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
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i2 : h={1,3,5,2,1}
o2 = {1, 3, 5, 2, 1}
o2 : List
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i3 : r={5,11,3,2}
o3 = {5, 11, 3, 2}
o3 : List
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i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
-- 0.0110657 seconds elapsed
6 19 19 7 3
o4 = ZZ <-- ZZ <-- ZZ <-- ZZ <-- ZZ
0 1 2 3 4
o4 : ChainComplex
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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
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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
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i7 : elapsedTime (h,U)=SVDComplex CR;
-- 0.00333896 seconds elapsed
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i8 : h
o8 = HashTable{-1 => 1}
0 => 3
1 => 5
2 => 2
3 => 1
o8 : HashTable
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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
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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
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i11 : errors=apply(toList(min CR+1..max CR),ell->CR.dd_ell-U_(ell-1)*Sigma.dd_ell*transpose U_ell);
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i12 : maximalEntry chainComplex errors
o12 = {6.21725e-15, 3.2685e-13, 1.38556e-13, 3.55271e-15}
o12 : List
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i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
-- 0.00773122 seconds elapsed
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i14 : hL === h
o14 = true
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i15 : SigmaL =source U;
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i16 : for i from min CR+1 to max CR list maximalEntry(SigmaL.dd_i -Sigma.dd_i)
o16 = {1.77636e-14, 2.27374e-13, 5.68434e-14, 1.42109e-14}
o16 : List
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i17 : errors=apply(toList(min C+1..max C),ell->CR.dd_ell-U_(ell-1)*SigmaL.dd_ell*transpose U_ell);
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i18 : maximalEntry chainComplex errors
o18 = {2.18492e-13, 8.83738e-14, 2.66009e-13, -infinity}
o18 : List
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