# Research Interests

## Current interests

### Linearization of matrix polynomials

Polynomial matrices are everywhere, and their solution via linearization has been a central research topic for more than 50 years. Such problems are involve matrices having polynomial entries, that is \[ P(\lambda)= \lambda ^{N}P_{N}+\lambda^{N-1}P_{N-1}+\cdots+\lambda P_1+P_0\>, \] where $P(\lambda)$ is an $m\times n$ matrix having polynomial entries. One classical approach is to find a larger linear pencil $\lambda B-A$ having the same eigenstructure as $P(\lambda)$, that is, having the same finite and infinite elementary divisors, and the same null structure. The construction of linearizations is a central topic of interest, especially for polynomials expressed in non-monomial bases. One other aspect that is of interest is the construction of new linearizations having Fiedler-like form, resulting in linearizations having the structure \[ \lambda B-A = \left[ \begin{array}{c|c} \lambda M_1+M_0&L^{T}_{\eta}(\lambda)\\\hline L_{\epsilon}(\lambda)&0 \end{array} \right]\>, \] where $L_{k}(\lambda)=\left[\begin{array}{cc}I_k&0\end{array}\right] -\lambda \left[\begin{array}{cc}0&I_k\end{array}\right]$. One can show that for particular choices of $M_1$ and $M_0$, such pencils strongly linearize $P(\lambda)$. Furthermore, such structured matrices allow for a particularly simple backward error analysis, showing that under certain conditions, the eigenstructure of $P(\lambda)$ can be recovered in a structurally backward stable way.

### Approximate greatest common divisors for alternative polynomial bases

In most works discussing the approximate greatest common divisor (AGCD) problem, the authors usually only discuss polynomials expressed in the classical monomial basis. However, it is often of interest in applications to compute AGCD's in other bases related to how the original problem is proposed. For example, the Bernstein basis is often used in computer aided geometric design problems, and in other areas the underlying polynomials may only be known by their values at distinct points. In this work we utilize linearizations of rectangular matrix polynomials expressed in the Lagrange basis to compute good approximations of the AGCD's.

### Rootfinding via Generalized Companion Matrix Pencils

For interpolants expressed in barycentric Lagrange form, the roots of the interpolant can be found via the eigenvalues of a related companion matrix pencil. I have been interested in reducing these companion matrices to a tridiagonal plus rank-one update structure, which makes the deflation of infinite eigenvalues possible in the first steps of the QZ algorithm.

## Past interests

### Mandelbrot Polynomials and Matrices

Mandelbrot polynomials are defined by $p_{0}(z)=0$ and $p_{k+1}(z)=z\,p_{k}^2(z)+1$ for $k\geq 0$. $p_k(z)$ has degree $2^{k-1}-1$. The roots of $p_{k}(z)$ are the periodic points of the Mandelbrot set. The roots of Mandelbrot polynomials can be found by finding the eigenvalues of a recursively constructed family of matrices $\mathbf{M}_k$ defined as follows, let $\mathbf{r}_k=\left[0,0,\cdots,1\right]$ and $\mathbf{c}_k=\left[1,0,\cdots,0\right]^T$ be row and column vectors of length $2^{k-1}-1$. Put $\mathbf{M}_{2}=-1$ and for $k>1$ \[ \mathbf{M}_{k+1}=\left[ \begin{array}{ccc} \mathbf{M}_k& &-\mathbf{c}_{k}\mathbf{r}_{k}\\ -\mathbf{r}_{k}&0& \\ &-\mathbf{c}_k&\mathbf{M}_k \end{array}\right] \]

For this family of matrices $\mathbf{M}_k$, we have $\det{\left(z\mathbf{I}-\mathbf{M}_k\right)}=p_k(z)$, and thus the eigenvalues of $\mathbf{M}_k$ are exactly the zeros of $p_k(z)$. One can go further, and discover the LU decomposition of the resolvent $\sigma\mathbf{I}-\mathbf{M}_k$ for which one linear solve $(\sigma\mathbf{I}-\mathbf{M}_k)\mathbf{x}=\mathbf{b}$ can be done in $O(n)$ operations, making the computation of eigenvalues using sparse eigenvalue techniques very attractive, even for large values of $k$.

### Hermite interpolation

Hermite interpolation is interpolation in the Hermite basis, this allows one to include derivative data as well as function values. One can also represent the polynomial in barycentric form, from which one can find roots and evaluate the polynomial efficiently.

### Wright ω function

One of the first projects that I worked on when I got to the University of Western Ontario. The goal for this project was to implement double precision code to evaluate the Wright omega function. The implementation of this function was my primary interest.

### Metabolic Modeling

After doing a summer project in 2006 titled: "Fast Algorithms for Model based Therapeutics" with Chris Hann in the Centre for Bioengineering at the University of Canterbury, I became interested with other projects in Metabolic simulation and Metabolic parameter estimation. I then wrote my Honours thesis on parameter estimation for differential equation models.

Double PendulumOne project that I was working on just before I left Canterbury was to build a measurable double pendulum. The purpose of this project was to show students empirical evidence of chaotic dynamics.