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Annotation of OpenXM_contrib/PHC/Ada/Schubert/sagbi_results, Revision 1.1

1.1     ! maekawa     1: Intersection of (n-d)*d random real (n-d-1)-planes in complex n-space.
        !             2:
        !             3: Root Counts :
        !             4:
        !             5: --------------------------------
        !             6:   n\d   2     3    4    5  6  7
        !             7: --------------------------------
        !             8:    4    2     1
        !             9:    5    5     5
        !            10:    6   14    42   14
        !            11:    7   42   462        42
        !            12:    8  132  6006
        !            13:    9  429
        !            14: --------------------------------
        !            15:
        !            16: Timings on SunOS mars 5.6 Generic sun4u sparc SUNW,Ultra-1 :
        !            17:
        !            18:   Set-up : syzygies, bracket expansions, lifting to set up SAGBI homotopy
        !            19:   Volume : dynamic lifting for triangulation and volume computation
        !            20:   Polcon : polyhedral continuation to solve complex random system
        !            21:   Fladef : flat deformation to solve random complex problem instance
        !            22:   Cheat  : cheater's homotopy from random complex to random float
        !            23:   Total  : total cpu time for solving the problem
        !            24:
        !            25: -----------------------------------------------------------------------------
        !            26:   d=2  :  n = 4   n = 5     n = 6     n = 7        n = 8          n = 9
        !            27: ----------------------------------------------------------------------------
        !            28: #vars  :    4       6         8        10           12             14
        !            29: #sols  :    2       5        14        42          132            429
        !            30: -----------------------------------------------------------------------------
        !            31: Set-up :   30ms  0s 60ms   0s170ms  0m 0s400ms   0m 0s790ms   0h 0m 1s580ms
        !            32: Volume :    0ms  0s 10ms   0s 40ms  0m 0s190ms   0m 1s310ms   0h 0m 8s890ms
        !            33: Polcon :   70ms  1s210ms   6s790ms  1m27s 10ms  10m16s880ms   1h15m 9s870ms
        !            34: Fladef :   60ms  1s290ms   6s820ms  1m 5s990ms   7m 5s120ms   1h 1m38s280ms
        !            35: Cheat  :  640ms  3s 90ms  26s740ms  2m33s570ms  22m24s270ms   2h14m33s590ms
        !            36: Total  :  800ms  5s660ms  40s570ms  5m 7s190ms  39m48s420ms   4h31m32s290ms
        !            37: -----------------------------------------------------------------------------
        !            38:   d=3  :  n = 4   n = 5     n = 6       n = 7        n = 8
        !            39: -----------------------------------------------------------------------------
        !            40: #vars  :    3       6         9          12           15
        !            41: #sols  :    1       5        42         462         6006
        !            42: -----------------------------------------------------------------------------
        !            43: Set-up :   30ms  0s140ms  0m 0s440ms
        !            44: Volume :    0ms  0s  0ms  0m 0s170ms
        !            45: Polcon :   20ms  0s600ms  1m 2s750ms
        !            46: Fladef :   10ms  0s540ms  0m50s590ms
        !            47: Cheat  :   30ms  1s830ms  2m47s910ms
        !            48: Total  :   90ms  3s120ms  4m41s910ms
        !            49: -----------------------------------------------------------------------------
        !            50:
        !            51: Observations:
        !            52:   1) setting up the equations and computing the volume is minor
        !            53:   2) cheater's homotopy is most time-consuming
        !            54:   3) the complexity of the problem increases rapidly as n and d grow