Annotation of OpenXM_contrib/PHC/Ada/Schubert/READ_ME, Revision 1.1
1.1 ! maekawa 1: Numerical Schubert Calculus with PHCv2.
! 2:
! 3: This directory is entirely new in this release of PHC.
! 4:
! 5: SAGBI homotopies are a concatenation of three homotopies :
! 6: 1) polyhedral homotopies to set up the start system;
! 7: 2) flat deformation using a Groebner basis for the Grassmannian;
! 8: 3) Cheater's homotopy towards real target system.
! 9:
! 10: There are three parts in this library:
! 11: 1) Groebner and SAGBI homotopies
! 12: 2) original Pieri homotopy algorithm : chain by chain
! 13: 3) poset-based organization of the Pieri homotopies
! 14: The test programs are grouped per part.
! 15:
! 16: --------------------------------------------------------------------------------
! 17: file name : short description
! 18: --------------------------------------------------------------------------------
! 19: brackets : representation/manipulation of brackets
! 20: brackets_io : input/output of brackets
! 21: bracket_monomials : monomials of brackets
! 22: bracket_monomials_io : input/output of bracket monomials
! 23: bracket_polynomials : polynomials in the brackets, complex coeff
! 24: bracket_polynomials_io : input/output for bracket polynomials
! 25: straightening_syzygies : implementation of straightening algorithm
! 26: bracket_expansions : expansion of brackets
! 27: chebychev_polynomials : Chebychev polynomials for the conjecture
! 28: osculating_planes : input for the Shapiro^2 conjecture
! 29: matrix_homotopies : management of homotopies between matrices
! 30: evaluated_minors : determinant computations
! 31: sagbi_homotopies : set up of the SAGBI homotopies
! 32: driver_for_sagbi_homotopies : driver to the SAGBI homotopies
! 33: --------------------------------------------------------------------------------
! 34: ts_brackets : test manipulation of brackets
! 35: ts_brackmons : test bracket monomials
! 36: ts_brackpols : test bracket polynomials
! 37: ts_straighten : test straightening algorithm
! 38: ts_expand : test expansion of brackets
! 39: ts_local : test localization
! 40: ts_cheby : test working with Chebychev polynomials
! 41: ts_shapiro : test generation of input planes
! 42: ts_detrock : test various root counts on (m,p)-system
! 43: ts_sagbi : calls the driver to the SAGBI homotopies
! 44: --------------------------------------------------------------------------------
! 45: pieri_trees : representation/creation of Pieri trees
! 46: pieri_trees_io : output of Pieri trees
! 47: pieri_root_counts : produces a list of Pieri chains
! 48: symbolic_minor_equations : symbolic form of equations in Pieri homotopy
! 49: numeric_minor_equations : evaluated equations in the Pieri homotopy
! 50: determinantal_systems : construction of equations from expansions
! 51: solve_pieri_leaves : compute solutions at leaves of Pieri tree
! 52: specialization_of_planes : set up the moving cycles
! 53: pieri_continuation : traces the paths defined by Pieri homotopy
! 54: plane_representations : conversion between vector and matrix reps
! 55: pieri_deformations : deforms down along the Pieri chains
! 56: ts_org_pieri : test of original Pieri homotopy algorithm
! 57: --------------------------------------------------------------------------------
! 58: localization_posets : posets of top-bottom pivots
! 59: localization_posets_io : output routines for localization posets
! 60: pieri_homotopies : construction of the Pieri homotopies
! 61: curves_into_grassmannian : representation of the q-curves
! 62: curves_into_grassmannian_io : output facilities for the q-curves
! 63: deformation_posets : posets of solution planes
! 64: ts_posets : test on creation of the posets
! 65: ts_defpos : test on deformation posets
! 66: drivers_for_input_planes : generators of input for Pieri algorithm
! 67: driver_for_pieri_homotopies : driver for the Pieri homotopies
! 68: driver_for_quantum_pieri : driver for counting q-maps
! 69: ts_pieri : test on the Pieri homotopies
! 70: mainenum : main driver as called by PHC
! 71: --------------------------------------------------------------------------------
! 72: The equations in the Groebner homotopies are set up by means of the
! 73: straightening algorithm. Because the Groebner homotopies are not efficient,
! 74: they are only auxiliary to the set up of the SAGBI homotopies, which arise
! 75: from expanding the brackets in the linear equations over the Grasmannian.
! 76:
! 77: Organization of the Pieri homotopies :
! 78:
! 79: The original Pieri Homotopy Algorithm is organized in a chain-by-chain
! 80: fashion, whereas the newer implementation uses posets.
! 81: The poset-oriented Pieri Homotopy Algorithm works better, but since most of
! 82: the basic packages is recycled, we can keep the original Pieri at low cost.
! 83: Both implementations have three distinct aspects:
! 84:
! 85: 1) combinatorial root count
! 86:
! 87: A. The Pieri tree delivers chains of increasing sequences of brackets.
! 88: Every chain models a sequence of nested subspaces, with lowering
! 89: dimensions towards the leaves of the tree. The set of input planes
! 90: is partitioned into two nonempty sets and one set containing only the
! 91: last input plane. For each of the two sets a Pieri tree is constructed.
! 92: A contribution to the root count is made by each pair of leaves that
! 93: satisfies Pieri's condition, for which a triple intersection of input
! 94: plane yields a start solution. The input to this combinatorial root
! 95: count is a partition of the list of codimensions. As output we obtain
! 96: a list of pairs of leaves that satisfy Pieri's condition.
! 97:
! 98: Ada soft : Pieri_Trees, Pieri_Trees_io, Pieri_Root_Counts
! 99:
! 100: B. The poset-oriented Pieri homotopy algorithm only needs one concept,
! 101: that of a localization pattern. This pattern is determined by a
! 102: couple of pivots indicating the topmost and bottommost nonzero entries.
! 103:
! 104: Ada soft : Localization_Posets
! 105:
! 106: 2) symbolic-numeric computation
! 107:
! 108: An intersection condition corresponds to the vanishing of all maximal
! 109: minors of a matrix which is not necessarily square.
! 110: Each matrix consists of two blocks, for coefficients and indeterminates.
! 111: The Laplace expansion into minors respects this division and is
! 112: written formally as a polynomial that is quadratic in brackets.
! 113: To obtain a polynomial system we evaluate the minors, this is the
! 114: numerical part in setting up the equations.
! 115: The zeros in the localization pattern are determined by the positioning
! 116: of the current node in the Pieri trees, but as far as the choice of the
! 117: ones is concerned, we can divide by the largest element in each column to
! 118: obtain a scaling and a numerically favorable representation of the equations.
! 119:
! 120: Only minor modifications were needed in the poset-oriented version.
! 121:
! 122: Ada soft : Symbolic_Minor_Equations, Numeric_Minor_Equations
! 123: Determinantal_Systems, Solve_Pieri_Leaves
! 124:
! 125: 3) deforming geometries
! 126:
! 127: A. The pairs of chains that end in pairs of leaves that satisfy Pieri's
! 128: condition form the backbone of the deformations in the Pieri homotopy
! 129: algorithm. The deformations start at the pairs of leaves with a triple
! 130: intersection and move down in the Pieri trees until they reach the
! 131: lowest node where the all intersection conditions are satisfied.
! 132: While moving down we move to larger subspaces which admits the intersection
! 133: with more input planes. In the case where all codimensions equal one,
! 134: at each node the current solution plane satisfies one more nontrivial
! 135: intersection requirement with an additional input plane.
! 136:
! 137: Ada soft : Pieri_Homotopies, Pieri_Continuation, Pieri_Deformations
! 138:
! 139: B. The continuous analogue to the localization posets are deformation posets.
! 140: These posets contain in their nodes the solution p-planes at that level.
! 141:
! 142: Ada soft : Deformation_Posets
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