1549301742-The_Theory_of_Difference_Schemes__Samarskii

510 Difference Methods for Solving Nonlinear Equations

is valid under the condition ll.f^11 (y)lle = 16q.

Indeed, in conformity with the maximu1n principle (for more detail

see Chapter 4, Section 2) problem (9) has for .f'(y) < 0 the majorant

V ( x) = f{ x ( 1 - :r) , [ [ V ( x) [ [ e < % K ,

where I\= % fl.f^11 (y)ffe · lft [[~,,so that

lfkt

1

lie< If VIie < 115 ff.f^11 (Y)fle · lit II~< q flt II~·

view, we deduce that

If kt! lie< ~ q ffq3 if2ek+i , ,

thereby confinning the quadratic law of the convergence of iterations with

the initial approximation y subject to the condition

0 0

q lfY - yffc < 1 or !J f[v fie < 1 ·

When f' (y) > -c 1 , c 1 > 0, an alternative esti1nate in the grid norm of the

space L instead of (10) is such that

( 11)

where

lfkt^111 < ll.f

11

(y)ffe flt 112 = q^1 flt 112 ,

• 2(b+c1) o+c1

4. 27rh
   b = h2 sm 2

1

and q1 = 2 lf.f

11

(Y)ffe ·

Other iterative n1ethods apply equally well to problen1 (2). An1ong

the1n the 1nethod with the recurrence relation

k+l k •. k

Y J:x = {)Yxx - (1-{)) f(y)

will be appreciated. Here the parameter {) is given by the formula

c* = inax lf'(y)[:

y

in so doing the iterations converge with the rate of a ge01netric progression

with clen01ninator (h = {), so that

It is worth noting here that the iterations converge no matter how the initial

approximation y is chosen, because {) < 1.