### Abstract

The problem of isolating all real roots of a square-free integer polynomial f(X) inside any given interval I_{0} 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 C_{0}(I) ⊆ C_{1}(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 < 2^{L}, 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 I_{0} 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).

Original language | English (US) |
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Title of host publication | ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation |

Pages | 319-326 |

Number of pages | 8 |

DOIs | |

State | Published - 2012 |

Event | 37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012 - Grenoble, France Duration: Jul 22 2012 → Jul 25 2012 |

### Other

Other | 37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012 |
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Country | France |

City | Grenoble |

Period | 7/22/12 → 7/25/12 |

### Fingerprint

### Keywords

- Continuous amortization
- Integral analysis
- Real root isolation
- Subdivision algorithm

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation*(pp. 319-326) https://doi.org/10.1145/2442829.2442875

**Near optimal tree size bounds on a simple real root isolation algorithm.** / Sharma, Vikram; Yap, Chee K.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation.*pp. 319-326, 37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012, Grenoble, France, 7/22/12. https://doi.org/10.1145/2442829.2442875

}

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

SN - 9781450312691

SP - 319

EP - 326

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

ER -