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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.16895 |
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| _version_ | 1866929689733365760 |
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| author | Tan, Songchen Miao, Keming Edelman, Alan Rackauckas, Christopher |
| author_facet | Tan, Songchen Miao, Keming Edelman, Alan Rackauckas, Christopher |
| contents | This paper demonstrates new methods and implementations of nonlinear solvers with higher-order of convergence, which is achieved by efficiently computing higher-order derivatives. Instead of computing full derivatives, which could be expensive, we compute directional derivatives with Taylor-mode automatic differentiation. We first implement Householder's method with arbitrary order for one variable, and investigate the trade-off between computational cost and convergence order. We find that the second-order variant, i.e., Halley's method, to be the most valuable, and further generalize Halley's method to systems of nonlinear equations and demonstrate that it can scale efficiently to large-scale problems. We further apply Halley's method on solving large-scale ill-conditioned nonlinear problems, as well as solving nonlinear equations inside stiff ODE solvers, and demonstrate that it could outperform Newton's method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16895 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scalable higher-order nonlinear solvers via higher-order automatic differentiation Tan, Songchen Miao, Keming Edelman, Alan Rackauckas, Christopher Numerical Analysis This paper demonstrates new methods and implementations of nonlinear solvers with higher-order of convergence, which is achieved by efficiently computing higher-order derivatives. Instead of computing full derivatives, which could be expensive, we compute directional derivatives with Taylor-mode automatic differentiation. We first implement Householder's method with arbitrary order for one variable, and investigate the trade-off between computational cost and convergence order. We find that the second-order variant, i.e., Halley's method, to be the most valuable, and further generalize Halley's method to systems of nonlinear equations and demonstrate that it can scale efficiently to large-scale problems. We further apply Halley's method on solving large-scale ill-conditioned nonlinear problems, as well as solving nonlinear equations inside stiff ODE solvers, and demonstrate that it could outperform Newton's method. |
| title | Scalable higher-order nonlinear solvers via higher-order automatic differentiation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2501.16895 |