### Abstract

A rigorous method to determine whether a polynomial f(x) is nonnegative for all x in an interval is proposed. Conventionally, a method which determines whether the multiplicity of every zero of f(x) is even has been used for this problem. The conventional method is, however, somewhat complicated because the Sturm function sequence must be generated repeatedly and the number of variations of sign must be investigated for each sequence. The method proposed in this paper determines whether f(x) + ε (ε is a sufficiently small positive number) is positive for all x in the interval. This method is simpler than the conventional one in the sense that only two Sturm function sequences are generated. Moreover, the determination can easily be made with the number of variations of sign of the function sequence made by concatenating two Sturm function sequences.

Original language | English |
---|---|

Pages (from-to) | 58-65 |

Number of pages | 8 |

Journal | Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi) |

Volume | 81 |

Issue number | 5 |

Publication status | Published - May 1998 |

Externally published | Yes |

### Fingerprint

### Keywords

- Nonnegativity of functions
- Positive real functions
- Sturm theorem

### ASJC Scopus subject areas

- Electrical and Electronic Engineering

### Cite this

**A test for nonnegativity of real polynomials.** / Nishi, Tetsuo; Takahashi, Norikazu.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A test for nonnegativity of real polynomials

AU - Nishi, Tetsuo

AU - Takahashi, Norikazu

PY - 1998/5

Y1 - 1998/5

N2 - A rigorous method to determine whether a polynomial f(x) is nonnegative for all x in an interval is proposed. Conventionally, a method which determines whether the multiplicity of every zero of f(x) is even has been used for this problem. The conventional method is, however, somewhat complicated because the Sturm function sequence must be generated repeatedly and the number of variations of sign must be investigated for each sequence. The method proposed in this paper determines whether f(x) + ε (ε is a sufficiently small positive number) is positive for all x in the interval. This method is simpler than the conventional one in the sense that only two Sturm function sequences are generated. Moreover, the determination can easily be made with the number of variations of sign of the function sequence made by concatenating two Sturm function sequences.

AB - A rigorous method to determine whether a polynomial f(x) is nonnegative for all x in an interval is proposed. Conventionally, a method which determines whether the multiplicity of every zero of f(x) is even has been used for this problem. The conventional method is, however, somewhat complicated because the Sturm function sequence must be generated repeatedly and the number of variations of sign must be investigated for each sequence. The method proposed in this paper determines whether f(x) + ε (ε is a sufficiently small positive number) is positive for all x in the interval. This method is simpler than the conventional one in the sense that only two Sturm function sequences are generated. Moreover, the determination can easily be made with the number of variations of sign of the function sequence made by concatenating two Sturm function sequences.

KW - Nonnegativity of functions

KW - Positive real functions

KW - Sturm theorem

UR - http://www.scopus.com/inward/record.url?scp=0032072516&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032072516&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032072516

VL - 81

SP - 58

EP - 65

JO - Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)

JF - Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)

SN - 1042-0967

IS - 5

ER -