Abstract
In order to determine the extinction probability of a Markovian binary tree, it is necessary to find the minimal nonnegative solution of a quadratic vector equation. We apply the modified Newton–Shamanskii method for solving the equation. We show that the sequence of vectors generated by the modified Newton–Shamanskii method is monotonically increasing and converges to the minimal nonnegative solution of the equation. Numerical experiments show the effectiveness of our method.
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Acknowledgments
The authors wish to show their gratitude to an anonymous referee for the helpful suggestions, which greatly improved the paper. This research was supported in part by National Natural Science Foundation of China under Grant 10901024.
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Guo, PC., Xu, SF. The modified Newton–Shamanskii method for the solution of a quadratic vector equation arising in Markovian binary trees. Calcolo 52, 317–325 (2015). https://doi.org/10.1007/s10092-014-0118-8
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DOI: https://doi.org/10.1007/s10092-014-0118-8
Keywords
- Markovian binary tree
- Quadratic vector equation
- Newton–Shamanskii method
- Minimal nonnegative solution
- Monotone convergence