MacaulayLab
Toolbox for solving systems of multivariate polynomial equations and rectangular multiparameter eigenvalue problems
Description:
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 domain in science and engineering. While both problem types seem unrelated at first, they are connected through the (block) Macaulay matrix and, consequently, solved via a 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. 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.
More Information:
Originates from the AI-Driven Data Science research challenge (FAIR 2019–2023).
Responsible Research Lead: Bart De Moor
Authors: Christof Vermeersch
Programming Language: Matlab
Categories: Optimization, Time series
Publications:
Vermeersch, C. and De Moor, B. (2021), “A Column Space Based Approach to Solve Systems of Multivariate Polynomial Equations”, IFAC-Papers-On-Line 54(9), 137-144.
Vermeersch, C. and De Moor, B. (2022), “Two Complementary Block Macaulay Matrix Algorithms to Solve Multiparameter Eigenvalue Problems”, Linear Algebra and its Applications 654, 177-209.
Vermeersch, C. and De Moor, B. (2023), “Recursive Algorithms to Update a Numerical Basis Matrix of the Null Space of the Block Row, (Banded) Block Toeplitz, and Block Macaulay Matrix”, SIAM Journal on Scientific Computing (SISC) 45(2), A596-A620.
Vermeersch, C. and De Moor, B. (2023), “Two Double Recursive Block Macaulay Matrix Algorithms to Solve Multiparameter Eigenvalue Problems”, IEEE Control Systems Letters 7, 319-324.