Jenkins–Traub algorithm: Difference between revisions

Revision as of 15:11, 27 July 2009

Revision as of 10:16, 3 August 2009

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The iteration uses the given ''P'' and <math>\scriptstyle P^{\prime}</math>. In contrast the third-stage of Jenkins-Traub

:<math>

:<math>s_{\lambda+1}=s_\lambda- \frac{P(s_\lambda)}{\bar H^{(\lambda+1)}(s_\lambda)}</math>

s_{\lambda+1}

:<math>s_{\lambda+1} =s_\lambda- \frac{P(s_\lambda)}{\bar H^{(\lambda+1)}(s_\lambda)}</math>

=s_\lambda-\frac{W^\lambda(s_\lambda)}{W^\lambda'(s_\lambda)}</math>

is precisely a Newton-Raphson iteration performed on certain [[rational functions]]. More precisely, Newton-Raphson is being performed on a sequence of rational functions

:<math>W^\lambda(z)=\frac{P(z)/}{H^{(\lambda(z)}</math>.

For <math>\lambda</math> sufficiently large,

:<math>P(z)/\bar H^{(\lambda)}(z)</math>

is as close as desired to a first degree polynomial

Line 141 ⟶ 144:

:<math>-\frac{P(s)}{\bar H^k(s)}</math>

is a current approximation of the smallest eigenvalue of <math>(M_X-s\cdot id)</math> resp.

:<math>s-\tfrac{P(s)}{\bar H^k(ss_k)}</math>

is an improved approximation of the smallest root of ''P(X)''.

Chosing instead

:<math>s-\tfrac{P(s)}{\bar H^{k+1}(s_k)}</math>

amounts to a second inverse power iteration with shift <math>s_k</math>

==Real coefficients==