New Power Method for Solving Eigenvalue Problems

TL;DR

This work introduces a novel power method that uses a matrix functional $\hat{F} = \hat{H}^\alpha e^{-\tau \hat{H}}$ (with $\alpha=1$) to selectively amplify eigencomponents, enabling the retrieval of arbitrary eigenvalues without deflation. The amplification is governed by the peak of $f(E)=E^\alpha e^{-\tau E}$ at $E_p=\alpha/\tau$, and convergence is analyzed via $R(E_p)=\frac{f(E_b)}{f(E_a)}$ and the turning point $E_{tp}=\frac{E_b-E_a}{\ln(E_b/E_a)}$, with a short-time operator form for matrices and a finite-difference diffusion interpretation for differential operators. The method is validated on a simple matrix, and quantum systems including a 1D box, a ring, a harmonic oscillator, and a 3D cubic box, showing that computed eigenvalues near $E_p$ agree with analytical values and obey predicted iteration counts. Overall, the approach extends power-method capabilities to target any eigenvalue, offering a simple yet effective tool for large-scale eigenproblems in physics and related fields.

Abstract

We present a new power method to obtain solutions of eigenvalue problems. The method can determine not only the dominant or lowest eigenvalues but also all eigenvalues without the need for a deflation procedure. The method uses a functional of an operator (or a matrix) to select or filter an eigenvalue. The method can freely select a solution by varying a parameter associated to an estimate of the eigenvalue. The convergence of the method is highly dependent on how closely the parameter to the eigenvalues. In this paper, numerical results of the method are shown to be in excellent agreement with the analytical ones.

Figures (10)

• Figure 1: $f(E)$ in Eq. (\ref{['eqn-fe']}) as a function of $E$ with $\tau = 1$ for three values of $\alpha$. The dotted brown lines are used to indicate the positions of the maximum values.
• Figure 2: Numerical values of $E$ for the matrix in Eq. (\ref{['eqn-splmatrix']}) at $k$th iteration with the iteration parameters $M = 100$ and $\psi^{(0)} = [0.7, 0.8, 0.4 ]^T$ for four values of $E_p$.
• Figure 3: Numerical values of $E$ for the matrix in Eq. (\ref{['eqn-splmatrix']}) at $k$th iteration with the parameters $M= 100$ and $E_p = 1.6$ for four different random initial vectors.
• Figure 4: Numerical values of $E$ for the matrix in Eq. (\ref{['eqn-splmatrix']}) at $k$th iteration with the parameters $E_p= 1.6$ and $\psi^{(0)} = [0.7, 0.8, 0.4]^T$ for four values of $M$.
• Figure 5: Eigenvalues of the matrix in Eq. (\ref{['eqn-splmatrix']}) computed using the iteration parameters $E_p = 1.6$ and $\psi^{(0)} = [0.7, 0.8, 0.4]^T$ for various values of $M$.