Steffensen's method: Difference between revisions

Revision as of 01:05, 30 October 2007

Revision as of 01:06, 30 October 2007

Line 7:

: <math>x_{k+1} = x_k + [I - F(f(x_k), x_k)]^{-1}(f(x_k) - x_k) \, </math>

for a [[Map (mathematics)|mapping]] f&fnof; on a [[Banach space]] ''X'' and ''F''(''x''<nowiki>'</nowiki>,x") a [[Indexed family|family]] of [[Bounded set|bounded]] [[linear operators]] associated with ''x''<nowiki>'</nowiki> and ''x''", having the properties

: <math>F(x',x'')(x'-x'')=F(x')- F(x'') \,</math>

Line 13:

and

: <math>||F(x,y)-F(y,z)|| \le K_0||(x-z)|| + K_1||(x-y)|| + K_1||(y-z)||. \,</math>

This process, given a sufficiently good initial approximation, converges quadratically to a fixed point.