# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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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.
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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