Editing Inverse iteration (section)

=== Speed of  convergence ===

Let us analyze the [[rate of convergence]] of the method.

The [[power method]] is known to [[Rate of convergence#Convergence speed for iterative methods|converge linearly]] to the limit, more precisely:

<math display="block"> \mathrm{Distance}( b^\mathrm{ideal},  b^{k}_\mathrm{Power~Method})=O \left(   \left| \frac{\lambda_\mathrm{subdominant} }{\lambda_\mathrm{dominant} } \right|^k \right), </math>

hence for the inverse iteration method similar result sounds as:

<math display="block"> \mathrm{Distance}( b^\mathrm{ideal},  b^{k}_\mathrm{Inverse~iteration})=O \left(   \left| \frac{\mu -\lambda_{\mathrm{closest~ to~ }\mu}   }{\mu - \lambda_{\mathrm{second~ closest~ to~} \mu} } \right|^k \right). </math>

This is a key formula for understanding the method's convergence. It shows that if <math>\mu</math> is chosen close enough to some  eigenvalue  <math>\lambda </math>, for example <math> \mu- \lambda = \epsilon </math> each iteration will improve the accuracy <math> |\epsilon| /|\lambda +\epsilon - \lambda_{\mathrm{closest~ to~} \lambda} |  </math>  times. (We use that for small enough <math>\epsilon</math> "closest to <math>\mu</math>" and "closest to <math>\lambda </math>" is the same.) For small enough  <math> |\epsilon|</math> it is approximately the same as  <math> |\epsilon| /|\lambda  - \lambda_{\text{closest to } \lambda}|  </math>. Hence if one is able to find <math>\mu </math>, such that the <math> \epsilon </math> will be small enough, then very few iterations may be satisfactory.