Erich L. Kaltofen
Clément Pernet
ABSTRACT
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.
Incorporating our two counts we develop a combined early termination algorithm. We then specialize the algorithm in [Boyer and Kaltofen, Proc. SNC 2014] for parametric linear system solving to the recovery of a vector of rational functions, 1/g(u) f(u), from its evaluations. Thus, if the rational function vector is the solution to a full rank linear system our early termination strategy applies and we may recover it from fewer evaluations than generalized Welch/Berlekamp decoding. If we allow evaluations at poles (roots of g) there are examples where the Cabay count is not sufficient to recover the rational function vector from just its evaluations. This problem is solved if in addition to indicating that an evaluation point is a pole, the black box gives information about the numerators of the solution at the evaluation point.
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Digital Library
- Brice B. Boyer and Erich Kaltofen. 2014. Numerical Linear System Solving With Parametric Entries By Error Correction. In SNC'14 Proc. 2014 Internat. Workshop on Symbolic-Numeric Comput., Jan Verschelde and Stephen M. Watt (Eds.). Association for Computing Machinery, New York, N. Y., 33--38. http://www.math.ncsu.edu/kaltofen/bibliography/14/BoKa14.pdf. Google ScholarDigital Library
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- Erich Kaltofen and Clément Pernet. 2013. Cauchy Interpolation with Errors in the Values. Submitted manuscript, 13 pages. (Dec. 2013).Google Scholar
- Erich Kaltofen and Zhengfeng Yang. 2013. Sparse multivariate function recovery from values with noise and outlier errors. In ISSAC 2013 Proc. 38th Internat. Symp. Symbolic Algebraic Comput., Manuel Kauers (Ed.). Association for Computing Machinery, New York, N. Y., 219--226. http://www.math.ncsu.edu/kaltofen/bibliography/13/KaYa13.pdf. Google ScholarDigital Library
- Erich Kaltofen and Zhengfeng Yang. 2014. Sparse Multivariate Function Recovery With a High Error Rate in Evaluations. In ISSAC 2014 Proc. 39th Internat. Symp. Symbolic Algebraic Comput., Katsusuke Nabeshima (Ed.). Association for Computing Machinery, New York, N. Y., 280--287. http://www.math.ncsu.edu/kaltofen/bibliography/14/KaYa14.pdf. Google ScholarDigital Library
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- Clément Pernet. 2014. High Performance Algebraic Reliable Computing. Habilitation Thesis. Univ. Joseph Fourier (Grenoble 1).Google Scholar
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Index Terms
- Early Termination in Parametric Linear System Solving and Rational Function Vector Recovery with Error Correction
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