# Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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## Rank deficiencies and bifurcation into affine subspaces for separable parameterized equationsHTML articles powered by AMS MathViewer

by Yun-Qiu Shen and Tjalling J. Ypma
Math. Comp. 85 (2016), 271-293 Request permission

## Abstract:

Many applications lead to separable parameterized equations of the form $F(y,\mu ,z) \equiv A(y, \mu )z+b(y, \mu )=0$, where $y \in \mathbb R^n$, $z \in \mathbb R^N$, $A(y, \mu ) \in \mathbb {R}^{(N+n) \times N}$, $b(y, \mu ) \in \mathbb {R}^{N+n}$ and $\mu \in \mathbb R$ is a parameter. Typically $N >>n$. Suppose bifurcation occurs at a solution point $(y^*,\mu ^*,z^*)$ of this equation. If $A(y^*, \mu ^*)$ is rank deficient, then the linear component $z$ bifurcates into an affine subspace at this point. We show how to compute such a point $(y,\mu ,z)$ by reducing the original system to a smaller separable system, while preserving the bifurcation, the rank deficiencies and a non-degeneracy condition. A numerical algorithm for solving the reduced system and examples illustrating the method are provided.
References
Similar Articles
• Yun-Qiu Shen
• Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063
• MR Author ID: 191125
• Email: yunqiu.shen@wwu.edu
• Tjalling J. Ypma
• Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063
• Email: tjalling.ypma@wwu.edu
• Received by editor(s): May 7, 2012
• Received by editor(s) in revised form: April 19, 2014
• Published electronically: June 2, 2015