We compute the singular value decomposition either by the iterated Projections or by the Laplacian method. In case the input consists of two chainComplexes we use the iterated Projection method, and identify the stable singular values.
i1 : needsPackage "RandomComplexes" o1 = RandomComplexes o1 : Package |
i2 : h={1,3,5,2,1} o2 = {1, 3, 5, 2, 1} o2 : List |
i3 : r={5,11,3,2} o3 = {5, 11, 3, 2} o3 : List |
i4 : elapsedTime C=randomChainComplex(h,r,Height=>4) -- 0.005468 seconds elapsed 6 19 19 7 3 o4 = ZZ <-- ZZ <-- ZZ <-- ZZ <-- ZZ 0 1 2 3 4 o4 : ChainComplex |
i5 : C.dd^2 6 19 o5 = 0 : ZZ <----- ZZ : 2 0 19 7 1 : ZZ <----- ZZ : 3 0 19 3 2 : ZZ <----- ZZ : 4 0 o5 : ChainComplexMap |
i6 : CR=(C**RR_53)[1] 6 19 19 7 3 o6 = RR <-- RR <-- RR <-- RR <-- RR 53 53 53 53 53 -1 0 1 2 3 o6 : ChainComplex |
i7 : elapsedTime (h,U)=SVDComplex CR; -- 0.00190757 seconds elapsed |
i8 : h o8 = HashTable{-1 => 1} 0 => 3 1 => 5 2 => 2 3 => 1 o8 : HashTable |
i9 : Sigma =source U 6 19 19 7 3 o9 = RR <-- RR <-- RR <-- RR <-- RR 53 53 53 53 53 -1 0 1 2 3 o9 : ChainComplex |
i10 : Sigma.dd_0 o10 = | 20.7789 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 18.3883 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 9.51991 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 7.19109 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 2.40088 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 6 19 o10 : Matrix RR <--- RR 53 53 |
i11 : errors=apply(toList(min CR+1..max CR),ell->CR.dd_ell-U_(ell-1)*Sigma.dd_ell*transpose U_ell); |
i12 : maximalEntry chainComplex errors o12 = {6.66134e-15, 1.56319e-13, 1.06581e-13, 1.24345e-14} o12 : List |
i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian); -- 0.00522536 seconds elapsed |
i14 : hL === h o14 = true |
i15 : SigmaL =source U; |
i16 : for i from min CR+1 to max CR list maximalEntry(SigmaL.dd_i -Sigma.dd_i) o16 = {1.06581e-14, 5.68434e-14, 8.52651e-14, 3.55271e-15} o16 : List |
i17 : errors=apply(toList(min C+1..max C),ell->CR.dd_ell-U_(ell-1)*SigmaL.dd_ell*transpose U_ell); |
i18 : maximalEntry chainComplex errors o18 = {3.02425e-13, 9.41469e-14, 3.3884e-13, -infinity} o18 : List |
The optional argument
The algorithm might fail if the condition numbers of the differential are too bad