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We give a self-contained randomized algorithm based on shifted inverse iteration which provably computes the eigenvalues of an arbitrary matrix $M\in\mathbb{C}^{n\times n}$ up to backward error $\delta\|M\|$ in $O(n^4+n^3\log^2(n/\delta)+\log(n/\delta)^2\log\log(n/\delta))$ floating point operations using $O(\log^2(n/\delta))$ bits of precision.
Das iterationsverfahren bei nicht selbstadjungierten linearen eigenwertaufgaben
Helmut Wielandt · 1944
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The calculation of the eigenvectors of a general complex matrix by inverse iteration
JM Varah · 1968
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The calculation of specified eigenvectors by inverse iteration
Gwendoline Peters and James H Wilkinson · 1971
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Global convergence of the QR algorithm for unitary matrices with some results for normal matrices
PJ Eberlein and CP Huang · 1975
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Inverse iteration, ill-conditioned equations and newton’s method
G Peters and James H Wilkinson · 1979
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Backward stability of the QR algorithm
Francoise Tisseur · 1996
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Applied numerical linear algebra
James W Demmel · 1997
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Computing an eigenvector with inverse iteration
Ilse CF Ipsen · 1997
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