225-238
Summary.
Wilkinson, in [1], has given a comprehensive account of the numerical
difficulties of working with polynomials on a floating point computer. The
object of this note is to attempt to rehabilitate the polynomial to a certain
extent. In particular it is shown here that polynomial deflation can be
performed satisfactorily by a method akin to `backward recursion'. Error
analyses and examples are given to illustrate the stability of the
process.
Received October 4, 1993 /
Revised version received July 14, 1993 相似文献
15.
Florian Jarre 《Numerische Mathematik》1994,68(1):81-94
Summary. We define the notion of self-concordance of
order two
for the restriction
of a logarithmic barrier function to a
given
line. Based on this notion we prove an inner approximation of the domain
of , as well as a lower bound of the distance from
a point $t$ to the minimum of .
These results provide the theoretical tools to
develop a simple and efficient search step for
finding the minimum of the barrier function along a given line.
The new bound on the size of the line-search step is better than
the optimal bound known
for the case of a self-concordant function (of order one).
We conclude with some numerical examples that
illustrate the promise of the new line-search step.
Received May 24, 1993 / Revised version received February 1994 相似文献
16.
Naoki Osada 《Numerische Mathematik》1996,73(4):521-531
Summary.
The ρ-algorithm of Wynn is an excellent device for
accelerating the convergence of some logarithmically convergent
sequences.
Until now a convergence theorem and an acceleration theorem for the
ρ-algorithm have not been obtained.
The purpose of this paper is to give an acceleration theorem for the
ρ-algorithm.
Moreover, it is proved that the ρ-algorithm cannot accelerate
linear convergence.
Numerical examples are given.
Received October 20, 1994 / Revised version received July 2,
1995 相似文献
17.
J. Matejaš 《Numerische Mathematik》2000,87(1):171-199
Summary. A quadratic convergence bound for scaled Jacobi iterates is proved provided the initial symmetric positive definite matrix has simple eigenvalues. The bound is expressed in terms of the off-norm of the scaled initial matrix and the minimum relative gap in the spectrum. The obtained result can be used to predict the stopping moment in the two-sided and especially in the one-sided Jacobi method. Received October 31, 1997 / Revised version received March 8, 1999 / Published online July 12, 2000 相似文献
18.
A. Melman 《Numerische Mathematik》1995,69(4):483-493
Summary.
A method is proposed for the solution of a secular equation, arising
in modified symmetric eigenvalue problems and in several other
areas.
This equation has singularities which make the application of standard root-finding
methods difficult.
In order to solve the equation, a class of transformations of variables is
considered, which transform the equation into one for which Newton's method
converges from any point in a certain given interval. In addition, the form of the
transformed equation suggests a convergence accelerating modification of
Newton's method. The same ideas are applied to the secant
method and numerical results are presented.
Received July 1, 1994 相似文献
19.
Summary. This analysis of convergence of a coupled FEM-IEM is based on our previous work on the FEM and the IEM for exterior Helmholtz
problems. The key idea is to represent both the exact and the numerical solution by the Dirichlet-to-Neumann operators that
they induce on the coupling hypersurface in the exterior of an obstacle. The investigation of convergence can then be related
to a spectral analysis of these DtN operators. We give a general outline of our method and then proceed to a detailed investigation
of the case that the coupling surface is a sphere. Our main goal is to explore the convergence mechanism. In this context,
we show well-posedness of both the continuous and the discrete models. We further show that the discrete inf-sup constants
have a positive lower bound that does not depend on the number of DOF of the IEM. The proofs are based on lemmas on the spectra
of the continuous and the discrete DtN operators, where the spectral characterization of the discrete DtN operator is given
as a conjecture from numerical experiments. In our convergence analysis, we show algebraic (in terms of N) convergence of arbitrary order and generalize this result to exponential convergence.
Received April 10, 1999 / Revised version received November 10, 1999 / Published online October 16, 2000 相似文献
20.
Summary. It is well known that any nonsingular M–matrix admits an LU factorization into M–matrices (with L and U lower and upper triangular respectively) and any singular M–matrix is permutation similar to an M–matrix which admits an
LU factorization into M–matrices. Varga and Cai establish necessary and sufficient conditions for a singular M–matrix (without
permutation) to allow an LU factorization with L nonsingular. We generalize these results in two directions. First, we find necessary and sufficient conditions for the existence
of an LU factorization of a singular M-matrix where L and U are both permitted to be singular. Second, we establish the minimal block structure that a block LU factorization of a singular
M–matrix can have when L and U are M–matrices.
Received November 21, 1994 / Revised version received August 4, 1997 相似文献