Newton–Krylov method
Nowadays, Newton–Krylov method is a topic that has captured the attention of many people around the world. With a significant impact on various areas of life, Newton–Krylov method has generated unprecedented debate, attracting supporters and critics alike. As Newton–Krylov method continues to loom large in the collective consciousness, his influence extends across multiple sectors, from politics to entertainment, and from technology to society at large. In this article, we will explore the various facets of Newton–Krylov method, examining its current relevance and potential future impact.
Newton–Krylov methods are numerical methods for solving non-linear problems using Krylov subspace linear solvers.[1][2]
Generalising the Newton method to systems of multiple variables, the iteration formula includes a Jacobian matrix. Solving this directly would involve calculation of the Jacobian's inverse, when the Jacobian matrix itself is often difficult or impossible to calculate.
It may be possible to solve the Newton iteration formula without the inverse using a Krylov subspace method, such as the Generalized minimal residual method (GMRES). (Depending on the system, a preconditioner might be required.) The result is a Newton–Krylov method.
The Jacobian itself might be too difficult to compute, but the GMRES method does not require the Jacobian itself, only the result of multiplying given vectors by the Jacobian. Often this can be computed efficiently via difference formulae. Solving the Newton iteration formula in this manner, the result is a Jacobian-Free Newton-Krylov (JFNK) method.
References
- ^ Knoll, D.A.; Keyes, D.E. (2004). "Jacobian-free Newton–Krylov methods: a survey of approaches and applications". Journal of Computational Physics. 193 (2): 357. CiteSeerX 10.1.1.636.3743. doi:10.1016/j.jcp.2003.08.010.
- ^ Kelley, C.T. (2003). Solving nonlinear equations with Newton's method (1 ed.). SIAM.
External links
 | This applied mathematics–related article is a stub. You can help Wikipedia by expanding it. |