Abstract
This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments.
For a square-free polynomial A (X) of degree d with L-bit integer coefficients, we use an amortization argument to show that all the roots, real or complex, can be isolated using at most O(dL + dlgd) Sturm probes. This extends Davenport’s result for the case of isolating all real roots.
We also show that a relatively straightforward algorithm, based on the classical subresultant PQS, allows us to evaluate the Sturm sequence of A(X) at rational Õ(dL)-bit values in time Õ(d 3 L); here the Õ-notation means we ignore logarithmic factors. Again, an amortization argument is used. We provide a family of examples to show that such amortization is necessary.
The work is supported by NSF Grant #CCF-043836. A preliminary version of this work appeared at the International Workshop on Symbolic-Numeric Computation (SNC 2005), Xi’an, China, July 19–21, 2005.
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Du, Z., Sharma, V., Yap, C.K. (2007). Amortized Bound for Root Isolation via Sturm Sequences. In: Wang, D., Zhi, L. (eds) Symbolic-Numeric Computation. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7984-1_8
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DOI: https://doi.org/10.1007/978-3-7643-7984-1_8
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