Newton–Krylov method

Newton–Krylov methods are numerical methods for solving non-linear problems using Krylov subspace linear solvers.[1][2]

The Newton method, when generalised to systems of multiple variables, includes the inverse of a Jacobian matrix in the iteration formula. Calculation of the inverse of the Jacobian matrix is bypassed by employing a Krylov subspace method, e.g. the Generalized minimal residual method (GMRES), to solve the iteration formula.

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.

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[1] 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.

[2] Kelley, C.T. (2003). Solving nonlinear equations with Newton's method (1 ed.). SIAM.