Numerical Schubert Calculus defining Schubert Induction with PHCv2.4.85

This directory is entirely new in release 2.2 of PHCpack,
it implements a geometric Littlewood-Richardson rule and defines
Littlewood-Richardson homotopies to solve Schubert problems.

Run "gprbuild induction.gpr" to make all test programs.
On windows, type "gprbuild induction.gpr -Xos=windows"
at the PowerShell prompt.
The "gprclean induction.gpr" removes all files created by gprbuild.

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file name                      : short description
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checker_boards                 : defines checker boards
checker_boards_io              : input/output of boards and permutations
checker_moves                  : operations to move black and white checkers
checker_localization_patterns  : pattern for matrix based on checker game
checker_posets                 : Littlewood-Richardson rule by checker game
checker_posets_io              : output of posets with moves in checker game
intersection_posets            : combine posets for several conditions
intersection_posets_io         : output of formal sums and equations
test_checker_games             : tests on checker games
ts_checkers                    : main interactive test program
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numeric_schubert_conditions    : numerical evaluation of Schubert conditions
symbolic_schubert_conditions   : symbolic definition of Schubert conditions
remember_symbolic_minors       : remember table for symbolic minors
start_flag_homotopies          : start solutions for flag homotopies
setup_flag_homotopies          : utilities to setup flag homotopies
moving_flag_homotopies         : defines homotopies in one checker game
flag_transformations           : transforming pairs of flags
test_homotopy_definitions      : tests setup of homotopies
ts_induce                      : main test program for homotopies
recondition_swap_homotopies    : reconditioning tools for swap homotopies
checker_homotopies             : defines Littlewood-Richardson homotopies
wrapped_path_trackers          : calling path trackers on natural homotopy
moving_flag_continuation       : trackers for moving flag homotopies
test_schubert_conditions       : tests conditions imposed by bracket and flag
ts_flagcond                    : main test on Schubert conditions
standard_solution_posets       : to store solutions at each node and level
dobldobl_solution_posets       : posets of double double solution lists
quaddobl_solution_posets       : posets of quad double solution lists
checker_poset_deformations     : tracking paths to connect the posets
resolve_schubert_problems      : code to resolve general Schubert problems
test_lr_homotopies             : tests for Littlewood-Richardson homotopies
ts_lrhom                       : main test on L-R homotopies
main_schubert_induction        : defines phc -e, enumerative geometry
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The program "ts_checkers" is based on the paper by Ravi Vakil
on "A geometric Littlewood-Richardson rule",
Annals of Mathematics 164(2): 489-512, 2006.
