共查询到20条相似文献,搜索用时 15 毫秒
1.
H. Woźniakowski 《Numerische Mathematik》1978,30(3):301-314
Summary We deal with the rounding error analysis of successive approximation iterations for the solution of large linear systemsA
x
=b. We prove that Jacobi, Richardson, Gauss-Seidel and SOR iterations arenumerically stable wheneverA=A
*>0 andA has PropertyA. This means that the computed resultx
k
approximates the exact solution with relative error of order A·A
–1 where is the relative computer precision. However with the exception of Gauss-Seidel iteration the residual vector Ax
k
–b is of order A2 A
–1 and hence the remaining three iterations arenot well-behaved.This work was partly done during the author's visit at Carnegie-Mellon University and it was supported in part by the Office of Naval Research under Contract N00014-76-C-0370; NR 044-422 and by the National Science Foundation under Grant MCS75-222-55 相似文献
2.
H. Woźniakowski 《Numerische Mathematik》1977,28(2):191-209
Summary This paper contains the rounding error analysis for the Chebyshev method for the solution of large linear systemsAx+g=0 whereA=A
* is positive definite. We prove that the Chebyshev method in floating point arithmetic is numerically stable, which means that the computed sequence {x
k} approximates the solution such that
x
k
– is of order AA
–1 where is the relative computer precision.We also point out that in general the Chebyshev method is not well-behaved, which means that the computed residualsr
k=Ax
k+g are of order A2A
–1.This work was supported in part by the Office of Naval Research under Contract N0014-67-0314-0010, NR 044-422 and by the National Science Foundation under Grant GJ32111 相似文献
3.
Guanggui Ding 《数学学报(英文版)》1995,11(4):417-421
IfT is an isomorphism ofL
(A, ) intoL
(B, ) which satisfies the condition T T
–11+, where (A, ) is a -finite measure space, thenT/T is close to an isometry with an error less than 4. 相似文献
4.
John Dye 《Integral Equations and Operator Theory》1989,12(1):12-22
Let {T1, ..., TN} be a finite set of linear contraction mappings of a Hilbert space H into itself, and let r be a mapping from the natural numbers N to {1, ..., N}. One can form Sn=Tr(n)...Tr(1) which could be described as a random product of the Ti's. Roughly, the Sn converge strongly in the mean, but additional side conditions are necessary to ensure uniform, strong or weak convergence. We examine contractions with three such conditions. (W): xn1, Txn1 implies (I-T)xn0 weakly, (S): xn1, Txn1 implies (I-T)xn0 strongly, and (K): there exists a constant K>0 such that for all x, (I-T)x2K(x2–Tx2).We have three main results in the event that the Ti's are compact contractions. First, if r assumes each value infinitely often, then Sn converges uniformly to the projection Q on the subspace i=
1
N
[x|Tix=x]. Secondly we prove that for such compact contractions, the three conditions (W), (S), and (K) are equivalent. Finally if S=S(T1, ..., TN) denotes the algebraic semigroup generated by the Ti's, then there exists a fixed positive constant K such that each element in S satisfies (K) with that K. 相似文献
5.
Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r
q
, whereq depends on the properties of the norm. We specify it in the case ofL
spaces, >1. 相似文献
6.
Frank Zorzitto 《Aequationes Mathematicae》1985,29(1):172-174
A discrete norm on an Abelian groupA is a non-negative function · A which satisfies the triangle inequality, is homogenous with respect to scaling ofA by and is bounded away from 0 onA/{0}.A countable Abelian group is discretely normed if and only if the group is free. 相似文献
7.
M. I. Khazan 《Journal of Mathematical Sciences》1986,35(1):2282-2292
One obtains estimates of the form, whereu. are generalized solutions of the equationsdu/dt=Au, du/dt=Bu whileA, B are non-linear,m-dissipative operators in a Banach space, and there exists an operatorP:D(A)D(B), such thatPw · W+BPw –Aw, uniformly on some setw. These results are applied to the investigation of the dependence of the solutions of the Cauchy, Dirichlet problems and of the problem with the boundary condition –du/dn=(u) for the equation u1=(u) on the continuous nondecreasing functions and.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 183–198, 1984.The author is sincerely grateful to O. A. Ladyzhenskaya and N. N. Ural'tseva for their interest in this paper and for useful discussions. 相似文献
8.
N. M. Ivochkina 《Journal of Mathematical Sciences》1978,9(5):681-696
We consider the integral We solve the problem of determination of necessary and sufficient conditions in order that (u) be independent of the values of u(x) inside a bounded domain . These conditions are written in the form of a set of differential equations for the functions f(x,u,¯p,Tij) on the set m{x; u+¯p+ Tij<}. For such functions (u) is represented in the form of a boundary integral.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 52, pp. 35–51, 1975. 相似文献
9.
B. P. Duggal 《Monatshefte für Mathematik》1991,112(4):265-270
LetB (H) denote the algebra of operators on the separable Hilbert spaceH. LetC
2 denote the (Hilbert) space of Hilbert-Schmidt operators onH, with norm .2 defined by S
2
2
=(S,S)=tr(SS
*). GivenA, B B (H), define the derivationC (A, B):B(H)B(H) byC(A, B)X=AX-XB. We show that C(A,B)X+S
2
2
=C(A,B)X
2
2
+S
2
2
holds for allXB(H) and for everySC
2 such thatC(A, B)S=0 if and only if
reducesA, ker
S reducesB, andA |
S and B| ker
S are unitarily equivalent normal operators. We also show that ifA, BB(H) are contractions andR(A, B)B(H)B(H) is defined byR(A, B)X=AXB-X, thenSC
2 andR(A, B)S=0 imply R(A,B)X+S
2
2
=R(A,B)X
2
2
+S
2
2
for allXB(H). 相似文献
10.
Izu Vaisman 《Annals of Global Analysis and Geometry》1990,8(2):137-145
Summary Let (M, J, g) be a compact complex 2-dimensional Hermitian manifold with the Kähler form , and the torsion 1-form defined by d = . In this note we obtain the Euler-Lagrange equations for the variational functionals defined by 2 and d2, whereg runs in the space of all the Hermitian metrics onM. In the first case, the extremals are precisely the Kähler metrics [Gd]. In the second case, we also write down a formula for the second variation.Communicated by J. Szenthe 相似文献
11.
James Weldon Demmel 《Numerische Mathematik》1987,51(3):251-289
Summary The condition number of a problem measures the sensitivity of the answer to small changes in the input. We call the problem ill-posed if its condition number is infinite. It turns out that for many problems of numerical analysis, there is a simple relationship between the condition number of a problem and the shortest distance from that problem to an ill-posed one: the shortest distance is proportional to the reciprocal of the condition number (or bounded by the reciprocal of the condition number). This is true for matrix inversion, computing eigenvalues and eigenvectors, finding zeros of polynomials, and pole assignment in linear control systems. In this paper we explain this phenomenon by showing that in all these cases, the condition number satisfies one or both of the diffrential inequalitiesm·2DM·2, where D is the norm of the gradient of . The lower bound on D leads to an upper bound 1/m(x) on the distance. fromx to the nearest ill-posed problem, and the upper bound on D leads to a lower bound 1/(M(X)) on the distance. The attraction of this approach is that it uses local information (the gradient of a condition number) to answer a global question: how far away is the nearest ill-posed problem? The above differential inequalities also have a simple interpretation: they imply that computing the condition number of a problem is approximately as hard as computing the solution of the problem itself. In addition to deriving many of the best known bounds for matrix inversion, eigendecompositions and polynomial zero finding, we derive new bounds on the distance to the nearest polynomial with multiple zeros and a new perturbation result on pole assignment. 相似文献
12.
B. T. Polyak 《Set-Valued Analysis》2001,9(1-2):159-168
Let f: XY be a nonlinear differentiable map, X,Y are Hilbert spaces, B(a,r) is a ball in X with a center a and radius r. Suppose f
(x) is Lipschitz in B(a,r) with Lipschitz constant L and f
(a) is a surjection: f
(a)X=Y; this implies the existence of >0 such that f
(a)*
yy, yY. Then, if r,/(2L), the image F=f(B(a,)) of the ball B(a,) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint x–a. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for small power control is convex. This leads to various results in optimal control. 相似文献
13.
R. W. Owens 《Numerische Mathematik》1977,29(1):83-91
Summary In this paper, overdetermined systems ofm linear equations inn unknowns are considered. With
m
equipped with a smooth strictly convex norm, ·, an iterative algorithm for finding the best approximate solution of the linear system which minimizes the ·-error is given. The convergence of the algorithm is established and numerical results are presented for the case when · is anl
p norm, 1<p<.Portions of this paper are taken from the author's Ph.D. thesis at Michigan State University 相似文献
14.
H. Kita 《Acta Mathematica Hungarica》1998,81(3):175-193
Let and be positive increasing convex functions defined on [0, ). Suppose satisfies the 2-condition, that is, (t)2 (C1t) for sufficiently large t, and has some nice properties. If -1(u)log(u+1) C2-1(u) for sufficiently large uthen we have S*(f) L CfL for all f L ([-, ])where S*(f) is the majorant function of partial sums of trigonometric Fourier series and fL is the Orlicz norm of f. This result is sharp. 相似文献
15.
Ariyadasa Aluthge 《Integral Equations and Operator Theory》1990,13(3):307-315
The distance formula Tt-I)–1=[Dist(, (T)]–1, (T), for hyponormal operators, is generalized top-hyponormal operators for 0<p<1. Several other results involving eigenspaces ofU and |T|, the joint point spectrum, and the spectral radius are also otained, where |T|=(T
*
T)1/2 andU is the unitary operator in the polar decomposition of thep-hyponormal operatorT=U|T|. 相似文献
16.
Paweŀ Hitczenko 《Probability Theory and Related Fields》1990,86(2):225-238
Summary Let (f
n
) be a martingale. We establish a relationship between exponential bounds for the probabilities of the typeP(|f
n
|>·T(f
n
)) and the size of the constantC
p
appearing in the inequality f
*
p
C
p
T
*(f)
p
, for some quasi-linear operators acting on martingales.This research was supported in part by NSF Grant, no. DMS-8902418On leave from Academy of Physical Education, Warsaw, Poland 相似文献
17.
In this note, the optimal L
2-error estimate of the finite volume element method (FVE) for elliptic boundary value problem is discussed. It is shown that u–u
h
0Ch
2|ln h|1/2f1,1 and u–u
h
0Ch
2f1,p
, p>1, where u is the solution of the variational problem of the second order elliptic partial differential equation, u
h
is the solution of the FVE scheme for solving the problem, and f is the given function in the right-hand side of the equation. 相似文献
18.
LetX be a complex Banach space andA: D(A)X a densely defined closed linear operator whose resolvent set contains the real line and for which (–A)–1 is bounded onR. We give a necessary and sufficient condition, in terms of the complex powers ofA and –A, for the existence of a decompositionX=X
+X
–, whereX
± are closed subspaces, invariant forA, the spectra of the reduced operatorsA
± are {(A);Im>0} and {(A);Im<0} respectively, and (–A
±)–1 is bounded forIm0.Finally we give an example of an operator in anL
p-type space for which the decomposition exists if 1<p<+ and does not exist ifp=1. 相似文献
19.
Jo A. M. Bollen 《Numerische Mathematik》1984,43(3):361-377
Summary In this paper we perform a round-off error analysis of descent methods for solving a liner systemAx=b, whereA is supposed to be symmetric and positive definite. This leads to a general result on the attainable accuracy of the computed sequence {x
i
} when the method is performed in floating point arithmetic. The general theory is applied to the Gauss-Southwell method and the gradient method. Both methods appear to be well-behaved which means that these methods compute an approximationx
i
to the exact solutionA
–1
b which is the exact solution of a slightly perturbed linear system, i.e. (A+A)x
i
=b, A of order A, where is the relative machine precision and · denotes the spectral norm. 相似文献
20.
The functional equation of multiplicative derivation is superstable on standard operator algebras 总被引:2,自引:0,他引:2
Peter Šemrl 《Integral Equations and Operator Theory》1994,18(1):118-122
LetX be a real or complex infinite dimensional Banach space andA a standard operator algebra onX. Denote byB(X) the algebra of all bounded linear operators onX. Let : + + be a function with the property lim
t (t)t
–1=0. Assume that a mappingD:A B(X) satisfies D(AB)–AD(B)–D(A)B<(A B) for all operatorsA, B D (no linearity or continuity ofD is assumed). ThenD is of the formD(A)=AT–TA for someTB(X).This work was supported by the Research Council of Slovenia 相似文献