Solving Gen. Eigen Probs w/ Real Sym Indefinite A & Definite B

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Discussion Overview

The discussion centers around solving a generalized eigenvalue problem involving a real symmetric indefinite matrix A and a real symmetric matrix B that is intended to be negative definite but is instead indefinite due to numerical issues. Participants explore the implications of these matrix properties on the efficiency of solving the eigenvalue problem and potential methods for regularizing B.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes a situation where a matrix B, expected to be negative definite, is instead indefinite due to numerical precision issues in the algorithm used to generate it.
  • Another participant questions the approach of maintaining the indefinite matrix B and suggests that it should be adjusted to ensure it is negative definite, implying that the current method is flawed.
  • A different participant defends the correctness of matrix B's results in the context of the larger problem, attributing the issue to finite precision and the numerical methods employed, which involve complex calculations that cannot be easily modified.
  • One participant proposes a mathematical optimization approach to shift B into a negative definite form while maintaining the integrity of the eigenvalue problem, suggesting a method to bound the error introduced by this adjustment.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and feasibility of modifying matrix B. While some argue for the need to ensure B is negative definite, others maintain that the current formulation of B is valid despite its indefinite nature.

Contextual Notes

The discussion highlights limitations related to numerical precision and the dependence on specific algorithmic implementations, which may affect the properties of the matrices involved. There are unresolved mathematical steps regarding how to effectively regularize B without compromising the eigenvalue problem.

Born2bwire
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I have a system that ideally creates a real symmetric negative definite matrix. However, due to the implementation of the algorithm and/or finite-precision of floating point, the matrix comes out indefinite. For example, in a 2700 square matrix, four eigenvalues are positive, the rest are negative. This is a problem because I want to solve a generalized eigenvalue problem and the fact that neither of my matrices are definite forces me to use very inefficient methods. If it was just a simple eigenvalue problem then I could just do a simple shift, but I need to solve for the problem

\mathbf{A}\mathbf{x}=\lambda\mathbf{B}\mathbf{x}

A is real symmetric indefinite and B should be real symmetric negative definite but limitations seem to push it to indefinite. I know I could do matrix multiplication by the transpose of the matrices but then B gets pushed to semi-definite which still is not valid for these methods (Looking at Lanczos which uses Cholesky decomposition on B if it is Hermitian definite).

Does anyone know of a way I could regularize B into a definite matrix without compromising the original eigenvalue problem?
 
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No offense but your problem is originated from a numerical algorithm and you want to keep it rigorous after this error without using B-\varepsilon I \prec 0? I don't see why, since it should come out negative definite anyway. Why are you continuing using the wrong B? Looks like another SeDuMi problem :)
 


There isn't anything inherently wrong with the matrix of B. The results derived from B are correct, in terms of the solutions afforded by B in the larger problem and the results I wish to achieve from the eigenvalues. There isn't anything that I can feasibly do modify in the creation of B. I believe that the main problem arises from finite precision and the numerical procedures used to generate the elements. I am already using double precision and there isn't anything I can do about the numerical methods since the elements involve a double surface integration that must be estimated via quadrature.
 


Yes I agree, but the matrix B should be negative definite when it comes out of the algorithm according to the information you provided. No? So pushing it back to negative definite is something you have to do anyway. Otherwise you just assume that it is almost correct and you are stuck with nonoptimized code. Why don't you just shift it and try to get an error bound for this or put it in your optimization procedure maybe like

\begin{align*}<br /> \min_x &amp; \ \ \epsilon \\<br /> s.t. &amp;\mathbf{A}\mathbf{x}=\lambda\left(\mathbf{B}-\epsilon I\right)\mathbf{x}\\<br /> &amp;\mathbf{B}-\epsilon I \prec 0\\<br /> &amp;\epsilon &gt; 0<br /> \end{align*}<br />
 

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