SVDComplex -- Compute the SVD decomposition of a chain complex over RR

Usage:

(h,U)=SVDComplex C or

(h,U)=SVDComplex(C,C')
Inputs:
- C, a complex, over RR_{53}
- C', a 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
- U, a map of complexes, a map C <- Sigma where the source is the chain complex of the singular value matrices and U is given by orthogonal matrices

Description

We compute the singular value decomposition either by the iterated Projections or by the Laplacian method. In case the input consists of two chain complexes 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)
 -- .0118577s elapsed

       6       19       19       7       3
o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
                                        
     0       1        2        3       4

o4 : Complex

i5 : C.dd^2

o5 = 0

o5 : ComplexMap

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 : Complex

i7 : elapsedTime (h,U)=SVDComplex CR;
 -- .00425676s 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 : Complex

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 complex errors

o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15}

o12 : List

i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
 -- .0643167s 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 = {1.77636e-14, 6.39488e-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 complex errors

o18 = {2.37144e-13, 1.42109e-13, 2.63678e-13, -infinity}

o18 : List

The optional argument

Caveat

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

Ways to use SVDComplex:

SVDComplex(Complex)
SVDComplex(Complex,Complex)

For the programmer

The object SVDComplex is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SVDComplexes.m2:1068:0.