random(DiffAlgElement,Ring) -- replaces the variables of the coefficient ring of a differential form or a vector field with random values

Synopsis

Function: random
Usage:

random(e,R)
Inputs:
- e, an instance of the type DiffAlgElement, a differential form or vector field with generic coefficients
- R, a ring, the ring where the random values are taken from. Default value is an integer
Optional inputs:
- Density => ..., default value 1,
- Height => ..., default value 10,
- MaximalRank => ..., default value false,
- UpperTriangular => ..., default value false,
Outputs:
- an instance of the type DiffAlgElement, the differential form or vector field e whose variables of the coefficient ring were evaluated at random values

Description

i1 : random newForm(2,2,1,"a")

o1 = - x dx dx  + (- 4x  + 3x  + x )dx dx  + (- 4x  + x  + 5x )dx dx
        1  0  1        0     1    2   0  2        0    1     2   1  2

o1 : DiffAlgForm

i2 : random(newField(2,2,"a"),QQ)

      22 2   1        5 2   18       28       23 2          43 2   23       1 2           5        9 2          11 2          
o2 = (--x  + -x x  - --x  - --x x  + --x x  + --x )ax  + (- --x  - --x x  + -x  + x x  - --x x  - --x )ax  + (- --x  + 4x x  -
      63 0   5 0 1   24 1    7 0 2    5 1 2    5 2   0      10 0    3 0 1   6 1    0 2   12 1 2   10 2   1      18 0     0 1  
     ----------------------------------------------------------------------------------------------------------------------------
       2   10       59       1 2
     3x  - --x x  - --x x  - -x )ax
       1   21 0 2   18 1 2   4 2   2

o2 : DiffAlgField

Options Density and Height are implemented

i3 : random(newForm(2,2,1,"a"),Density => .2)

o3 = (- 4x  - 8x )dx dx  + (- 6x  + 4x )dx dx
          1     2   0  1        0     2   0  2

o3 : DiffAlgForm

i4 : random(newForm(2,2,1,"a"),Height => 100)

o4 = (58x  + 73x  - 54x )dx dx  + (- 56x  + 31x  + 48x )dx dx  + (32x  + 38x  - 4x )dx dx
         0      1      2   0  1         0      1      2   0  2       0      1     2   1  2

o4 : DiffAlgForm

Caveat

This routine depends on the Macaulay2 method random which is not implemented in some rings. The option random(...,Density=>...) applied to a non-generic form (such us 2a logarithmic form) may return a form equal to 0