By Hyeong In Choi, Rida T. Farouki, Chang Yong Han, Hwan Pyo Moon (auth.), Deepak Kapur (eds.)
This booklet constitutes completely refereed post-conference complaints of the eighth Asian Symposium on laptop arithmetic, ASCM 2007, held in Singapore in December 2007.
The 22 revised complete papers and five revised poster papers provided including three invited lectures have been conscientiously chosen in the course of rounds of reviewing and development from sixty five submissions. The papers are prepared in topical sections on algorithms and implementations, numerical tools and purposes, cryptology, and computational logic.
Read or Download Computer Mathematics: 8th Asian Symposium, ASCM 2007, Singapore, December 15-17, 2007. Revised and Invited Papers PDF
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Additional info for Computer Mathematics: 8th Asian Symposium, ASCM 2007, Singapore, December 15-17, 2007. Revised and Invited Papers
Proof. It is suﬃcient to prove the case where n = 2. e (x) e2 (x) When the determinant D(x) = 1 is not identically equal to 0, e1 (x) e2 (x) there is α ∈ I such that D(α) = 0. Therefore, (e1 (α), e1 (α)) and (e2 (α), e2 (α)) are linearly independent over R. Hence, there exist c1 , c2 ∈ R such that (−f (α), −f (α)) = c1 (e1 (α), e1 (α))+c2 (e2 (α), e2 (α)). Therefore, f (x)+c1 e1 (x)+ c2 e2 (x) has a multiple zero at α ∈ I. Next, we consider the case where D(x) is identically equal to 0. Changing the indexes, if necessary, we can assume that e1 is not the zero polynomial since e1 or e2 is not constant.
Optimization by Vector Space Methods. , Chichester (1969) 10. : A note on a nearest polynomial with a given root. ACM SIGSAM Bulletin 39(2), 53–60 (2005) 11. : Eﬃcient algorithms for computing the nearest polynomial with constrained roots. In: Proc. 1998 International Symposium on Symbolic and Algebraic Computation (ISSAC 1998), pp. 236–243 (1998) 12. : Eﬃcient algorithms for computing the nearest polynomial with a real root and related problems. In: Proc. 1999 International Symposium on Symbolic and Algebraic Computation (ISSAC 1999), pp.
A metric d on F is given such that the topology of F induced by d is the ordinal topology of Rn . Here, we identify F with Rn through the map F n f (x) + i=1 ci ei (x) → (c1 , . . , cn ) ∈ Rn . We consider the following problem. Problem 1. Find f˜ ∈ F such that f˜ has a multiple zero in I and d(f, f˜) is minimal. Clearly, there is no solution to Problem 1 if there is no polynomial in F having a multiple zero in I. If there is a polynomial in F having a multiple zero in I, there is a solution to Problem 1.
Computer Mathematics: 8th Asian Symposium, ASCM 2007, Singapore, December 15-17, 2007. Revised and Invited Papers by Hyeong In Choi, Rida T. Farouki, Chang Yong Han, Hwan Pyo Moon (auth.), Deepak Kapur (eds.)