Аннотация
Consider a symmetric matrix \$A(v)ın\RR^nn\$ depending on a vector
\$vın\RR^n\$ and satisfying the property \$A(v)=A(v)\$ for any
\$\alphaın\RR0\$. We will here study the problem of finding
\$(łambda,v)ın\RR\RR^n\backslash\0\\$ such that \$(łambda,v)\$ is an
eigenpair of the matrix \$A(v)\$ and we propose a generalization of inverse
iteration for eigenvalue problems with this type of eigenvector nonlinearity.
The convergence of the proposed method is studied and several convergence
properties are shown to be analogous to inverse iteration for standard
eigenvalue problems, including local convergence properties. The algorithm is
also shown to be equivalent to a particular discretization of an associated
ordinary differential equation, if the shift is chosen in a particular way. The
algorithm is adapted to a variant of the Schrödinger equation known as the
Gross-Pitaevskii equation. We use numerical simulations toillustrate the
convergence properties, as well as the efficiency of the algorithm and the
adaption.
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