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. The Jacobian matrix itself is often lengthy to evaluate analytically or expensive to evaluate computationally. Evaluation and inversion of the Jacobian matrix is bypassed by employing a 'matrix-free' 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.

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External links

Open Source code for MATLAB and Fortran90. [1]

[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.

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