Operator Equations In Hilbert Spaces Research Articles

Remarks on the generalized overrelaxation and the extrapolated Jacobi methods for operator equations in Hilbert space

The purpose of these remarks is fourfold: First, we amplify the implication made in our paper [5] concerning the necessary and sufficient conditions for the convergence of the generalized overrelaxation method (go-method). Theorem 1 below, which is equivalent to the Main Theorem in [5], states these conditions explicitly. Second, using our results in Section 1, we obtain in Section 2 similar explicit necessary and sufficient conditions for the convergence of the extrapolated Jacobi method studied in [6]. Third, the results of Sections 1 and 2 are extended to certain classes of non-self-adjoint and non-K-symmetric operators. Fourth, we obtain an estimate for the rate of convergence of the go-method as well as for the extrapolated Jacobi method and their spatial cases. In particular, as a special case of our result for the gomethod, we get an estimate for the rate of convergence of the Gauss-Seidel method obtained recently by Krasnoselsky and Stetzenko [2]. In what follows, our notation and arguments are those used in [S]. We refer the reader to [l, 3,4, 5, 61 f or references dealing with the overrelaxation methods for finite matrices.

On the solution of operator equations in Hilbert space

On the solution of operator equations in Hilbert space

Approximate solution of linear operator equations in Hilbert space by the method of least squares. II

Approximate solution of linear operator equations in Hilbert space by the method of least squares. II

Approximate solution of linear operator equations in Hilbert space by the method of least squares. I

Approximate solution of linear operator equations in Hilbert space by the method of least squares. I

The Generalized Overrelaxation Method for the Approximate Solution of Operator Equations in Hilbert Space

Previous article Next article The Generalized Overrelaxation Method for the Approximate Solution of Operator Equations in Hilbert SpaceW. V. PetryshynW. V. Petryshynhttps://doi.org/10.1137/0110052PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Julius Albrecht, Fehlerabschätzungen bei Relaxationsverfahren zur numerischen Auflösung linearer Gleichungssysteme, Numer. Math., 3 (1961), 188–201 10.1007/BF01386019 MR0146957 0111.12602 CrossrefGoogle Scholar[2] Hans Bückner, A special method of successive approximations for Fredholm integral equations, Duke Math. J., 15 (1948), 197–206 MR0025290 0030.39201 CrossrefISIGoogle Scholar[3] Yu. L. Daleckii˘, Integration and differentiation of functions of Hermitian operators depending on a parameter, Uspehi Mat. Nauk (N.S.), 12 (1957), 182–186 MR0086276 Google Scholar[4] W. Dück, Eine Fehlerabschätzung zum Einzelschrittverfahren bei linearen Gleichungssystemen, Numer. Math., 1 (1959), 73–77 MR0102910 0084.34504 CrossrefGoogle Scholar[5A] Gene H. Golub and , Richard S. Varga, Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods. I, Numer. Math., 3 (1961), 147–156 10.1007/BF01386013 MR0145678 0099.10903 CrossrefGoogle Scholar[5B] Gene H. Golub and , Richard S. Varga, Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods. II, Numer. Math., 3 (1961), 157–168 10.1007/BF01386014 MR0145679 0099.10903 CrossrefGoogle Scholar[6] G. J. Habetler and , E. L. Wachspress, Symmetric successive overrelaxation in solving diffusion difference equations, Math. Comp., 15 (1961), 356–362 MR0129139 0102.11403 CrossrefGoogle Scholar[7] P. R. Halmos, Introduction to Hilbert space, Chelsea, New York, 1957 0079.12404 Google Scholar[8] Einar Hille and , Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957xii+808 MR0089373 0078.10004 Google Scholar[9] Alston S. Householder, On the convergence of matrix iterations, Rep. ORNL 1883, Oak Ridge National Laboratory, Oak Ridge, Tenn., 1955, 47– MR0075677 Google Scholar[10] H. B. Keller, Numerical Methods, Lecture Notes, New York University, 1959, Ch. 2 Google Scholar[11] A. T. Lonseth, The propagation of error in linear problems, Trans. Amer. Math. Soc., 62 (1947), 193–212 MR0022315 0032.03203 CrossrefISIGoogle Scholar[12] W. V. Petryshyn, Masters Thesis, Direct and Iterative Methods for the Solution of Linear Operator Equations in Hilbert Space, Doctoral Thesis, Columbia University, 1961 Google Scholar[13] Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons Inc., New York, 1958xvi+423 MR0098966 0081.10202 Google Scholar[14] Johannes Weissinger, Verallgemeinerungen des Seidelschen Iterationsverfahrens, Z. Angew. Math. Mech., 33 (1953), 155–163 MR0056375 0050.12401 CrossrefGoogle Scholar[15] G. Wiarda, Integralgleichungen unter besonderer Berücksichtung der Anwendungen, Leipzig, 1930 Google Scholar[16] David Young, Iterative methods for solving partial difference equations of elliptic type, Trans. Amer. Math. Soc., 76 (1954), 92–111 MR0059635 0055.35704 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 10, Issue 4| 1962Journal of the Society for Industrial and Applied Mathematics History Submitted:18 January 1962Published online:13 July 2006 InformationCopyright © 1962 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0110052Article page range:pp. 675-690ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

Direct and Iterative Methods for the Solution of Linear Operator Equations in Hilbert Space

Direct and Iterative Methods for the Solution of Linear Operator Equations in Hilbert Space

Direct and iterative methods for the solution of linear operator equations in Hilbert space

Introduction. During the last two decades the problem of justifying existing methods and finding new ones for the solution of the equation Lu = f, where f is a given vector in Hilbert space and L is a given operator, has been investigated by many authors(2). This investigation led to the development of a group of direct methods of Ritz, Galerkin, least squares, and moments; and the iterative methods of steepest descent or gradient method, the method with minimal residuals, and their variants. These methods have a long history and have been studied and extensively applied by earlier authors to integral and differential equations. Only recently, however, the Hilbert space operator theory has been used for their study. Thus, using the variational principle, Mikhlin [16] proved the convergence of Ritz method for a self-adjoint positive definite operator L and the convergence of the method of least squares for an invertible operator L. In case L is of the form L = A + B, where A is self-adjoint and positive definite and B satisfies some additional conditions, the method of Galerkin was investigated by Mikhlin [16], Polsky [19], and others [9; 13]. For similar differential operators the method of moments was studied by Kravchuk [11], Polsky [19], and Zdanov [22]. The convergence and the estimate of error of the gradient method for selfadjoint and positive definite operators L have been studied by Kantorovich [7] and Hayes [6]. The latter considered also a slightly more general problem. In case L is a finite, symmetric, and positive definite matrix the method with minimal residuals was developed by Krasnoselsky and Krein [10]. A powerful method related to direct methods was also developed by Murray [18]. The purpose of this paper is to extend the study and the applicability of these methods to a larger class of linear operator equations(3) than those considered by the above authors and to present these seemingly distinct methods in a more unified manner. In our investigation we do not use the variational principle

Functional equations involving a parameter

1. The present note concerns the examination of nonlinear functional equations depending on a parameter. We investigate here the iterative method described in paper [1] and [2], which is a generalization of Newton's classical method. Another abstract formalism for Newton's method has been given first by L. V. Kantorovich (for references see [4]) and applied by him to the examination of operator equations in Banach spaces. The main point here is the application of the majorant method, which was used by Kantorovich [4] and also in paper [3]. The results stated here make it possible to find an error estimation of the exact solution in the case when the solution of a suitable approximate equation is given. An application to approximate solutions of operator equations in Hilbert space will be given in another note. Let X and M be two Banach spaces, and let F(x, ,u) be a nonlinear continuous functional defined on the space X+M, where x and ,u are in some closed spheres in X, M with centres x0, ,uo, respectively. Consider the nonlinear functional equation