The Shifted Inverse Power Method. Suppose that μis not an eigenvalue of A. (a) Show that the ite

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The Shifted Inverse Power Method. Suppose that μis not an...

The Shifted Inverse Power Method. Suppose that $\mu$ is not an eigenvalue of $A$. (a) Show that the iterative system $\mathbf{u}^{(k+1)}=(A-\mu \mathrm{I})^{-1} \mathbf{u}^{(k)}$ converges to the eigenvector of $A$ corresponding to the eigenvalue $\lambda^*$ that is closest to $\mu$. Explain how to find the eigenvalue $\lambda^*$. (b) What is the rate of convergence of the algorithm? (c) What happens if $\mu$ is an eigenvalue?

The Shifted Inverse Power Method. Suppose that $\mu$ is not an eigenvalue of $A$.
(a) Show that the iterative system $\mathbf{u}^{(k+1)}=(A-\mu \mathrm{I})^{-1} \mathbf{u}^{(k)}$ converges to the eigenvector of $A$ corresponding to the eigenvalue $\lambda^*$ that is closest to $\mu$. Explain how to find the eigenvalue $\lambda^*$. (b) What is the rate of convergence of the algorithm? (c) What happens if $\mu$ is an eigenvalue?

Key Concepts

Effect of Choosing a Shift Equal to an Eigenvalue

If the shift ? is chosen to be an eigenvalue of A, the matrix (A ? ?I) becomes singular, meaning its inverse does not exist in the usual sense. This situation undermines the algorithm since the iterative process relies on the computation of (A ? ?I)?¹. In such a case, the method fails to function as intended because the singularity prevents the correct isolation of the desired eigenpair.

Convergence Rate

The rate of convergence in the Shifted Inverse Power Method is determined by the separation between the reciprocal of the eigenvalue difference corresponding to the desired eigenvalue and those corresponding to the other eigenvalues. More specifically, the convergence is linear with a factor that depends on the ratio of the distance of the closest eigenvalue (in the shifted spectrum) to the next closest eigenvalue. The closer the desired eigenvalue is to the shift relative to the others, the faster the convergence.

Shifted Inverse Power Method

This method is a variant of the basic inverse power method that involves modifying the matrix A by subtracting a scalar times the identity matrix. The idea is to shift the spectrum of A so that an eigenvalue near the chosen shift ? becomes dominant in the transformed problem. This maneuver makes the iterative process converge to the eigenvector corresponding to the eigenvalue of A that is closest to ?, as the inverse of (A ? ?I) amplifies the component in the direction of that eigenvector.

Eigenvalue and Eigenvector

In the context of this method, an eigenvalue is a scalar such that there exists a nonzero vector (the eigenvector) for which A times that vector equals the eigenvalue times the same vector. The Shifted Inverse Power Method seeks the eigenpair (eigenvalue and eigenvector) for which the eigenvalue is closest to a prescribed shift ?. The method iterates by repeatedly applying the inverse of the shifted matrix to an initial vector, aligning it increasingly with the desired eigenvector.

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