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A general view of minimally extended systems for simple bifurcation points. (English) Zbl 0915.65042
The development of the minimally extended systems method [cf. A. Griewank and G. W. Reddien, SIAM J. Numer. Anal. 21, 176-185 (1984; Zbl 0536.65031); V. Janovskij, Computing 43, No. 1, 27-36 (1989; Zbl 0695.65033)] for calculation of simple bifurcation points is suggested. In this method the simple bifurcation point \((x^{*},\lambda^{*})\) of the equation \(F(x,\lambda)=0,\quad (x\in \mathbb{R}^{n};\lambda\in \mathbb{R}\) is bifurcation parameter) becomes as regular point \(z^{*}=(x^{*},\lambda^{*},0)\) of the extended system \[ G(x,\lambda,\mu):=[ {F(x,\lambda)+\mu d \atop f(x,\lambda)}]=0 \] \((d\in \mathbb{R}^{n}, f:\mathbb{R}^{n}\times \mathbb{R}\rightarrow \mathbb{R}^{2},\) the Jacobian \(\partial G(x^{*},\lambda^{*},0)\in \mathbb{R}^{(n+2)\times(n+2)}\) is nonsingular) and for its calculation superlinearly convergent Newton type methods can be applied.

MSC:
65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
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