Iterative Methods for the Solution of a Linear Operator Equation in Hilbert Space: A SurveyIn this expository work we shall conduct a survey of iterative techniques for solving the linear operator equations Ax=y in a Hilbert space. Whenever convenient these iterative schemes are given in the context of a complex Hilbert space -- Chapter II is devoted to those methods (three in all) which are given only for real Hilbert space. Thus chapter III covers those methods which are valid in a complex Hilbert space except for the two methods which are singled out for special attention in the last two chapters. Specifically, the method of successive approximations is covered in Chapter IV, and Chapter V consists of a discussion of gradient methods. While examining these techniques, our primary concern will be with the convergence of the sequence of approximate solutions. However, we shall often look at estimates of the error and the speed of convergence of a method. |
Contents
INTRODUCTION | 1 |
ITERATIVE METHODS IN REAL HILBERT SPACES | 8 |
SUCCESSIVE APPROXIMATION METHODS | 55 |
Copyright | |
4 other sections not shown
Other editions - View all
Common terms and phrases
Altman Au,Ku Au,u Banach space bounded and positive bounded below operator bounded linear operator bounded operator Browder and Petryshyn choice complex Hilbert space condition conjugate direction method conjugate gradient method continuously invertible converges strongly converges weakly corollary defined denoted discuss due to Petryshyn eigenvalue ergodic theorem error estimate Figueiredo and Karlovitz geometric progression Hayes 71 Hence I+T₁)y inequality inf Ax-y inner product inverse iterative method iterative scheme Jacobi methods Kantorovic Kpd operator Krasnoselskii least as fast Lemma linear operator equations Math method of steepest method of successive Nauk Neumann series Oblomskaja obtain Petryshyn 116 positive bounded progression with ratio Proof rate of convergence real Hilbert space real number recursion formula Russian Samanskii Section self-adjoint operator sequence shows solution of linear solvable steepest descent successive approximations T¹y techniques Theorem 11 tion unique solution vector z,Az χελ хо