Sidi's generalized secant method: Difference between revisions

Revision as of 14:34, 16 March 2013

Revision as of 14:39, 16 March 2013

Line 30:

== Convergence ==

Sidi showed that if the function <math>f</math> is (''k''+1)-times [[Smooth function|continuously differentiable]] in an [[open interval]] <math>I</math> containing <math>\alpha</math> (i.e. <math>f \in C^k (I)</math>), and the initial estimates <math>x_1 , \dots , x_{k+1}</math> are chosen close enough to <math>\alpha</math>, then the sequence <math>\{ x_i \}</math> convergenceconverges to <math>\alpha</math> (i.e. the following [[Limit of a sequence|limit]] holds: <math>\lim\limits_{n \to \infty} x_n = \alpha</math>).

Sidi furthermore showed that the sequence [[Rate of convergence|converges]] to <math>\alpha</math> of order <math>\psi_k</math>, i.e.