### Abstract

We bound the condition number of the Jacobian in pseudo-arclength continuation problems, and we quantify the effect of this condition number on the linear system solution in a Newton-GMRES solve. Pseudo-arclength continuation solves parameter dependent nonlinear equations G(u, λ) = 0 by introducing a new parameter s, which approximates arclength, and viewing the vector x = (u, λ) as a function of s. In this way simple fold singularities can be computed directly by solving a larger system F(x, s) = 0 by simple continuation in the new parameter s. It is known that the Jacobian F_{x} of F with respect to x = (u, λ) is nonsingular if the path contains only regular points and simple fold singularities. We introduce a new characterization of simple folds in terms of the singular value decomposition, and we use it to derive a new bound for the norm of F_{x}^{-1}. We also show that the convergence rate of GMRES in a Newton step for F(x, s) = 0 is essentially the same as that of the original problem G(u, λ) = 0. In particular, we prove that the bounds on the degrees of the minimal polynomials of the Jacobians F_{x} and G_{u} differ by at most 2. We illustrate the effectiveness of our bounds with an example from radiative transfer theory.

Original language | English (US) |
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Pages (from-to) | 263-276 |

Number of pages | 14 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 45 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2007 |

### All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Keywords

- Eigenvalues
- Fold point
- GMRES
- Limit point
- Pseudo-arclength continuation
- Rank-one update
- Simple fold
- Singular vectors
- Singularity
- Turning point

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## Cite this

*SIAM Journal on Numerical Analysis*,

*45*(1), 263-276. https://doi.org/10.1137/060654384