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SVDComplexes :: laplacians

laplacians -- compute the laplacians of a chain complex

Synopsis

Description

For a chain complex over RR defined by matrices A_i=C.dd_i the i-th laplacian is defined by delta#i = transpose(A_i)*A_i+A_{i+1}*transpose A_{i+1}.

i1 : needsPackage "RandomComplexes"

o1 = RandomComplexes

o1 : Package
i2 : setRandomSeed "a good example";
i3 : h={2,3,5,3}

o3 = {2, 3, 5, 3}

o3 : List
i4 : r={4,3,5}

o4 = {4, 3, 5}

o4 : List
i5 : C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)

       6       10       13       8
o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
                                
     0       1        2        3

o5 : ChainComplex
i6 : C.dd^2

           6          13
o6 = 0 : ZZ  <----- ZZ   : 2
                0

           10          8
     1 : ZZ   <----- ZZ  : 3
                 0

o6 : ChainComplexMap
i7 : D=disturb(C**RR_53,1e-4)

         6         10         13         8
o7 = RR    <-- RR     <-- RR     <-- RR
       53        53         53         53
                                      
     0         1          2          3

o7 : ChainComplex
i8 : delta=laplacians D

o8 = HashTable{0 => | 40968300  -18096200 12519600 -6333510 -8778030  -9612320  |                                                                                                  }
                    | -18096200 18114000  -6139670 1920640  2483360   2721690   |
                    | 12519600  -6139670  10491100 9492560  4486190   -7276010  |
                    | -6333510  1920640   9492560  25114500 3648080   4334860   |
                    | -8778030  2483360   4486190  3648080  31627800  -25178700 |
                    | -9612320  2721690   -7276010 4334860  -25178700 28913500  |
               1 => | 2681610000  2770930000  -3585750000 2744420000  -2747230000 -21682100  1458620000 1217550000  390139000   2132970000  |
                    | 2770930000  5842380000  -8192680000 4634410000  -2551080000 -974718000 578651000  3309520000  2353860000  1651270000  |
                    | -3585750000 -8192680000 14739600000 -6.772e9    3555930000  2588810000 998672000  -2932580000 -3974070000 435634000   |
                    | 2744420000  4634410000  -6.772e9    3960790000  -2681910000 -718188000 745035000  2301530000  1657420000  1569670000  |
                    | -2747230000 -2551080000 3555930000  -2681910000 2916820000  29046000   -1.466e9   -872977000  -279468000  -2021300000 |
                    | -21682100   -974718000  2588810000  -718188000  29046000    730696000  840960000  17571600    -793548000  1062580000  |
                    | 1458620000  578651000   998672000   745035000   -1.466e9    840960000  1898870000 897709000   -631797000  2598700000  |
                    | 1217550000  3309520000  -2932580000 2301530000  -872977000  17571600   897709000  3080210000  1256510000  2039920000  |
                    | 390139000   2353860000  -3974070000 1657420000  -279468000  -793548000 -631797000 1256510000  1439290000  -425770000  |
                    | 2132970000  1651270000  435634000   1569670000  -2021300000 1062580000 2598700000 2039920000  -425770000  3825220000  |
               2 => | 1458050000  -565025000  -2399920000 164373000   1463470000  -1218450000 1338170000  -332270000  3104280000  -1940640000 -1143350000 -103255000  -657007000  |
                    | -565025000  695447000   1558820000  -367865000  -1064420000 936635000   -1290720000 -170515000  -1888480000 1369690000  121074000   118554000   624418000   |
                    | -2399920000 1558820000  5831910000  -825802000  -2828350000 3050640000  -3519650000 -333619000  -6412780000 4252030000  1702200000  484021000   1717320000  |
                    | 164373000   -367865000  -825802000  2027060000  380169000   -1042440000 2322330000  1443550000  2124670000  -1564960000 1181680000  -1765030000 -1188790000 |
                    | 1463470000  -1064420000 -2828350000 380169000   3472310000  -815682000  2487760000  844816000   3498870000  -2414930000 -311741000  -543504000  -2098280000 |
                    | -1218450000 936635000   3050640000  -1042440000 -815682000  2586970000  -2054670000 250788000   -3675360000 2338900000  1093810000  217784000   306884000   |
                    | 1338170000  -1290720000 -3519650000 2322330000  2487760000  -2054670000 4378470000  1921520000  5550280000  -3873580000 833202000   -1786720000 -2488120000 |
                    | -332270000  -170515000  -333619000  1443550000  844816000   250788000   1921520000  2448470000  775936000   -852074000  1931870000  -1928530000 -1857690000 |
                    | 3104280000  -1888480000 -6412780000 2124670000  3498870000  -3675360000 5550280000  775936000   9723090000  -6185740000 -1059630000 -592912000  -2395660000 |
                    | -1940640000 1369690000  4252030000  -1564960000 -2414930000 2338900000  -3873580000 -852074000  -6185740000 4230560000  235472000   1113840000  1977350000  |
                    | -1143350000 121074000   1702200000  1181680000  -311741000  1093810000  833202000   1931870000  -1059630000 235472000   2668450000  -1578040000 -1066020000 |
                    | -103255000  118554000   484021000   -1765030000 -543504000  217784000   -1786720000 -1928530000 -592912000  1113840000  -1578040000 3536380000  1930360000  |
                    | -657007000  624418000   1717320000  -1188790000 -2098280000 306884000   -2488120000 -1857690000 -2395660000 1977350000  -1066020000 1930360000  2277050000  |
               3 => | 373590000  238254000  -50812600  -437085000 74181500   -41332800 37454200   -393315000 |
                    | 238254000  698531000  -480720000 -382529000 -162466000 200661000 144062000  -388569000 |
                    | -50812600  -480720000 464199000  75246600   279787000  -77707800 -17571400  353222000  |
                    | -437085000 -382529000 75246600   891490000  -44759100  141150000 -407019000 316331000  |
                    | 74181500   -162466000 279787000  -44759100  282194000  112362000 -8247220   195236000  |
                    | -41332800  200661000  -77707800  141150000  112362000  381327000 -46660400  149778000  |
                    | 37454200   144062000  -17571400  -407019000 -8247220   -46660400 357298000  129866000  |
                    | -393315000 -388569000 353222000  316331000  195236000  149778000 129866000  925279000  |

o8 : HashTable
i9 : L0=(SVD delta#0)_0, L1=(SVD delta#1)_0,L2=(SVD delta#2)_0,L3=(SVD delta#3)_0

o9 = ({60648900}, {28210500000}, {28210500000}, {2056900000})
      {55489200}  {9617270000 }  {9617270000 }  {1028620000}
      {29990300}  {3132530000 }  {3132530000 }  {754460000 }
      {9100710 }  {60649000   }  {2056900000 }  {484906000 }
      {.327165 }  {55489300   }  {1028620000 }  {49026600  }
      {.0102018}  {29990300   }  {754460000  }  {3.7497    }
                  {9100740    }  {484906000  }  {2.9526    }
                  {38.8744    }  {49026600   }  {1.35595   }
                  {21.2473    }  {51.0793    }
                  {5.85738    }  {29.3294    }
                                 {20.8771    }
                                 {3.48837    }
                                 {2.04772    }

o9 : Sequence
i10 : commonEntries(L0,L1)

o10 = ({0, 1, 2, 3}, {3, 4, 5, 6})

o10 : Sequence
i11 : commonEntries(L1,L2)

o11 = ({0, 1, 2}, {0, 1, 2})

o11 : Sequence
i12 : commonEntries(L2,L3)

o12 = ({3, 4, 5, 6, 7}, {0, 1, 2, 3, 4})

o12 : Sequence

Caveat

See also

Ways to use laplacians :

For the programmer

The object laplacians is a method function.