Approximate method for separation of solutions of nonlinear operator equations

Abstract

The basic theory for the analysis of approximate methods of solution of nonlinear operator and functional equations is presented in the monographs [i-4], which also give extensive bibliography on these topics.The process of solving nonlinear equations with many isolated solutions consists of two stages: separation of solutions and iterative refinement of solutions. While the topic of refinement of solutions has been studied in considerable detail, the very difficult topic of separation of isolated solutions of nonlinear equations has been insufficiently studied.Separation of isolated solutions has been considered in [5] for the case when the nonlinear operators generating the exact equation and the sequence of approximate equations act in the same Hilbert space. In the present paper, we consider separation of isolated solutions when the nonlinear operators generating the exact and the approximate equations act in different spaces. The sequence of approximate equations is constructed by projection methods [2][3][4]. The solutions of the exact equation are separated using a local theorem of the iterative method [6].Thus, following [2, 3], we consider the operator equation r'u : = u -~u = 0 (1)

Using reserves for computation optimization to improve the integration of rapidly oscillating functions

“…To obtain the majorant round-off error estimate (E t ) proposed by the c.a. of the solution of problem (1), Wilkinson's estimates [18] are used for round-off errors in the basic arithmetic operations on the assumption that computations are carried out in the floating point mode and the results of arithmetic operations are rounded off by the classical rule to t binary digits in normalized mantissa.…”

Some approaches to the analysis of computation optimization problems

“…Some results in this direction are published in [7]. which considers the separation of isolated solutions for the case when the operators T and T n with fairly general properties act in different spaces.…”

“…For the case when twice Fr6chet differentiable operators T and T n act in the same Hilbert space H, a method of solving this problem has been proposed in [5]. According to this method, the exact or approximate solutions of Eq.…”

“…(4) and (5). We accordingly consider a sequence of linear restrictions of the projectors Pn, each projecting from X onto the corresponding X n. In this case, closeness of Eqs.…”

“…Let {Xn} be a sequence of closed subspaces of X. Denote fin = f/ C3 X n and assume that a continuous operator 7"n with values in X n is defined on each fin, i.e., Tn:fl n C X n --, X n, and is close in some sense to the operator T. Then the equations Tnu n : = u n -TnU n = 0 (n = 1,2 .... ) (5) are regarded as approximating equations for Eq. (4), and their solutions are declared approximate solutions of Eq.…”