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)
-- 0.00559409 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;
-- 0.00176564 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);
-- 0.00393523 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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