sparseCoeffs -- Converts ring elements into vectors over the coefficient ring

Synopsis

Usage:

sparseCoeffs L
Inputs:
- L, an instance of the type NCRingElement, or a List of NCRingElements
Optional inputs:
- Monomials => a list, default value null, a list of monomials to use as a basis for computing coordinate vectors
Outputs:
- a matrix,

Description

This method converts a list of ring elements to coordinate vectors - returned as a matrix - relative to a list of monomials. If the user does not supply a monomial list, the list is taken to be the monomials occurring in the elements of the list (with repetition).

i1 : A=QQ{a, b, c, d, e, f, g, h}

o1 = A

o1 : NCPolynomialRing

i2 : F = a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2;

i3 : sparseCoeffs(F)

o3 = | 1 |
     | 1 |
     | 1 |
     | 1 |
     | 1 |
     | 1 |
     | 1 |
     | 1 |

              8        1
o3 : Matrix QQ  <--- QQ

i4 : bas = flatten entries basis(2,A);

i5 : #bas

o5 = 64

i6 : sparseCoeffs(F,Monomials=>bas)

o6 = | 1 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 1 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 1 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 1 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 1 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 1 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 1 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 0 |
     | 1 |

              64        1
o6 : Matrix QQ   <--- QQ

i7 : sparseCoeffs(toList (10:F),Monomials=>bas)

o7 = | 1 1 1 1 1 1 1 1 1 1 |
     | 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 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 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 |
     | 1 1 1 1 1 1 1 1 1 1 |
     | 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 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 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 |
     | 1 1 1 1 1 1 1 1 1 1 |
     | 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 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 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 |
     | 1 1 1 1 1 1 1 1 1 1 |
     | 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 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 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 |
     | 1 1 1 1 1 1 1 1 1 1 |
     | 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 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 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 |
     | 1 1 1 1 1 1 1 1 1 1 |
     | 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 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 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 |
     | 1 1 1 1 1 1 1 1 1 1 |
     | 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 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 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 |
     | 1 1 1 1 1 1 1 1 1 1 |

              64        10
o7 : Matrix QQ   <--- QQ

Ways to use sparseCoeffs :

sparseCoeffs(List)
sparseCoeffs(NCRingElement)

For the programmer

The object sparseCoeffs is a method function with options.