ReduceOrder.txt:

  This file has the main code. It is organized into a number of sections:

    Absolute Factorization:

	Contains algorithms for absolute factorization from the ISSAC paper with
	efficiency improves from the paper for the ISSAC-JSC special issue. To use
	these improvements, you have to download the file "RightFactors.txt" as well.
	This is because the newer code passes data to LREtools[RightFactors] about
	which combinations it should skip. The file "RightFactors.txt" is essentially
	the code of RightFactors, with a small modification: it skips the combinations
	that _Env_RightFactors_data tells it to skip. (Without loading "RightFactors.txt"
	the code should still work, but it'll be slower because the information
	in _Env_RightFactors_data would not be used. This will only be noticeable on
	larger examples though).

    Reduce Order 3 --> 2:

	This contains the implementation of the algorithm from the ISSAC paper
	to solve 3rd order recurrences in terms of order 2.

	If you use the optional argument `projective` then simplifies (up to
	projective equivalence) the output L2, in which case the transformations
	are not returned.

    Reduce Orders {3,4} --> 2

	This contains the implementation of the algorithm for order 4 from the
	the paper for the ISSAC-JSC special issue.

    Simplify an operator of order 2:

	Given an operator L of order 2, find an operator of hopefully-minimal size
	that is gauge-equivalent (or optionally: projectively equivalent) to L.

    Tools:

	Symmetric product (to compute examples.
	ProjectiveHom, given two operators L1,L2 of the same order, this code
	tries to find r for which SymProd(L1, tau-r) is gauge-equivalent to L2.
	It then returns r and the gauge-transformation.
	For this to work you need to load the file Hom.txt

    Conic:

	Solve a quadratic equation of the form a*X^2 + b*y^2 + c*Z^2 = 0 where a,b,c
	are rational functions over Q.  For more see:
	www.math.fsu.edu/~hoeij/files/ConicProgram




Hom.txt:

	Implementation by Yongjae Cha for "Hom" which computes the following:
	 Given L1, L2 in D,  compute a basis of operators G in C(x)[tau]
	 for which G(V(L1)) is contained in V(L2),  where V(L) = solution space of L.
	 Here tau denotes the shift operator that sends f(x) to f(x+1).
	For more  (Yongjae's slides/abstract/implementation)  see:
	https://www.math.fsu.edu/~hoeij/papers/Hom/index.html
	



RightFactors.txt:

	The code of LREtools[RightFactors] from  www.math.fsu.edu/~hoeij/algorithms/RFactors
	with a small modification: if _Env_RightFactors_data contains data about which
	combinations to skip, then use that data.
	


ReduceOrder.mw

         Maple Wprksheet that includes the examples from papers "Algorithms for 2-Solvable Difference Equations" and "Solving Third Order Linear Difference Equations in Terms of Second Order Equations". 



ReduceOrder.pdf
        
         pdf version of the worksheet