In case the field is RR we use the SVD normal form to compute the pseudo inverse. In case of QQ we compute the pseudo inverse directly over QQ.
i1 : needsPackage "RandomComplexes" o1 = RandomComplexes o1 : Package |
i2 : setRandomSeed "a pretty good example"; |
i3 : h={2,3,1}
o3 = {2, 3, 1}
o3 : List
|
i4 : r={2,3}
o4 = {2, 3}
o4 : List
|
i5 : C=randomChainComplex(h,r,Height=>11,ZeroMean=>true)
4 8 4
o5 = ZZ <-- ZZ <-- ZZ
0 1 2
o5 : ChainComplex
|
i6 : C.dd
4 8
o6 = 0 : ZZ <------------------------------------- ZZ : 1
| -4 -18 -6 -14 -10 -18 -16 -28 |
| 8 16 22 -12 -20 -14 -8 6 |
| 6 15 15 -3 -9 -3 0 12 |
| 2 7 4 3 1 4 4 9 |
8 4
1 : ZZ <----------------------- ZZ : 2
| -5 22 19 -41 |
| -54 26 55 -66 |
| 31 -26 -29 25 |
| 33 50 -23 36 |
| -20 -16 24 -47 |
| 2 -22 -1 -10 |
| -12 -6 3 9 |
| 25 -16 -30 43 |
o6 : ChainComplexMap
|
i7 : CQ=C**QQ
4 8 4
o7 = QQ <-- QQ <-- QQ
0 1 2
o7 : ChainComplex
|
i8 : CR=C**RR_53
4 8 4
o8 = RR <-- RR <-- RR
53 53 53
0 1 2
o8 : ChainComplex
|
i9 : CRplus = pseudoInverse CR
4 8 4
o9 = RR <-- RR <-- RR
53 53 53
-2 -1 0
o9 : ChainComplex
|
i10 : CQplus = pseudoInverse CQ
4 8 4
o10 = QQ <-- QQ <-- QQ <-- 0
-2 -1 0 1
o10 : ChainComplex
|
i11 : CRplus.dd
4 8
o11 = -2 : RR <------------------------------------------------------------------------------------------------- RR : -1
53 | .0130658 -.00710013 .00910891 .00617097 .00616535 .00510154 -.0108838 -.00160172 | 53
| .00363228 .00323956 -.00408194 .0104004 -.00386761 -.00421635 -.00125529 -.00211786 |
| -.00265724 .00380405 -.00358933 -.0016806 -.00115101 -.00166389 .00291106 -.000639162 |
| -.0129721 .00195488 -.00650979 -.000997146 -.00888455 -.0055727 .0094004 .00387383 |
8 4
-1 : RR <--------------------------------------------------- RR : 0
53 | -.00131432 .00348016 .0024824 .000742313 | 53
| -.00687747 .00658955 .00601697 .0027222 |
| -.00148997 .00975584 .0063005 .00142258 |
| -.00652617 -.0059618 -.00161923 .00136167 |
| -.00521185 -.00944196 -.00410162 .000619358 |
| -.008322 -.00701722 -.00171373 .00179488 |
| -.00718333 -.00422171 -.00037803 .00173283 |
| -.0116078 .00168319 .00449225 .00365066 |
o11 : ChainComplexMap
|
i12 : CQplus.dd
4 8
o12 = -2 : QQ <----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- QQ : -1
| 22455062/1718617773 -305060062/42965444325 391368268/42965444325 88379471/14321814775 264897193/42965444325 219189842/42965444325 -467627593/42965444325 -2219954/1385982075 |
| 2080832/572872591 46396386/14321814775 -58460804/14321814775 148952386/14321814775 -55391154/14321814775 -60385726/14321814775 -17977996/14321814775 -978438/461994025 |
| -1522260/572872591 54480883/14321814775 -51405662/14321814775 -24069267/14321814775 -16484537/14321814775 -23829853/14321814775 41691712/14321814775 -295289/461994025 |
| -22294079/1718617773 83992087/42965444325 -279696193/42965444325 -14280946/14321814775 -381728443/42965444325 -239433617/42965444325 403892368/42965444325 5369054/1385982075 |
8 4
-1 : QQ <---------------------------------------------------------------- QQ : 0
| -6884/5237683 18228/5237683 1182/476153 3888/5237683 |
| -36022/5237683 34514/5237683 2865/476153 14258/5237683 |
| -7804/5237683 51098/5237683 3000/476153 7451/5237683 |
| -34182/5237683 -31226/5237683 -771/476153 7132/5237683 |
| -27298/5237683 -49454/5237683 -1953/476153 3244/5237683 |
| -2564/308099 -2162/308099 -48/28009 553/308099 |
| -37624/5237683 -22112/5237683 -180/476153 9076/5237683 |
| -60798/5237683 8816/5237683 2139/476153 19121/5237683 |
4
0 : QQ <----- 0 : 1
0
o12 : ChainComplexMap
|
i13 : (CQplus**RR_53).dd
4 8
o13 = -2 : RR <------------------------------------------------------------------------------------------------- RR : -1
53 | .0130658 -.00710013 .00910891 .00617097 .00616535 .00510154 -.0108838 -.00160172 | 53
| .00363228 .00323956 -.00408194 .0104004 -.00386761 -.00421635 -.00125529 -.00211786 |
| -.00265724 .00380405 -.00358933 -.0016806 -.00115101 -.00166389 .00291106 -.000639162 |
| -.0129721 .00195488 -.00650979 -.000997146 -.00888455 -.0055727 .0094004 .00387383 |
8 4
-1 : RR <--------------------------------------------------- RR : 0
53 | -.00131432 .00348016 .0024824 .000742313 | 53
| -.00687747 .00658955 .00601697 .0027222 |
| -.00148997 .00975584 .0063005 .00142258 |
| -.00652617 -.0059618 -.00161923 .00136167 |
| -.00521185 -.00944196 -.00410162 .000619358 |
| -.008322 -.00701722 -.00171373 .00179488 |
| -.00718333 -.00422171 -.00037803 .00173283 |
| -.0116078 .00168319 .00449225 .00365066 |
4
0 : RR <----- 0 : 1
53 0
o13 : ChainComplexMap
|
i14 : CRplus.dd^2
4 4
o14 = -2 : RR <----------------------------------------------------------- RR : 0
53 | -4.06576e-20 -8.13152e-20 -4.06576e-20 5.0822e-21 | 53
| -3.81165e-20 1.35525e-20 1.35525e-20 9.74088e-21 |
| -3.38813e-21 3.55754e-20 1.94818e-20 2.96462e-21 |
| 2.37169e-20 3.38813e-20 2.03288e-20 -1.01644e-20 |
o14 : ChainComplexMap
|
i15 : CQplus.dd^2
4 4
o15 = -2 : QQ <----- QQ : 0
0
8
-1 : QQ <----- 0 : 1
0
o15 : ChainComplexMap
|
Pseudo inverses frequently exist also over finite fields.
i16 : Fp=ZZ/nextPrime 10^3 o16 = Fp o16 : QuotientRing |
i17 : Cp=C**Fp
4 8 4
o17 = Fp <-- Fp <-- Fp
0 1 2
o17 : ChainComplex
|
i18 : Cpplus=pseudoInverse Cp
4 8 4
o18 = Fp <-- Fp <-- Fp <-- 0
-2 -1 0 1
o18 : ChainComplex
|
i19 : Cpplus.dd
4 8
o19 = -2 : Fp <---------------------------------------------- Fp : -1
| -503 196 106 -420 -383 278 -1 308 |
| -447 -399 -485 370 -440 -273 157 266 |
| -56 -279 92 145 65 351 358 -70 |
| 166 39 -496 -313 -356 126 276 -330 |
8 4
-1 : Fp <--------------------------- Fp : 0
| -33 -170 -193 -108 |
| 384 -230 -455 -340 |
| 441 -18 -244 -235 |
| 445 475 -353 -86 |
| 478 -364 -160 22 |
| 426 68 -87 -121 |
| -76 390 55 -140 |
| -161 -357 -65 -391 |
4
0 : Fp <----- 0 : 1
0
o19 : ChainComplexMap
|
i20 : Cpplus.dd^2
4 4
o20 = -2 : Fp <----- Fp : 0
0
8
-1 : Fp <----- 0 : 1
0
o20 : ChainComplexMap
|
i21 : arePseudoInverses(Cp,Cpplus) o21 = true |
Over finite fields the algorithm can fail if toDo what happens.