Numerical Schubert Calculus in Macaulay2
Anton Leykin, Abraham Mart\'in del Campo, Frank Sottile, Ravi Vakil,, Jan Verschelde
TL;DR
The paper introduces a Macaulay2 package that implements numerical algorithms for solving Schubert problems on Grassmannians, utilizing homotopy continuation methods with dual implementations for efficiency and flexibility.
Contribution
It provides the first integrated implementation of Pieri and Littlewood-Richardson homotopy algorithms in Macaulay2, combining scripting and compiled code for numerical Schubert calculus.
Findings
Two independent implementations of each algorithm in Macaulay2 and PHCpack.
Efficient numerical solutions for Schubert problems on Grassmannians.
Enhanced computational tools for algebraic geometry research.
Abstract
The Macaulay2 package NumericalSchubertCalculus provides methods for the numerical computation of Schubert problems on Grassmannians. It implements both the Pieri homotopy algorithm and the Littlewood-Richardson homotopy algorithm. Each algorithm has two independent implementations in this package. One is in the scripting language of Macaulay2 using the package NumericalAlgebraicGeometry, and the other is in the compiled code of PHCpack.
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Taxonomy
TopicsPolynomial and algebraic computation · Mathematics and Applications · Advanced Differential Equations and Dynamical Systems