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From "Mikkel Meyer Andersen (JIRA)" <j...@apache.org>
Subject [jira] Commented: (MATH-435) Efficient matrix power
Date Tue, 09 Nov 2010 06:55:23 GMT
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[ https://issues.apache.org/jira/browse/MATH-435?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=12929987#action_12929987
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Mikkel Meyer Andersen commented on MATH-435:
--------------------------------------------

Of course numerical instability still occurs for large powers. One possibility is to extract
scalars of the type 10eX for integer X, i.e. both positive and negative, and then X together
with the scaled result. This requires a bit more bookkeeping, but might be a good idea. Of
course, if the matrix is to be used directly, this doesn't change much, but for further processing
this actually might save the day.

> Efficient matrix power
> ----------------------
>
>                 Key: MATH-435
>                 URL: https://issues.apache.org/jira/browse/MATH-435
>             Project: Commons Math
>          Issue Type: Improvement
>            Reporter: Mikkel Meyer Andersen
>            Assignee: Mikkel Meyer Andersen
>         Attachments: MATH435-patch1
>
>   Original Estimate: 0.07h
>  Remaining Estimate: 0.07h
>
> For symmetric matrices A it is easy to find A^n also for large n by making an eigenvalue/-vector
decomposition.
> In general, if the structure of the matrix is not know and the n'th power is needed,
A*A*...*A is way too inefficient. By using a binary representation and powers of 2, powers
can be found far faster similar to finding 5^14 as 5^14 = 5^8 * 5^4 = ((5^2)^2)^2 * (5^2)^2
= x3 * x2 where x1 = 5^2, x2 = x1^2, and x3 = x2^2, thus saving a lot of computations.

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