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Power method

Assume A is an NXN matrix, diagonalizable, with eigenvalues tex2html_wrap_inline352 and associated eigenvectors tex2html_wrap_inline354 . The power method computes

eqnarray178

The argument for why this process converges follows easily if we think of the initial guess tex2html_wrap_inline356 . Then

eqnarray195

This argument shows that the iterations tex2html_wrap_inline358 approximate the true eigenvalue with an error of tex2html_wrap_inline360 . This is the same error (in norm) for the approximation of the true eigenvector by tex2html_wrap_inline362 .

If two eigenvalues are close in absolute value, the power iteration will converge slowly. In this case, it may be advantageous to shift the eigenvalues. In other words, perform the power method on a matrix tex2html_wrap_inline364 , with tex2html_wrap_inline366 chosen to speed up the calculation. The eigenvalue you compute is, of course, shifted also.

Another method for speeding up the calculation is Aitken's extrapolation.

eqnarray204

Finally, to obtain other eigenvalues (say the second) FOR SYMMETRIC MATRICES is to apply the power method to the matrix tex2html_wrap_inline368 where tex2html_wrap_inline370 is the normlized eigenvector. You should convince yourself that tex2html_wrap_inline298 has the same eigenvectors as A, and the same eigenvalues, except that tex2html_wrap_inline374 has been replaced by 0.



E. Bruce Pitman
Wed Oct 28 09:33:45 EST 1998