Smallest eigenvalue without inverse - Computational Science Stack Exchange most recent 30 from scicomp.stackexchange.com 2019-10-20T03:32:23Z https://scicomp.stackexchange.com/feeds/question/21733 https://creativecommons.org/licenses/by-sa/4.0/rdf https://scicomp.stackexchange.com/q/21733 11 Smallest eigenvalue without inverse Justin Solomon https://scicomp.stackexchange.com/users/5129 2016-01-02T21:19:55Z 2016-01-04T18:08:13Z <p>Suppose $A\in\mathbb{R}^{n\times n}$ is a symmetric, positive definite matrix. $A$ is big enough that it's expensive to solve $Ax=b$ directly.</p> <p><strong>Is there an iterative algorithm for finding the smallest eigenvalue of $A$ that doesn't involve inverting $A$ in each iteration?</strong></p> <p>That is, I'd have to use an iterative algorithm like conjugate gradients to solve $Ax=b$, so repeatedly applying $A^{-1}$ seems like an expensive "inner loop." I only need a single eigenvector.</p> <p>Thanks!</p> https://scicomp.stackexchange.com/questions/21733/-/21742#21742 13 Answer by GoHokies for Smallest eigenvalue without inverse GoHokies https://scicomp.stackexchange.com/users/17294 2016-01-04T10:53:49Z 2016-01-04T12:59:26Z <ol> <li><p>Compute the largest-magnitude eigenvalue $\lambda_{max}$ of $A$ (with, say, <code>eigs('lm')</code>).</p></li> <li><p>Then compute the largest magnitude (negative) eigenvalue $\hat{\lambda}_{max}$ of $M = A - \lambda_{max}I$ (again, through a standard call to <code>eigs('lm')</code>).</p></li> <li><p>Observe that $\hat{\lambda}_{max} + \lambda_{\max} = \lambda_{min}(A)$. The reason why this holds is explained <a href="https://math.stackexchange.com/questions/271864/power-iteration-smallest-eigenvalue">here</a>.</p></li> <li><p>Find your eigenvector $v$ by solving $(A - \lambda_{min} I) v = 0$.</p></li> </ol>