### Abstract

We describe a new algorithm Miranda for isolating the simple zeros of a function f : R_{n} → Rn within a box B0 ⊆ R_{n}. The function f and its partial derivatives must have interval forms, but need not be polynomial. Our subdivision-based algorithm is “effective” in the sense that our algorithmic description also specifies the numerical precision that is sufficient to certify an implementation with any standard BigFloat number type. The main predicate is the Moore-Kioustelidis (MK) test, based on Miranda's Theorem (1940). Although the MK test is well-known, this paper appears to be the first synthesis of this test into a complete root isolation algorithm. We provide a complexity analysis of our algorithm based on intrinsic geometric parameters of the system. Our algorithm and complexity analysis are developed using 3 levels of description (Abstract, Interval, Effective). This methodology provides a systematic pathway for achieving effective subdivision algorithms in general.

Original language | English (US) |
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Title of host publication | ISSAC 2019 - Proceedings of the 2019 ACM International Symposium on Symbolic and Algebraic Computation |

Publisher | Association for Computing Machinery |

Pages | 355-362 |

Number of pages | 8 |

ISBN (Electronic) | 9781450360845 |

DOIs | |

State | Published - Jul 8 2019 |

Event | 44th ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2019 - Beijing, China Duration: Jul 15 2019 → Jul 18 2019 |

### Publication series

Name | Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC |
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### Conference

Conference | 44th ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2019 |
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Country | China |

City | Beijing |

Period | 7/15/19 → 7/18/19 |

### Fingerprint

### Keywords

- Certified Computation
- Complexity Analysis
- Effective Certified Algorithm
- Miranda Theorem
- Moore-Kioustelidis Test
- Root Isolation
- Subdivision Algorithms
- System of Real Equations

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*ISSAC 2019 - Proceedings of the 2019 ACM International Symposium on Symbolic and Algebraic Computation*(pp. 355-362). (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC). Association for Computing Machinery. https://doi.org/10.1145/3326229.3326270

**Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis.** / Xu, Juan; Yap, Chee.

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

*ISSAC 2019 - Proceedings of the 2019 ACM International Symposium on Symbolic and Algebraic Computation.*Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC, Association for Computing Machinery, pp. 355-362, 44th ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2019, Beijing, China, 7/15/19. https://doi.org/10.1145/3326229.3326270

}

TY - GEN

T1 - Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis

AU - Xu, Juan

AU - Yap, Chee

PY - 2019/7/8

Y1 - 2019/7/8

N2 - We describe a new algorithm Miranda for isolating the simple zeros of a function f : Rn → Rn within a box B0 ⊆ Rn. The function f and its partial derivatives must have interval forms, but need not be polynomial. Our subdivision-based algorithm is “effective” in the sense that our algorithmic description also specifies the numerical precision that is sufficient to certify an implementation with any standard BigFloat number type. The main predicate is the Moore-Kioustelidis (MK) test, based on Miranda's Theorem (1940). Although the MK test is well-known, this paper appears to be the first synthesis of this test into a complete root isolation algorithm. We provide a complexity analysis of our algorithm based on intrinsic geometric parameters of the system. Our algorithm and complexity analysis are developed using 3 levels of description (Abstract, Interval, Effective). This methodology provides a systematic pathway for achieving effective subdivision algorithms in general.

AB - We describe a new algorithm Miranda for isolating the simple zeros of a function f : Rn → Rn within a box B0 ⊆ Rn. The function f and its partial derivatives must have interval forms, but need not be polynomial. Our subdivision-based algorithm is “effective” in the sense that our algorithmic description also specifies the numerical precision that is sufficient to certify an implementation with any standard BigFloat number type. The main predicate is the Moore-Kioustelidis (MK) test, based on Miranda's Theorem (1940). Although the MK test is well-known, this paper appears to be the first synthesis of this test into a complete root isolation algorithm. We provide a complexity analysis of our algorithm based on intrinsic geometric parameters of the system. Our algorithm and complexity analysis are developed using 3 levels of description (Abstract, Interval, Effective). This methodology provides a systematic pathway for achieving effective subdivision algorithms in general.

KW - Certified Computation

KW - Complexity Analysis

KW - Effective Certified Algorithm

KW - Miranda Theorem

KW - Moore-Kioustelidis Test

KW - Root Isolation

KW - Subdivision Algorithms

KW - System of Real Equations

UR - http://www.scopus.com/inward/record.url?scp=85069752003&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85069752003&partnerID=8YFLogxK

U2 - 10.1145/3326229.3326270

DO - 10.1145/3326229.3326270

M3 - Conference contribution

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

SP - 355

EP - 362

BT - ISSAC 2019 - Proceedings of the 2019 ACM International Symposium on Symbolic and Algebraic Computation

PB - Association for Computing Machinery

ER -