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Is there a name and common technique for such equations, where $A$ and $B$ are matrices and $x$ a vector?

$Ax+f(\lambda)Bx=g(\lambda)x$.

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from the way this nonlinear eigenvalue problem looks, i presume both $\lambda$ and $x$ are "unknowns"? –  Suvrit Jan 4 '13 at 12:35
    
@Suvrit: yes, $\lambda$ is a scalar. –  Felix Goldberg Jan 4 '13 at 12:51
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up vote 5 down vote accepted

Akin to my comment, this equation can be called a nonlinear generalized eigenvalue problem. Usually, $f$ and $g$ are polynomials in $\lambda$, but more general nonlinearities might be allowed. In general, I doubt there will be robust, globally convergent method for this equation that gets all the solutions. The talk or this paper might be good starting points (see especially the paper).

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