SVDHomology -- Estimate the homology of a chainComplex over RR with the SVD decomposition

Usage:

(h,h1)=SVDHomology C or

(h,h1)=SVDHomology(C,C')
Inputs:
- C, a chain complex, over RR_{53}
- C', a chain complex, in a lower precision
Optional inputs:
- Strategy => a symbol, default value Projection, Laplacian or Projection for the method used
- Threshold => a real number, default value .0001, the relative threshold used to detect the zero singular values
Outputs:
- h, a hash table, the dimensions of the homology groups HH C
- h1, a hash table, information about the singular values

Description

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

i2 : h={1,3,5,2}

o2 = {1, 3, 5, 2}

o2 : List

i3 : r={4,3,3}

o3 = {4, 3, 3}

o3 : List

i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
 -- .00337739s elapsed

       5       10       11       5
o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
                                
     0       1        2        3

o4 : ChainComplex

i5 : C.dd^2

           5          11
o5 = 0 : ZZ  <----- ZZ   : 2
                0

           10          5
     1 : ZZ   <----- ZZ  : 3
                 0

o5 : ChainComplexMap

i6 : CR=(C**RR_53)

         5         10         11         5
o6 = RR    <-- RR     <-- RR     <-- RR
       53        53         53         53
                                      
     0         1          2          3

o6 : ChainComplex

i7 : elapsedTime (h,h1)=SVDHomology CR
 -- .00106244s elapsed

o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, )           })
                1 => 3             2 => (37.9214, 30.3707, 4.44441e-15)
                2 => 5             3 => (14.972, 8.57847, 2.56036e-15)
                3 => 2

o7 : Sequence

i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
 -- .00180359s elapsed

o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.49161e-14)      })
                1 => 3             1 => (1.71747, 922.381, 6.04463e-13)
                2 => 5             2 => (922.381, 73.5901, 1.85733e-13)
                3 => 2             3 => (73.5901, , 5.10357e-14)

o8 : Sequence

i9 : hL === h

o9 = true

i10 : (h1#1_1)^2, hL1#1_0, (h1#1_1)^2-hL1#1_0

o10 = (1.71747, 1.71747, -3.83693e-13)

o10 : Sequence

i11 : (h1#2_1)^2, hL1#2_0, (h1#2_1)^2-hL1#2_0

o11 = (922.381, 922.381, -9.09495e-13)

o11 : Sequence

i12 : (h1#3_1)^2, hL1#3_0, (h1#3_1)^2-hL1#3_0

o12 = (73.5901, 73.5901, -1.42109e-14)

o12 : Sequence

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

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

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     
      -----------------------------------------------------------------------
      |
      |
      |
      |
      |

                 5         10
o15 : Matrix RR    <-- RR
               53        53

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, 4.44441e-15)
                 2 => 5             3 => (14.972, 8.57847, 2.56036e-15)
                 3 => 2

o16 : Sequence

i17 : hd === h

o17 = true

i18 : hd1 === h1

o18 = true

Caveat

The algorithm might fail if the condition numbers of the differential are too bad

Ways to use SVDHomology:

SVDHomology(ChainComplex)
SVDHomology(ChainComplex,ChainComplex)

For the programmer

The object SVDHomology is a method function with options.

The source of this document is in SVDComplexes.m2:1166:0.

SVDHomology -- Estimate the homology of a chainComplex over RR with the SVD decomposition

Description

Caveat

See also

Ways to use SVDHomology:

For the programmer