# laplacians -- compute the laplacians of a chain complex

## Synopsis

• Usage:
delta=laplacians C
• Inputs:
• C, , defined over RR
• Outputs:
• delta, , of the laplacians

## 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