18 research outputs found
Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time
We exhibit a randomized algorithm which given a square complex
matrix with and , computes with high probability
invertible and diagonal such that and
in
arithmetic operations on a floating point machine with bits of precision. Here is the number of arithmetic
operations required to multiply two complex matrices numerically
stably, with for every , where
is the exponent of matrix multiplication. The algorithm is a variant
of the spectral bisection algorithm in numerical linear algebra (Beavers and
Denman, 1974). This running time is optimal up to polylogarithmic factors, in
the sense that verifying that a given similarity diagonalizes a matrix requires
at least matrix multiplication time. It significantly improves best previously
provable running times of arithmetic operations for
diagonalization of general matrices (Armentano et al., 2018), and (w.r.t.
dependence on ) arithmetic operations for Hermitian matrices
(Parlett, 1998).
The proof rests on two new ingredients. (1) We show that adding a small
complex Gaussian perturbation to any matrix splits its pseudospectrum into
small well-separated components. This implies that the eigenvalues of the
perturbation have a large minimum gap, a property of independent interest in
random matrix theory. (2) We rigorously analyze Roberts' Newton iteration
method for computing the matrix sign function in finite arithmetic, itself an
open problem in numerical analysis since at least 1986. This is achieved by
controlling the evolution the iterates' pseudospectra using a carefully chosen
sequence of shrinking contour integrals in the complex plane.Comment: 78 pages, 3 figures, comments welcome. Slightly edited intro from
previous version + explicit statement of forward error Theorem (Corolary
1.7). Minor corrections mad
Semi-classical Analysis and Pseudospectra
We prove an approximate spectral theorem for non-self-adjoint operators and
investigate its applications to second order differential operators in the
semi-classical limit. This leads to the construction of a twisted FBI
transform. We also investigate the connections between pseudospectra and
boundary conditions in the semi-classical limit
Mean left-right eigenvector self-overlap in the real Ginibre ensemble
We study analytically the Chalker-Mehlig mean diagonal overlap
between left and right eigenvectors associated with a complex
eigenvalue of matrices in the real Ginibre ensemble (GinOE). We
first derive a general finite expression for the mean overlap and then
investigate several scaling regimes in the limit . While
in the generic spectral bulk and edge of the GinOE the limiting expressions for
are found to coincide with the known results for the complex
Ginibre ensemble (GinUE), in the region of eigenvalue depletion close to the
real axis the asymptotic for the GinOE is considerably different. We also study
numerically the distribution of diagonal overlaps and conjecture that it is the
same in the bulk and at the edge of both the GinOE and GinUE, but essentially
different in the depletion region of the GinOE.Comment: 23 pages, 7 figure
Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization
We present a randomized, inverse-free algorithm for producing an approximate
diagonalization of any matrix pencil . The bulk of the
algorithm rests on a randomized divide-and-conquer eigensolver for the
generalized eigenvalue problem originally proposed by Ballard, Demmel, and
Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer
approach can be formulated to succeed with high probability as long as the
input pencil is sufficiently well-behaved, which is accomplished by
generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas,
Kulkarni, and Srivastava [Foundations of Computational Mathematics 2022]. In
particular, we show that perturbing and scaling regularizes its
pseudospectra, allowing divide-and-conquer to run over a simple random grid and
in turn producing an accurate diagonalization of in the backward error
sense. The main result of the paper states the existence of a randomized
algorithm that with high probability (and in exact arithmetic) produces
invertible and diagonal such that and in at most
operations, where is the asymptotic complexity of matrix
multiplication. This not only provides a new set of guarantees for highly
parallel generalized eigenvalue solvers but also establishes nearly matrix
multiplication time as an upper bound on the complexity of exact arithmetic
matrix pencil diagonalization.Comment: 58 pages, 8 figures, 2 table
Propriétés spectrales des opérateurs non-auto-adjoints aléatoires
In this thesis we are interested in the spectral properties of random non-self-adjoint operators. Weare going to consider primarily the case of small random perturbations of the following two types of operators: 1. a class of non-self-adjoint h-differential operators Ph, introduced by M. Hager [32], in the semiclassical limit (hâ0); 2. large Jordan block matrices as the dimension of the matrix gets large (Nââ). In case 1 we are going to consider the operator Ph subject to small Gaussian random perturbations. We let the perturbation coupling constant ÎŽ be e (-1/Ch) †Ύ â©œ h(k), for constants C, k > 0 suitably large. Let â be the closure of the range of the principal symbol. Previous results on the same model by M. Hager [32], W. Bordeaux-Montrieux [4] and J. Sjöstrand [67] show that if ÎŽ âȘą e(-1/Ch) there is, with a probability close to 1, a Weyl law for the eigenvalues in the interior of the pseudospectrumup to a distance âȘą (-h ln ÎŽ h) 2/3 to the boundary of â. We will study the one- and two-point intensity measure of the random point process of eigenvalues of the randomly perturbed operator and prove h-asymptotic formulae for the respective Lebesgue densities describing the one- and two-point behavior of the eigenvalues in â. Using the density of the one-point intensity measure, we will give a complete description of the average eigenvalue density in â describing as well the behavior of the eigenvalues at the pseudospectral boundary. We will show that there are three distinct regions of different spectral behavior in â. The interior of the of the pseudospectrum is solely governed by a Weyl law, close to its boundary there is a strong spectral accumulation given by a tunneling effect followed by a region where the density decays rapidly. Using the h-asymptotic formula for density of the two-point intensity measure we will show that two eigenvalues of randomly perturbed operator in the interior of â exhibit close range repulsion and long range decoupling. In case 2 we will consider large Jordan block matrices subject to small Gaussian random perturbations. A result by E.B. Davies and M. Hager [16] shows that as the dimension of the matrix gets large, with probability close to 1, most of the eigenvalues are close to a circle. They, however, only state a logarithmic upper bound on the number of eigenvalues in the interior of that circle. We study the expected eigenvalue density of the perturbed Jordan block in the interior of thatcircle and give a precise asymptotic description. Furthermore, we show that the leading contribution of the density is given by the Lebesgue density of the volume form induced by the PoincarĂ©metric on the disc D(0, 1).Dans cette thĂšse, nous nous intĂ©ressons aux propriĂ©tĂ©s spectrales des opĂ©rateurs non-auto-adjoints alĂ©atoires. Nous allons considĂ©rer principalement les cas des petites perturbations alĂ©atoires de deux types des opĂ©rateurs non-auto-adjoints suivants :1. une classe dâopĂ©rateurs non-auto-adjoints h-diffĂ©rentiels Ph, introduite par M. Hager [32],dans la limite semiclassique (hâ0); 2. des grandes matrices de Jordan quand la dimension devient grande (Nââ). Dans le premier cas nous considĂ©rons lâopĂ©rateur Ph soumis Ă de petites perturbations alĂ©atoires. De plus, nous imposons que la constante de couplage ÎŽ vĂ©rifie e (-1/Ch) †Ύ â©œ h(k), pour certaines constantes C, k > 0 choisies assez grandes. Soit â lâadhĂ©rence de lâimage du symbole principal de Ph. De prĂ©cĂ©dents rĂ©sultats par M. Hager [32], W. Bordeaux-Montrieux [4] et J. Sjöstrand [67] montrent que, pour le mĂȘme opĂ©rateur, si lâon choisit ÎŽ âȘą e(-1/Ch), alors la distribution des valeurs propres est donnĂ©e par une loi de Weyl jusquâĂ une distance âȘą (-h ln ÎŽ h) 2/3 du bord de â. Nous Ă©tudions la mesure dâintensitĂ© Ă un et Ă deux points de la mesure de comptage alĂ©atoire des valeurs propres de lâopĂ©rateur perturbĂ©. En outre, nous dĂ©montrons des formules h-asymptotiques pour les densitĂ©s par rapport Ă la mesure de Lebesgue de ces mesures qui dĂ©crivent le comportement dâun seul et de deux points du spectre dans â. En Ă©tudiant la densitĂ© de la mesure dâintensitĂ© Ă un point, nous prouvons quâil y a une loi de Weyl Ă lâintĂ©rieur du pseudospectre,une zone dâaccumulation des valeurs propres dĂ»e Ă un effet tunnel prĂšs du bord du pseudospectre suivi par une zone oĂč la densitĂ© dĂ©croĂźt rapidement. En Ă©tudiant la densitĂ© de la mesure dâintensitĂ© Ă deux points, nous prouvons que deux valeurs propres sont rĂ©pulsives Ă distance courte et indĂ©pendantes Ă grande distance Ă lâintĂ©rieur de â. Dans le deuxiĂšme cas, nous considĂ©rons des grands blocs de Jordan soumis Ă des petites perturbations alĂ©atoires gaussiennes. Un rĂ©sultat de E.B. Davies et M. Hager [16] montre que lorsque la dimension de la matrice devient grande, alors avec probabilitĂ© proche de 1, la plupart des valeurs propres sont proches dâun cercle. De plus, ils donnent une majoration logarithmique du nombre de valeurs propres Ă lâintĂ©rieur de ce cercle. Nous Ă©tudions la rĂ©partition moyenne des valeurs propres Ă lâintĂ©rieur de ce cercle et nous en donnons une description asymptotique prĂ©cise. En outre, nous dĂ©montrons que le terme principal de la densitĂ© est donnĂ© par la densitĂ© par rapport Ă la mesure de Lebesgue de la forme volume induite par la mĂ©trique de PoincarĂ© sur la disque D(0, 1)