[{"page":"147-156","status":"public","author":[{"first_name":"Sven","full_name":"Köhler, Sven","last_name":"Köhler"},{"first_name":"Martin","last_name":"Ziegler","full_name":"Ziegler, Martin"}],"user_id":"15415","date_updated":"2022-01-06T06:57:18Z","language":[{"iso":"eng"}],"title":"On the Stability of Fast Polynomial Arithmetic","date_created":"2021-10-15T09:57:36Z","abstract":[{"lang":"eng","text":"Operations on univariate dense polynomials—multiplication, division with remainder, multipoint\r\nevaluation—constitute central primitives entering as build-up blocks into many higher applications and\r\nalgorithms. Fast Fourier Transform permits to accelerate them from naive quadratic to running time\r\nO(n·polylogn), that is softly linear in the degree n of the input. This is routinely employed in complexity\r\ntheoretic considerations and, over integers and finite fields, in practical number theoretic calculations.\r\nThe present work explores the benefit of fast polynomial arithmetic over the field of real numbers\r\nwhere the precision of approximation becomes crucial. To this end, we study the computability of the\r\nabove operations in the sense of Recursive Analysis as an effective refinement of continuity. This theo-\r\nretical worst-case stability analysis is then complemented by an empirical evaluation: We use GMP and\r\nthe iRRAM to find the precision required for the intermediate calculations in order to achieve a desired\r\noutput accuracy."}],"year":"2008","type":"conference","citation":{"bibtex":"@inproceedings{Köhler_Ziegler_2008, title={On the Stability of Fast Polynomial Arithmetic}, booktitle={Proc. 8th Conference on Real Numbers and Computers}, author={Köhler, Sven and Ziegler, Martin}, year={2008}, pages={147–156} }","mla":"Köhler, Sven, and Martin Ziegler. “On the Stability of Fast Polynomial Arithmetic.” *Proc. 8th Conference on Real Numbers and Computers*, 2008, pp. 147–56.","ama":"Köhler S, Ziegler M. On the Stability of Fast Polynomial Arithmetic. In: *Proc. 8th Conference on Real Numbers and Computers*. ; 2008:147-156.","chicago":"Köhler, Sven, and Martin Ziegler. “On the Stability of Fast Polynomial Arithmetic.” In *Proc. 8th Conference on Real Numbers and Computers*, 147–56, 2008.","apa":"Köhler, S., & Ziegler, M. (2008). On the Stability of Fast Polynomial Arithmetic. *Proc. 8th Conference on Real Numbers and Computers*, 147–156.","short":"S. Köhler, M. Ziegler, in: Proc. 8th Conference on Real Numbers and Computers, 2008, pp. 147–156.","ieee":"S. Köhler and M. Ziegler, “On the Stability of Fast Polynomial Arithmetic,” in *Proc. 8th Conference on Real Numbers and Computers*, 2008, pp. 147–156."},"_id":"26243","department":[{"_id":"63"}],"publication":"Proc. 8th Conference on Real Numbers and Computers"}]