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| A '''nonlinear eigenproblem''' is a generalization of an ordinary [[Eigenvalue, eigenvector and eigenspace|eigenproblem]] to equations that depend [[nonlinearly]] on the eigenvalue. Specifically, it refers to equations of the form:
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| :<math>A(\lambda) \mathbf{x} = 0 , \,</math>
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| where '''x''' is a [[vector (mathematics)|vector]] (the nonlinear "eigenvector") and ''A'' is a [[matrix (mathematics)|matrix]]-valued [[function (mathematics)|function]] of the number <math>\lambda</math> (the nonlinear "eigenvalue"). (More generally, <math>A(\lambda)</math> could be a [[linear map]], but most commonly it is a finite-dimensional, usually square, matrix.) ''A'' is usually required to be a [[holomorphic]] function of <math>\lambda</math> (in some [[domain (mathematics)|domain]]).
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| For example, an ordinary linear eigenproblem <math>B\mathbf{v} = \lambda \mathbf{v}</math>, where ''B'' is a square matrix, corresponds to <math>A(\lambda) = B - \lambda I</math>, where ''I'' is the [[identity matrix]].
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| One common case is where ''A'' is a [[polynomial matrix]], which is called a '''polynomial eigenvalue problem'''. In particular, the specific case where the polynomial has [[degree of a polynomial|degree]] two is called a [[quadratic eigenvalue problem]], and can be written in the form:
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| :<math>A(\lambda) \mathbf{x} = ( A_2 \lambda^2 + A_1 \lambda + A_0) \mathbf{x} = 0 , \,</math>
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| in terms of the constant square matrices ''A''<sub>0,1,2</sub>. This can be converted into an ordinary linear generalized eigenproblem of twice the size by defining a new vector <math>\mathbf{y} = \lambda \mathbf{x}</math>. In terms of '''x''' and '''y''', the quadratic eigenvalue problem becomes:
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| :<math>\begin{pmatrix} -A_0 & 0 \\ 0 & I \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix} = \lambda
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| \begin{pmatrix} A_1 & A_2 \\ I & 0 \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix}
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| , </math>
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| where ''I'' is the identity matrix. More generally, if ''A'' is a matrix polynomial of degree ''d'', then one can convert the nonlinear eigenproblem into a linear (generalized) eigenproblem of ''d'' times the size.
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| Besides converting them to ordinary eigenproblems, which only works if ''A'' is polynomial, there are other methods of solving nonlinear eigenproblems based on the [[Jacobi-Davidson algorithm]] or based on [[Newton's method]] (related to [[inverse iteration]]).
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| ==References==
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| * [[Françoise Tisseur]] and Karl Meerbergen, "The quadratic eigenvalue problem," ''SIAM Review'' '''43''' (2), 235-286 (2001).
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| * Gene H. Golub and Henk A. van der Vorst, "Eigenvalue computation in the 20th century," ''Journal of Computational and Applied Mathematics'' '''123''', 35-65 (2000).
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| * Philippe Guillaume, "Nonlinear eigenproblems," ''SIAM J. Matrix. Anal. Appl.'' '''20''' (3), 575-595 (1999).
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| * Axel Ruhe, "Algorithms for the nonlinear eigenvalue problem," ''SIAM Journal on Numerical Analysis'' '''10''' (4), 674-689 (1973).
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| [[Category:Linear algebra]]
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