Early Termination in Parametric Linear System Solving and Rational Function Vector Recovery with Error Correction
Erich Kaltofen, Clément Pernet, Arne Storjohann, Cleveland Alexander Waddell · 2017
Consider solving a black box linear system, A(u) x = b(u), where the entries are polynomials in u over a field K, and A(u) is full rank. The solution, x = 1/g(u) f(u), where g is always the least common monic denominator, can be found by evaluating the system at distinct points ξl in K. The solution can be recovered even if some evaluations are erroneous. In [Boyer and Kaltofen, Proc. SNC 2014] the problem is solved with an algorithm that generalizes Welch/Berlekamp decoding of an algebraic Reed-Solomon code. Their algorithm requires the sum of a degree bound for the numerators plus a degree bound for the denominator of the solution. It is possible that the degree bounds input to their algorithm grossly overestimate the actual degrees. We describe an algorithm that given the same inputs uses possibly fewer evaluations to compute the solution. We introduce a second count for the number of evaluations required to recover the solution based on work by Stanley Cabay. The Cabay count includes bounds for the highest degree polynomial in the coefficient matrix and right side vector, but does not require solution degree bounds. Instead our algorithm iterates until the Cabay termination criterion is reached. At this point our algorithm returns the solution. Assuming we have the actual degrees for all necessary input parameters, we give the criterion that determines when the Cabay count is fewer than the generalized Welch/Berlekamp count.