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

Synopsis

• Function: random

• Usage:

  f = random(F,G)

• 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 => ..., default value 10, get a random homogeneous element from a graded ring
• 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} | 24             0       0   |
     {-2} | -29x+19y       -36     0   |
     {-3} | -29x2-8xy-22y2 19x-10y -30 |

             3      3
o2 : Matrix R  <-- R

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

o3 = | 1944060808 5893633895  15562508275 12208320525 31350869218 
     | 8984899069 27238708403 71925510511 56423403296 144894847741
     | 106709069  323500272   854222648   670112044   1720842325  
     ------------------------------------------------------------------------
     79232613553  |
     366190722085 |
     4349060833   |

              3       6
o3 : Matrix ZZ  <-- ZZ

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

o4 = | 4 0 0 0 8 2 0 0 3 0 |
     | 0 8 3 8 0 8 0 7 0 0 |
     | 0 0 1 0 0 0 0 1 7 0 |

              3       10
o4 : Matrix ZZ  <-- ZZ

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

o5 = | 0 9 2 7 1 7 |
     | 0 0 5 3 7 5 |
     | 0 0 0 0 0 8 |
     | 0 0 0 0 2 9 |
     | 0 0 0 0 0 4 |
     | 0 0 0 0 0 0 |

              6       6
o5 : Matrix ZZ  <-- ZZ

Caveat

See also

• setRandomSeed -- set starting point for random number generator