### Abstract

A verified integration algorithm is proposed for calculating s-dimensional integrals over a finite domain using numerical computations. To construct an efficient verified numerical integrator, the truncation error and the rounding error need to be considered. It has been known that interval arithmetic is one of the most efficient methods of evaluating the rounding error. However, it is much slower than pure floating-point arithmetic, so that in an inclusion algorithm for integrals, the computational effort by the interval arithmetic tends to become a large part. To overcome this problem, an algorithm for evaluating the rounding error using floating-point computations is proposed. The proposed algorithm is based on calculating a priori error bounds for function evaluations and an accurate sum algorithm. With the use of the proposed algorithm and a inclusion algorithm for evaluating the truncation error, we propose an automatic inclusion algorithm. Numerical examples are presented for illustrating the effectiveness of the proposed algorithm.

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

Pages (from-to) | 156-167 |

Number of pages | 12 |

Journal | Reliable Computing |

Volume | 15 |

Issue number | 2 |

Publication status | Published - 2011 Jun |

### Fingerprint

### Keywords

- Numerical integration
- Rounding error
- Verification

### ASJC Scopus subject areas

- Software
- Applied Mathematics
- Computational Mathematics

### Cite this

*Reliable Computing*,

*15*(2), 156-167.

**A note on a verified automatic integration algorithm.** / Yamanaka, Naoya; Kashiwagi, Masahide; Oishi, Shinichi; Ogita, Takeshi.

Research output: Contribution to journal › Article

*Reliable Computing*, vol. 15, no. 2, pp. 156-167.

}

TY - JOUR

T1 - A note on a verified automatic integration algorithm

AU - Yamanaka, Naoya

AU - Kashiwagi, Masahide

AU - Oishi, Shinichi

AU - Ogita, Takeshi

PY - 2011/6

Y1 - 2011/6

N2 - A verified integration algorithm is proposed for calculating s-dimensional integrals over a finite domain using numerical computations. To construct an efficient verified numerical integrator, the truncation error and the rounding error need to be considered. It has been known that interval arithmetic is one of the most efficient methods of evaluating the rounding error. However, it is much slower than pure floating-point arithmetic, so that in an inclusion algorithm for integrals, the computational effort by the interval arithmetic tends to become a large part. To overcome this problem, an algorithm for evaluating the rounding error using floating-point computations is proposed. The proposed algorithm is based on calculating a priori error bounds for function evaluations and an accurate sum algorithm. With the use of the proposed algorithm and a inclusion algorithm for evaluating the truncation error, we propose an automatic inclusion algorithm. Numerical examples are presented for illustrating the effectiveness of the proposed algorithm.

AB - A verified integration algorithm is proposed for calculating s-dimensional integrals over a finite domain using numerical computations. To construct an efficient verified numerical integrator, the truncation error and the rounding error need to be considered. It has been known that interval arithmetic is one of the most efficient methods of evaluating the rounding error. However, it is much slower than pure floating-point arithmetic, so that in an inclusion algorithm for integrals, the computational effort by the interval arithmetic tends to become a large part. To overcome this problem, an algorithm for evaluating the rounding error using floating-point computations is proposed. The proposed algorithm is based on calculating a priori error bounds for function evaluations and an accurate sum algorithm. With the use of the proposed algorithm and a inclusion algorithm for evaluating the truncation error, we propose an automatic inclusion algorithm. Numerical examples are presented for illustrating the effectiveness of the proposed algorithm.

KW - Numerical integration

KW - Rounding error

KW - Verification

UR - http://www.scopus.com/inward/record.url?scp=80053902842&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80053902842&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:80053902842

VL - 15

SP - 156

EP - 167

JO - Reliable Computing

JF - Reliable Computing

SN - 1385-3139

IS - 2

ER -