We compute the singular value decomposition either by the iterated Projections or by the Laplacian method. In case of the projection method we record in h1 the last two nonzero singular values and first singular value expected to be really zero.
In case of the Laplacian method we record in h1 the smallest common Eigenvalues of the neighboring Laplacians, and the first Eigenvalue expected to be zero.
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}
o2 = {1, 3, 5, 2}
o2 : List
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i3 : r={4,3,3}
o3 = {4, 3, 3}
o3 : List
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i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
-- 0.00478094 seconds elapsed
5 10 11 5
o4 = ZZ <-- ZZ <-- ZZ <-- ZZ
0 1 2 3
o4 : ChainComplex
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i5 : C.dd^2
5 11
o5 = 0 : ZZ <----- ZZ : 2
0
10 5
1 : ZZ <----- ZZ : 3
0
o5 : ChainComplexMap
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i6 : CR=(C**RR_53)
5 10 11 5
o6 = RR <-- RR <-- RR <-- RR
53 53 53 53
0 1 2 3
o6 : ChainComplex
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i7 : elapsedTime (h,h1)=SVDHomology CR
-- 0.00123679 seconds elapsed
o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, ) })
1 => 3 2 => (37.9214, 30.3707, 7.11744e-15)
2 => 5 3 => (14.972, 8.57847, 3.25752e-15)
3 => 2
o7 : Sequence
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i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
-- 0.0025883 seconds elapsed
o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14) })
1 => 3 1 => (1.71747, 922.381, 2.51496e-13)
2 => 5 2 => (922.381, 73.5901, 1.81323e-13)
3 => 2 3 => (73.5901, , 2.82914e-13)
o8 : Sequence
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i9 : hL === h
o9 = true
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i10 : (h1#1_1)^2, hL1#1_0, (h1#1_1)^2-hL1#1_0
o10 = (1.71747, 1.71747, -3.9968e-15)
o10 : Sequence
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i11 : (h1#2_1)^2, hL1#2_0, (h1#2_1)^2-hL1#2_0
o11 = (922.381, 922.381, -1.13687e-13)
o11 : Sequence
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i12 : (h1#3_1)^2, hL1#3_0, (h1#3_1)^2-hL1#3_0
o12 = (73.5901, 73.5901, 2.84217e-14)
o12 : Sequence
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i13 : D=disturb(C,1e-3,Strategy=>Discrete)
5 10 11 5
o13 = RR <-- RR <-- RR <-- RR
53 53 53 53
0 1 2 3
o13 : ChainComplex
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i14 : C.dd_1
o14 = | -1 -1 -5 -3 -4 -2 3 -3 7 -1 |
| -5 -2 -1 5 -3 1 5 4 3 0 |
| 1 -3 5 5 0 3 4 3 -9 -3 |
| 0 -3 -4 -2 -5 -1 6 -3 4 -3 |
| -1 -2 3 5 1 3 3 4 -5 0 |
5 10
o14 : Matrix ZZ <-- ZZ
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i15 : D.dd_1
o15 = | -.999 -1.001 -4.995 -2.997 -3.996 -2.002 2.997 -3.003 6.993 -.999
| -5.005 -2.002 -.999 4.995 -2.997 1.001 5.005 3.996 3.003 0
| .999 -3.003 5.005 4.995 0 3.003 4.004 3.003 -8.991 -3.003
| 0 -2.997 -3.996 -2.002 -4.995 -1.001 6.006 -3.003 4.004 -3.003
| -1.001 -2.002 2.997 5.005 1.001 2.997 3.003 3.996 -4.995 0
-----------------------------------------------------------------------
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5 10
o15 : Matrix RR <-- RR
53 53
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i16 : (hd,hd1)=SVDHomology(CR,D,Threshold=>1e-2)
o16 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, ) })
1 => 3 2 => (37.9214, 30.3707, 7.11744e-15)
2 => 5 3 => (14.972, 8.57847, 3.25752e-15)
3 => 2
o16 : Sequence
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i17 : hd === h
o17 = true
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i18 : hd1 === h1
o18 = true
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