Numerical Schubert Calculus via the Littlewood-Richardson Homotopy Algorithm
Anton Leykin, Abraham Martin del Campo, Frank Sottile, Ravi, Vakil, Jan Verschelde
TL;DR
The paper introduces a numerical algorithm based on the Littlewood-Richardson homotopy for solving Schubert problems on Grassmannians, enabling the computation of large solution sets efficiently.
Contribution
It presents a new optimal formulation of Schubert problems in local coordinates and demonstrates an implementation capable of handling tens of thousands of solutions.
Findings
Successfully solves large-scale Schubert problems with tens of thousands of solutions
Develops a novel formulation of Schubert problems in local Stiefel coordinates
Provides an efficient numerical continuation algorithm for geometric problems
Abstract
We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this algorithm is our new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Our implementation can solve problem instances with tens of thousands of solutions.
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