# 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

#### 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.

### Want to reply to this thread?

"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