random(Module,Module) -- make a random module map

Synopsis

Usage:
f = random(F,G)
Function: random
Inputs:
- F, a module, a free module
- G, a module, a free module
Optional inputs:
- MaximalRank => default value false, whether to ensure that the resulting map has maximal rank: designed mostly for use with matrices of numbers: for polynomial rings, returns inhomogeneous results
- Density => a real number, default value 1, the proportion of entries to set
- UpperTriangular => a Boolean value, default value false, whether to set just entries strictly above the diagonal
- Height => ..., -- random element of a type
Outputs:
- f, a matrix, a random, graded, degree 0 map, from G to F

Description

i1 : R = ZZ/101[x,y];

i2 : random(R^{1,2,3},R^{1,2,3})

o2 = {-1} | 46            0     0  |
     {-2} | -4x-32y       -16   0  |
     {-3} | 13x2-48xy+6y2 3x-2y 12 |

             3       3
o2 : Matrix R  <--- R

i3 : random(ZZ^3,ZZ^6,MaximalRank=>true)

o3 = | 274707616080  1591144264  423238820  6443175196   1109615730 
     | 704192910335  4078782111  1084941802 16516609034  2844418879 
     | 5460329713322 31626980089 8412666293 128070205971 22055696242
     ------------------------------------------------------------------------
     54482082347   |
     139660838957  |
     1082933692671 |

              3        6
o3 : Matrix ZZ  <--- ZZ

i4 : random(ZZ^3,ZZ^10,Density=>.3)

o4 = | 0 0 0 0 0 0 0 0 5 0 |
     | 0 0 0 2 0 0 0 0 0 8 |
     | 3 7 0 0 0 8 0 0 0 7 |

              3        10
o4 : Matrix ZZ  <--- ZZ

i5 : random(ZZ^6,ZZ^6,UpperTriangular=>true)

o5 = | 0 4 8 9 2 2 |
     | 0 0 4 6 3 7 |
     | 0 0 0 4 2 5 |
     | 0 0 0 0 6 9 |
     | 0 0 0 0 0 4 |
     | 0 0 0 0 0 0 |

              6        6
o5 : Matrix ZZ  <--- ZZ

Caveat

Over a polynomial ring, specifying MaximalRank=>true will yield a non-homogeneous matrix.

random(Module,Module) -- make a random module map

Synopsis

Description

Caveat

See also