Macaulaylab

What is Macaulaylab?

MacaulayLab is a MATLAB toolbox that features algorithms to solve systems of multivariate polynomial equations and rectangular multiparameter eigenvalue problems, which are encountered in various application domains in science and engineering. While both problems seem unrelated at first glance, they are connected through the (block) Macaulay matrix and, consequently, solved via similar numerical linear algebra methodology. In that sense, this toolbox is quite unique; the combined nature of the two types of problems allows for one tool(box) to tackle them both.

The toolbox uses only numerical linear algebra tools; hence, it relies on the decades of advancements in that field to tackle these important problems in a numerically robust and computationally efficient approach. Multidimensional realization problems in the null space or column space of the (block) Macaulay matrix constructed from the coefficients/coefficient matrices result in eigenvalue problems, the eigenvalues and eigenvectors of which yield the solutions of the problems. MacaulayLab contains recursive and sparse techniques to retrieve the solutions and is implemented without depending on a particular polynomial basis or monomial ordering; the user can choose what suits the application best. Next to the implementation of these algorithms, the toolbox also includes a database with test problems.

Features

• Solve systems of multivariate polynomial equations and multiparameter eigenvalue problems via iterative, recursive, and sparse algorithms.
• Use the standard monomial basis, the Chebyshev basis, or a user-defined custom polynomial basis to tackle the problems.
• Determine upper bounds on the number of solutions.

More information / contact

The toolbox has been fully developed by Christof Vermeersch and can be downloaded from the website.