Volume 13, Issue 2
An Algorithm that Localizes and Counts the Zeros of a $C^2$-Function

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 320-333.

Published online: 2020-03

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• Abstract

We describe an algorithm that localizes the zeros of a given real $C^2$-function $f$ on an interval $[a,b]$. The algorithm generates a sequence of subintervals which contain a single zero of $f$. In particular, the exact number of zeros of $f$ on $[a,b]$ can be determined in this way. Apart from $f$, the only additional input of the algorithm is an upper and a lower bound for $f''$. We also show how the intervals determined by the algorithm can be further refined until they are contained in the basin of attraction of the Newton method for the corresponding zero.

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30C15

norbert.hungerbuehler@math.ethz.ch (Norbert Hungerbühler)

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@Article{NMTMA-13-320, author = {Hungerbühler , Norbert and Wu , Rui}, title = {An Algorithm that Localizes and Counts the Zeros of a $C^2$-Function}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {2}, pages = {320--333}, abstract = {

We describe an algorithm that localizes the zeros of a given real $C^2$-function $f$ on an interval $[a,b]$. The algorithm generates a sequence of subintervals which contain a single zero of $f$. In particular, the exact number of zeros of $f$ on $[a,b]$ can be determined in this way. Apart from $f$, the only additional input of the algorithm is an upper and a lower bound for $f''$. We also show how the intervals determined by the algorithm can be further refined until they are contained in the basin of attraction of the Newton method for the corresponding zero.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0077}, url = {http://global-sci.org/intro/article_detail/nmtma/15450.html} }
TY - JOUR T1 - An Algorithm that Localizes and Counts the Zeros of a $C^2$-Function AU - Hungerbühler , Norbert AU - Wu , Rui JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 320 EP - 333 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0077 UR - https://global-sci.org/intro/article_detail/nmtma/15450.html KW - Number of zeros on an interval. AB -

We describe an algorithm that localizes the zeros of a given real $C^2$-function $f$ on an interval $[a,b]$. The algorithm generates a sequence of subintervals which contain a single zero of $f$. In particular, the exact number of zeros of $f$ on $[a,b]$ can be determined in this way. Apart from $f$, the only additional input of the algorithm is an upper and a lower bound for $f''$. We also show how the intervals determined by the algorithm can be further refined until they are contained in the basin of attraction of the Newton method for the corresponding zero.

Norbert Hungerbühler & Rui Wu. (2020). An Algorithm that Localizes and Counts the Zeros of a $C^2$-Function. Numerical Mathematics: Theory, Methods and Applications. 13 (2). 320-333. doi:10.4208/nmtma.OA-2019-0077
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