Annotation of OpenXM_contrib/PHC/Ada/Schubert/READ_ME, Revision 1.1.1.1
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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