Newton–Krylov method
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. However, it may be possible to solve the iteration formula using a Krylov subspace method, such as the Generalized minimal residual method (GMRES). (Depending on the system, a preconditioner might be required.) The GMRES method does not require knowledge of the Jacobian itself, only the result of multiplying given vectors by the Jacobian. Often this can be computed efficiently via difference formulae. 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
- Open Source code for MATLAB and Fortran90 with details on equations. [1]
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