TY - GEN

T1 - Near optimal tree size bounds on a simple real root isolation algorithm

AU - Sharma, Vikram

AU - Yap, Chee K.

PY - 2012

Y1 - 2012

N2 - The problem of isolating all real roots of a square-free integer polynomial f(X) inside any given interval I0 is a fundamental problem. EVAL is a simple and practical exact numerical algorithm for this problem: it recursively bisects I0, and any sub-interval I ⊆ I0, until a certain numerical predicate C0(I) ⊆ C1(I) holds on each I. We prove that the size of the recursion tree is O(d(L + r + log d)) where f has degree d, its coefficients have absolute values < 2L, and I0 contains r roots of f. In the range L ≥ d, our bound is the sharpest known, and provably optimal. Our results are closely paralleled by recent bounds on EVAL by Sagraloff-Yap (ISSAC 2011) and Burr-Krahmer (2012). In the range L ≤ d, our bound is incomparable with those of Sagraloff-Yap or Burr-Krahmer. Similar to the Burr-Krahmer proof, we exploit the technique of "continuous amortization" from Burr-Krahmer-Yap (2009), namely to bound the tree size by an integral I0 G(x)dx over a suitable "charging function" G(x). We give an application of this feature to the problem of ray-shooting (i.e., finding smallest root in a given interval).

AB - The problem of isolating all real roots of a square-free integer polynomial f(X) inside any given interval I0 is a fundamental problem. EVAL is a simple and practical exact numerical algorithm for this problem: it recursively bisects I0, and any sub-interval I ⊆ I0, until a certain numerical predicate C0(I) ⊆ C1(I) holds on each I. We prove that the size of the recursion tree is O(d(L + r + log d)) where f has degree d, its coefficients have absolute values < 2L, and I0 contains r roots of f. In the range L ≥ d, our bound is the sharpest known, and provably optimal. Our results are closely paralleled by recent bounds on EVAL by Sagraloff-Yap (ISSAC 2011) and Burr-Krahmer (2012). In the range L ≤ d, our bound is incomparable with those of Sagraloff-Yap or Burr-Krahmer. Similar to the Burr-Krahmer proof, we exploit the technique of "continuous amortization" from Burr-Krahmer-Yap (2009), namely to bound the tree size by an integral I0 G(x)dx over a suitable "charging function" G(x). We give an application of this feature to the problem of ray-shooting (i.e., finding smallest root in a given interval).

KW - Continuous amortization

KW - Integral analysis

KW - Real root isolation

KW - Subdivision algorithm

UR - http://www.scopus.com/inward/record.url?scp=84874994827&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874994827&partnerID=8YFLogxK

U2 - 10.1145/2442829.2442875

DO - 10.1145/2442829.2442875

M3 - Conference contribution

AN - SCOPUS:84874994827

SN - 9781450312691

T3 - Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

SP - 319

EP - 326

BT - ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

T2 - 37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012

Y2 - 22 July 2012 through 25 July 2012

ER -