In mathematics, particularly in computer algebra, Abramov's algorithm computes all rational solutions of a linear recurrence equation with polynomial coefficients. The algorithm was published by Sergei A. Abramov in 1989.[1][2]
Universal denominator
The main concept in Abramov's algorithm is a universal denominator. Let be a field of characteristic zero. The dispersion of two polynomials is defined as
where
denotes the set of non-negative integers. Therefore the dispersion is the maximum
such that the polynomial
and the
-times shifted polynomial
have a common factor. It is
if such a
does not exist. The dispersion can be computed as the largest non-negative integer root of the resultant
.
[3][4] Let
be a recurrence equation of order
with polynomial coefficients
, polynomial right-hand side
and rational sequence solution
. It is possible to write
for two relatively prime polynomials
. Let
and
where
denotes the
falling factorial of a function. Then
divides
. So the polynomial
can be used as a denominator for all rational solutions
and hence it is called a universal denominator.
[5] Algorithm
Let again be a recurrence equation with polynomial coefficients and a universal denominator. After substituting for an unknown polynomial and setting the recurrence equation is equivalent to
As the
cancel this is a linear recurrence equation with polynomial coefficients which can be solved for an unknown polynomial solution
. There are
algorithms to find polynomial solutions. The solutions for
can then be used again to compute the rational solutions
.
[2] algorithm rational_solutions is
input: Linear recurrence equation .
output: The general rational solution if there are any solutions, otherwise false.
Solve for general polynomial solution
if solution exists then
return general solution
else
return false
end if
Example
The homogeneous recurrence equation of order
over
has a rational solution. It can be computed by considering the dispersion
This yields the following universal denominator:
and
Multiplying the original recurrence equation with
and substituting
leads to
This equation has the polynomial solution
for an arbitrary constant
. Using
the general rational solution is
for arbitrary
.
References
- ^ Abramov, Sergei A. (1989). "Rational solutions of linear differential and difference equations with polynomial coefficients". USSR Computational Mathematics and Mathematical Physics. 29 (6): 7–12. doi:10.1016/s0041-5553(89)80002-3. ISSN 0041-5553.
- ^ a b Abramov, Sergei A. (1995). "Rational solutions of linear difference and q -difference equations with polynomial coefficients". Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95. pp. 285–289. doi:10.1145/220346.220383. ISBN 978-0897916998. S2CID 15424889.
- ^ Man, Yiu-Kwong; Wright, Francis J. (1994). "Fast polynomial dispersion computation and its application to indefinite summation". Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94. pp. 175–180. doi:10.1145/190347.190413. ISBN 978-0897916387. S2CID 2192728.
- ^ Gerhard, Jürgen (2005). Modular Algorithms in Symbolic Summation and Symbolic Integration. Lecture Notes in Computer Science. Vol. 3218. doi:10.1007/b104035. ISBN 978-3-540-24061-7. ISSN 0302-9743.
- ^ Chen, William Y. C.; Paule, Peter; Saad, Husam L. (2007). "Converging to Gosper's Algorithm". arXiv:0711.3386 [math.CA].
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