pseudoInverse -- compute the pseudoInverse of a chain complex

Usage:

Cplus = pseudoInverse C
Inputs:
- C, a complex, an approximate complex over an field
Optional inputs:
- Strategy (missing documentation) => a symbol, default value Projection, Laplacian or Projection for the method used
- Threshold (missing documentation) => a real number, default value .0001, the relative threshold used to detect the zero singular values
Outputs:
- Cplus, a complex, the pseudo inverse complex

Description

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";
 -- setting random seed to 1187585617994339702977441899450513636117846

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 : Complex

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 : ComplexMap

i7 : CQ=C**QQ

       4       8       4
o7 = QQ  <-- QQ  <-- QQ
                      
     0       1       2

o7 : Complex

i8 : CR=C**RR_53

         4         8         4
o8 = RR    <-- RR    <-- RR
       53        53        53
                          
     0         1         2

o8 : Complex

i9 : CRplus = pseudoInverse CR

         4         8         4
o9 = RR    <-- RR    <-- RR
       53        53        53
                          
     -2        -1        0

o9 : Complex

i10 : CQplus = pseudoInverse CQ

        4       8       4
o10 = QQ  <-- QQ  <-- QQ
                       
      -2      -1      0

o10 : Complex

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 : ComplexMap

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 |

o12 : ComplexMap

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  |

o13 : ComplexMap

i14 : CRplus.dd^2

               4                                                                 4
o14 = -2 : RR    <---------------------------------------------------------- RR    : 0
             53     | -1.15196e-19 -1.35525e-20 2.03288e-20 2.5411e-20   |     53
                    | -4.23516e-21 1.86347e-20  8.47033e-21 4.23516e-21  |
                    | 2.32934e-20  6.77626e-21  -2.5411e-21 -4.65868e-21 |
                    | 8.80914e-20  2.71051e-20  0           -1.69407e-20 |

o14 : ComplexMap

i15 : CQplus.dd^2

o15 = 0

o15 : ComplexMap

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 : Complex

i18 : Cpplus=pseudoInverse Cp

        4       8       4
o18 = Fp  <-- Fp  <-- Fp
                       
      -2      -1      0

o18 : Complex

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 |

o19 : ComplexMap

i20 : Cpplus.dd^2

o20 = 0

o20 : ComplexMap

i21 : arePseudoInverses(Cp,Cpplus)

o21 = true

Caveat

Over finite fields the algorithm can fail.

Ways to use pseudoInverse:

pseudoInverse(Complex)

For the programmer

The object pseudoInverse is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SVDComplexes.m2:1296:0.

pseudoInverse -- compute the pseudoInverse of a chain complex

Description

Caveat

See also

Ways to use pseudoInverse:

For the programmer