共查询到20条相似文献,搜索用时 31 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.
Takeaki Yamazaki 《Integral Equations and Operator Theory》2002,43(2):237-247
In 1951, Heinz showed the following useful norm inequality:If A, B0and XB(H), then AXB
r
X1–r
A
r
XB
r
holds for r [0, 1]. In this paper, we shall show the following two applications of this inequality:Firstly, by using Furuta inequality, we shall show an extension of Cordes inequality. And we shall show a characterization of chaotic order (i.e., logAlogB) by a norm inequality.Secondly, we shall study the condition under which
, where
is Aluthge transformation ofT. Moreover we shall show a characterization of normaloid operators (i.e.,r(T)=T) via Aluthge transformation. 相似文献
5.
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
6.
Yu. Lyubich 《Integral Equations and Operator Theory》1995,23(2):232-244
If X is a real Banach space, then the inequality x defines so-called hyperbolic cone in E=X. We develop a relevant version of Perron-Frobenius-Krein-Rutman theory. 相似文献
7.
S. Gratton 《BIT Numerical Mathematics》1996,36(3):523-530
LetA be anm × n, m n full rank real matrix andb a real vector of sizem. We give in this paper an explicit formula for the condition number of the linear least squares problem (LLSP) defined by min Ax–b2,x
n
. Let and be two positive real numbers, we choose the weighted Frobenius norm [A, b]
F
on the data and the usual Euclidean norm on the solution. A straightforward generalization of the backward error of [9] to this norm is also provided. This allows us to carry out a first order estimate of the forward error for the LLSP with this norm. This enables us to perform a complete backward error analysis in the chosen norms.Finally, some numerical results are presented in the last section on matrices from the collection of [5]. Three algorithms have been tested: the QR factorization, the Normal Equations (NE), the Semi-Normal Equations (SNE). 相似文献
8.
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
9.
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
10.
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
11.
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
12.
Summary Ann×n complex matrixB is calledparacontracting if B21 and 0x[N(I-B)]Bx2<x2. We show that a productB=B
k
B
k–1
...B
1 ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofB in terms of the subdominant singular values of theB
i
's and in terms of the angles between certain subspaces. Our results here extend earlier results due to Halperin and due to Smith, Solomon and Wagner. We also determine necessary and sufficient conditions forn numbers in the interval [0, 1] to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product. In the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the SOR iteration matrix associated with ann×n hermitian positive semidefinite matrixA none of whose diagonal entries vanish.The work of this author was supported in part by NSF Research Grant No. MCS-8400879 相似文献
13.
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
14.
W. A. Kirk 《manuscripta mathematica》1979,30(1):89-102
It is proved that if D is a bounded open subset of a uniformly convex Banach space X and
is a continuous mapping which is a local pseudo-contraction (e.g., locally nonexpansive) on D, then T has a fixed point in D if there exists xD such that z–Tz相似文献
15.
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
16.
B. I. Korenblyum 《Mathematical Notes》1971,10(1):456-458
Our main result is the following: iff (z) is in the space H2, and F(z) is its outer part, then F(n)H2F(n)H2(n=1,2,...), the left side being finite if the right side is finite. Under certain essential restrictions, this. inequality was proved by B. I. Korenblyum and V. S. Korolevich [1].Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 53–56, July, 1971. 相似文献
17.
Ryszard Rudnicki 《Integral Equations and Operator Theory》1996,24(3):320-327
A class of Markov operators appearing in biomathematics is investigated. It is proved that these operators are asymptotic stable inL
1, i.e. lim
n
P
n f=0 forfL
1 and f(x) dx=0. 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.
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
20.
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