Newton–Krylov method
Newton–Krylov methodsarenumerical methodsfor solving non-linear problems using Krylov subspace linear solvers.[1][2]
Generalising theNewton methodto systems of multiple variables, the iteration formula includes aJacobian 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 aKrylov subspacemethod, such as theGeneralized minimal residual method(GMRES). (Depending on the system, apreconditionermight 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[edit]
- ^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.CiteSeerX10.1.1.636.3743.doi:10.1016/j.jcp.2003.08.010.
- ^Kelley, C.T. (2003).Solving nonlinear equations with Newton's method(1 ed.).SIAM.
External links[edit]
- Open source code (MATLAB/Octave, Fortran90), further description of the method[1]
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