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Fastest quotient iteration with variational principles for self-adjoint eigenvalue problems

Marko Huhtanen, Vesa Kotila, Pauliina Uusitalo

Abstract

For the generalized eigenvalue problem, a quotient function is devised for estimating eigenvalues in terms of an approximate eigenvector. This gives rise to an infinite family of quotients, all entirely arguable to be used in estimation. Although the Rayleigh quotient is among them, one can suggest using it only in an auxiliary manner for choosing the quotient for near optimal results. In normal eigenvalue problems, for any approximate eigenvector, there always exists a "perfect" quotient exactly giving an eigenvalue. For practical estimates in the self-adjoint case, an approximate midpoint of the spectrum is a good choice for reformulating the eigenvalue problem yielding apparently the fastest quotient iterative method there exists. No distinction is made between estimating extreme or interior eigenvalues. Preconditioning from the left results in changing the inner-product and affects the estimates accordingly. Preconditioning from the right preserves self-adjointness and can hence be performed without any restrictions. It is used in variational methods for optimally computing approximate eigenvectors.

Fastest quotient iteration with variational principles for self-adjoint eigenvalue problems

Abstract

For the generalized eigenvalue problem, a quotient function is devised for estimating eigenvalues in terms of an approximate eigenvector. This gives rise to an infinite family of quotients, all entirely arguable to be used in estimation. Although the Rayleigh quotient is among them, one can suggest using it only in an auxiliary manner for choosing the quotient for near optimal results. In normal eigenvalue problems, for any approximate eigenvector, there always exists a "perfect" quotient exactly giving an eigenvalue. For practical estimates in the self-adjoint case, an approximate midpoint of the spectrum is a good choice for reformulating the eigenvalue problem yielding apparently the fastest quotient iterative method there exists. No distinction is made between estimating extreme or interior eigenvalues. Preconditioning from the left results in changing the inner-product and affects the estimates accordingly. Preconditioning from the right preserves self-adjointness and can hence be performed without any restrictions. It is used in variational methods for optimally computing approximate eigenvectors.
Paper Structure (7 sections, 12 theorems, 67 equations, 4 figures, 4 algorithms)

This paper contains 7 sections, 12 theorems, 67 equations, 4 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Quotient function for eigenvalue estimation
  3. Midpoint rule for estimates in self-adjoint eigenvalue problems
  4. Preconditioning and variational principles for ge-nerating approximate eigenvectors
  5. Estimating several eigenvalues
  6. Numerical experiments
  7. Conclusions

Key Result

theorem 1

The closure of the image of the quotient function is the disc of radius $\frac{\|Mx-{\rm rq}_{M,N}(x) Nx\|_P}{\|Nx\|_P}$ centred at ${\rm rq}_{M,N}(x)$. The limit of the quotient function at infinity is ${\rm rq}_{M,N}(x)$ while the limit at ${\rm rq}_{M,N}(x)$ is the circle of radius $\frac{\|Mx-{\

Figures (4)

  • Figure 1: The graph of the quotient function of Example \ref{['kungfood']}. The eigenvalues of $M$ are depicted vertically with 'x' on the $y$-axis. The Rayleigh quotient is $5$. With Algorithm 1 we attain $5.1333$. The value of the quotient function with $\mu$ being the exact midpoint \ref{['optikko']} is $5.183$ and is depicted with 'o'.
  • Figure 2: Convergence of the eigenpairs for the fluid flow generalized eigenvalue problem of Example \ref{['ekaesim']}. First thee itrations with Algorithm \ref{['alg:dese']} are executed. Thereafter two iterations with Algorithm \ref{['alg:factsparpo']} suffices for convergence. The left panel shows the difference between the eigenvalue approximation and the eigenvalue computed with Matlab's eigs command. The right panel displays the respective loss of linear dependency of the vectors $w_1$ and $w_2$.
  • Figure 3: Convergence of the eigenpairs for the waveguide problem of Example \ref{['wawee']}. First four itrations with Algorithm \ref{['alg:dese']} are executed. Thereafter two iterations with Algorithm \ref{['alg:factspar']} are needed for $\sigma_2([w_1,w_2])<10^{-10}$. The left panel shows the difference between the eigenvalue approximation and the eigenvalue computed with Matlab's eigs command. The right panel displays the respective loss of linear dependency of the vectors $w_1$ and $w_2$.
  • Figure 4: Plot of the eigenvector inside the Z-formed waveguide corresponding to the eigenvalue $\lambda_1 \approx 8.8961$.

Theorems & Definitions (33)

  • definition 1
  • Example 1
  • Example 2
  • theorem 1
  • proof
  • corollary 1
  • Example 3
  • definition 2
  • theorem 2
  • proof
  • ...and 23 more