Changes between Initial Version and Version 2 of Ticket #16742


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Timestamp:
08/19/14 11:29:42 (8 years ago)
Author:
ketzu
Comment:

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  • Ticket #16742

    • Property Keywords matrix weak-popov-form added
    • Property Priority changed from major to minor
    • Property Component changed from PLEASE CHANGE to performance
    • Property Milestone changed from sage-6.3 to sage-6.4
  • Ticket #16742 – Description

    initial v2  
     1The target of this ticket is to enhance the function weak_popov of the matrix interface.
     2The function should transform the matrix in weak popov form, it will use mulders-storjohann algorithm and should be much faster than the current implementation but will not work for polynomials over a fraction field only for polynomial rings over finite fields.
     3
     4Short description of weak popov form: Let R be an ordered Ring and Amxn a matrix over R. The leading position of a row is called the position i in [1,m) such that the order of A[i,_] is maximal within the row. If there are multiple entries with the maximum order, the highest i is the leading position (the furthest to the right in the matrix). A is in weak popov form if all leading positions are different (zero lines ignored).
     5
     6The function will implement this only for polynomial rings, the order function is the degree of the polynomial. Example:
     7
     8[x2+1, x]
     9
     10[x, x+1]
     11
     12is in weak popov form: Row 1 has the degrees 2 and 1, the leading position is for i=0, row 2 has two times degree 1 so the higher i is chosen with i=1.
     13
     14[x2+1, x]
     15
     16[x,0]
     17
     18is NOT in weak popov form, row 1 has now degrees 1 and -1, so the leading position is i=0 as in row 1.