# Quadratic eigenvalue problem and solution (solved in Mathematica)

#### joshmccraney

Hi PF!

Given the quadratic eigenvalue problem $Q(\lambda) \equiv (\lambda^2 M + \lambda D + K)\vec x = \vec 0$ where $K,D,M$ are $n\times n$ matrices, $\vec x$ a $1\times n$ vector, the eigenvalues $\lambda$ must solve $\det Q(\lambda)=0$.

When computing this, I employ a linearization technique, a simple matrix pencil, outlined page 6 here: http://www.ma.man.ac.uk/~ftisseur/talks/talk_X13.pdf

At the very end of the technique, I have $n$ vectors of length $1\times 2n$, call these $\vec \xi_i$. Evidently $\vec \xi_i = [\vec x_i,\lambda \vec x_i]$, implying $\vec x_i$ is a $1 \times n$ vector (see hyperlink, it's fairly simple). When I'm solving in Mathematica, once I have each $\vec \xi_i$ I cut it in half, and use an algebra solver, solving for $\lambda_i$ via the equations $\vec x_i = \lambda_i \vec x_i$. I loop through this for $i = 1,n$. Most of the time I recover good solutions that output eigenvalues equivalent to $\det Q(\lambda)=0$. However, occasionally it misses a few, likely because the solver's precision order is too high (remember, its solving $n$ algebraic equations $\vec x_i = \lambda_i \vec x_i$, which should all be linearly dependent).

To obtain a solution, should I turn down the precision of the solver, or is this ill-advised?

Last edited:
Related Linear and Abstract Algebra News on Phys.org

#### fresh_42

Mentor
2018 Award
It is always a problem to solve an equation, here $\det Q(\lambda)=0$ numerically. A little variation of inputs and you miss the zero result. Stability is a problem here. If you don't want to analyze the entire algorithm w.r.t. stability, it's probably best to try a few settings as you mentioned and see what you end up with.

#### joshmccraney

It is always a problem to solve an equation, here $\det Q(\lambda)=0$ numerically. A little variation of inputs and you miss the zero result. Stability is a problem here. If you don't want to analyze the entire algorithm w.r.t. stability, it's probably best to try a few settings as you mentioned and see what you end up with.
Thanks! Just wanted to make sure I'm not doing something insane.

"Quadratic eigenvalue problem and solution (solved in Mathematica)"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving