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Scaled Toda-like flows
Abstract
This paper discusses the class of isospectral flows , where ∘ denotes the Hadamard product and [., .] is the Lie bracket. The presence of A allows arbitrary and independent scaling for each element in the matrix X. The time-1 mapping of the scaled Toda-like flow still enjoys a QR-like iteration. The scaled structure includes the classical Toda flow, Brockett's double bracket flow, and other interesting flows as special cases. Convergence proof is thus unified and simplified. The effect of scaling on a variety of applications is demonstrated by examples.
... The best known examples are perhaps the isospectral flows [83], in particular the Toda flow [80] with its connection to the QR algorithm [78,34], and the work by Brockett [13] who produced a gradient flow that diagonalize matrices. There are also other examples, known to specialists but less known among most practitioners of numerical linear algebra [72,73,9,28,10,27,64,32,23,46,47,25,41]. Overviews and further references are available in survey papers [83,24,26,81]. ...
... From (25) we then obtain the explicit formulation of the flow aṡ ...
The space of probability distributions can be thought of as an infinite-dimensional Riemannian manifold. Classically, the Riemannian structure comes in two flavors: Wasserstein and Fisher-Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry–the differential geometric approach to statistics. Both Riemannian structures naturally restrict to the submanifold of multivariate Gaussian distributions, thereby inducing Riemannian structures on the space of covariance matrices.
Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher-Rao geometries are discussed. The link to matrices is obtained by considering OMT and information geometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the QR, Cholesky, spectral, and singular value decompositions. We also give a coherent description of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples.
The paper is a combination of previously known and original results. As a survey it covers the Riemannian description of OMT and polar decompositions (both smooth and linear category), entropy gradient flows, and the Fisher-Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. The original contributions include new gradient flows associated with various matrix decompositions, new geometric interpretations of previously studied isospectral flows, and a new proof of the polar decomposition of matrices based on convergence of a finite-dimensional lifted entropy gradient flow.
... When D is a diagonal matrix with distinct eigenvalues, the solution L(t) tends to a diagonal matrix L(zx~). If D = diag(n, n -1,..., 1) then the double-bracket flow becomes the Toda flow (1.3) (see [5]). The previous two flows are particular cases of the more general scaled Toda-like flow in which B(L) = T o L, where o denotes the Hadamard product on matrices and T is a skew-symmetric matrix characterizing the flow (see [5]). ...
... If D = diag(n, n -1,..., 1) then the double-bracket flow becomes the Toda flow (1.3) (see [5]). The previous two flows are particular cases of the more general scaled Toda-like flow in which B(L) = T o L, where o denotes the Hadamard product on matrices and T is a skew-symmetric matrix characterizing the flow (see [5]). In [2] Calvo et al. have shown that the classical numerical methods gives unsatisfactory results and that the isospectrality of L(t), generally, is lost. ...
This paper deals with the numerical solution of the Lax system L′ = [B(L),L], L(0) = L0 (∗), where L0 is a constant symmetric matrix, B(·) maps symmetric matrices into skew-symmetric matrices, and [B(L),L] is the commutator of B(L) and L. Here two different procedures, based on the approach recently proposed by Calvo, Iserles and Zanna (the MGLRK methods), are suggested. Such an approach is a computational form for the Flaschka formulation of (∗). Our numerical procedures consist in solving (∗) by a Runge-Kutta method, then, a single step of a Gauss-Legendre Runge-Kutta (GLRK) method may be applied to the Flaschka formulation of (∗). In the first procedure we compute the approximation of the Lax system by a continuous explicit RK method, instead, the second procedure computes the approximation of the Lax system by a GLRK method (the same method used for the Flaschka system). The computational costs have been derived and compared with the ones of the MGLRK methods. Finally, several numerical tests and computational comparisons will be shown.
... Since we are considering a unitary transformation, the spectrum of the Hamiltonian is preserved 1 . Thus, the SRG is related to so-called isospectral flows, a class of transformations that has been studied extensively in the mathematics literature (see for example Refs.[67][68][69][70][71]).The flow equation(10.8) is the most practical way of implementing an SRG evolution: We can obtainĤ(s) by integrating Eq. (10.8) numerically, without explicitly constructing the unitary transformation itself. Formally, we can also obtainÛ (s) by rearranging Eq. (10.7) into d dsÛ (s) =η(s)Û(s) . ...
We present a pedagogical discussion of Similarity Renormalization Group (SRG) methods, in particular the In-Medium SRG (IMSRG) approach for solving the nuclear many-body problem. These methods use continuous unitary transformations to evolve the nuclear Hamiltonian to a desired shape. The IMSRG, in particular, is used to decouple the ground state from all excitations and solve the many-body Schr\"odinger equation. We discuss the IMSRG formalism as well as its numerical implementation, and use the method to study the pairing model and infinite neutron matter. We compare our results with those of Coupled cluster theory, Configuration-Interaction Monte Carlo, and the Self-Consistent Green's Function approach. The chapter concludes with an expanded overview of current research directions, and a look ahead at upcoming developments.
... Here, the S-ordering operator ensures that the flow parameters appearing in the integrands are always in descending order, > > ¼ s s 1 2 . Since our continuous unitary transformation preserves the spectrum of the Hamiltonian and any other observable of interest, it is an example of a so-called isospectral flow, a class of transformations which has been studied extensively in the mathematics literature (see, e.g., [87][88][89] [48,61,63]). Note that the label diagonal does not need to mean strict diagonality here, but rather refers to a desired structure that the Hamiltonian will assume in the limit ¥ s . ...
We present a pedagogical introduction to the In-Medium Similarity Renormalization Group (IM-SRG) framework for ab initio calculations of nuclei. The IM-SRG performs continuous unitary transformations of the nuclear many-body Hamiltonian in second-quantized form, which can be implemented with polynomial computational effort. Through suitably chosen generators, it is possible to extract eigenvalues of the Hamiltonian in a given nucleus, or drive the Hamiltonian matrix in configuration space to specific structures, e.g., band- or block-diagonal form. Exploiting this flexibility, we describe two complementary approaches for the description of closed- and open-shell nuclei: The first is the Multireference IM-SRG (MR-IM-SRG), which is designed for the efficient calculation of nuclear ground-state properties. The second is the derivation of nonempirical valence-space interactions that can be used as input for nuclear Shell model (i.e., configuration interaction (CI)) calculations. This IM-SRG+Shell model approach provides immediate access to excitation spectra, transitions, etc., but is limited in applicability by the factorial cost of the CI calculations. We review applications of the MR-IM-SRG and IM-SRG+Shell model approaches to the calculation of ground-state properties for the oxygen, calcium, and nickel isotopic chains or the spectroscopy of nuclei in the lower $sd$ shell, respectively, and present selected new results, e.g., for the ground- and excited state properties of neon isotopes.
... In [49], a parameterized curve is constructed in S 2 passing through the iterates generated by the fixed-point iteration, the SDA and the Newton's method with some additional conditions. Finding a smooth curve with a specific structure that passes through a sequence of iterates generated by some numerical algorithm is a popular topic studied by many researchers, especially in the study of the so-called Toda flow that links matrices/matrix pairs generated by QR/QZ-algorithm [15,16,17,18,19,69]. The Toda flow is the solution of a nonlinear ordinary differential matrix equation in which the eigenvalues are preserved, but the eigenvectors are changed in t. ...
We construct a nonlinear differential equation of matrix pairs
$(\mathcal{M}(t),\mathcal{L}(t))$ that is invariant (the
\textbf{Structure-Preserving Property}) in the class of symplectic matrix pairs
\begin{align*}
\mathbb{S}_{\mathcal{S}_1,\mathcal{S}_2}=\left\{\left(\mathcal{M},\mathcal{L}\right)|
\ \mathcal{M}=\left[% \begin{array}{cc}
X_{12} & 0
X_{22} & I \end{array}% \right]\mathcal{S}_2, \mathcal{L}=\left[%
\begin{array}{cc}
I & X_{11}
0 & X_{21} \end{array}% \right]\mathcal{S}_1\right.\nonumber \left. \text{
and }X=\left[% \begin{array}{cc}
X_{11} & X_{12}
X_{21} & X_{22} \end{array}% \right]\text{is Hermitian}\right\} \end{align*}
for certain fixed symplectic matrices $\mathcal{S}_1$ and $\mathcal{S}_2$. Its
solution also preserves invariant subspaces on the whole orbit (the
\textbf{Eigenvector-Preserving Property}). Such a flow is called a
\textit{structure-preserving flow} and is governed by a Riccati differential
equation (RDE). In addition, Radon's lemma leads to an explicit form.
Therefore, blow-ups for the structure-preserving flows may happen at a finite
$t$. To continue, we then utilize the Grassmann manifolds to extend the domain
of the structure-preserving flow to the whole $\mathbb{R}$ subtracting some
isolated points.
... Essentially, an optimization over a group of permutations P n is relaxed to an optimization over an orthogonal group O n , using the double-bracket continuous-time gradient flows. Additional reading can be found in [80,24,104,57]. ...
... 1. The continuous realization processes (18) or (19) have the advantages that the desired form to which matrices are reduced can be almost arbitrary, and that if a desired form is not attainable then the limit point of the di erential system gives a way of measuring the distance from the best reduced matrices to the nearest matrices that have the desired form. ...
... In recent years there has been a growing interest in studying matrix differential systems whose solutions evolve on a smooth manifold such as the manifold of orthogonal or symplectic matrices (see, for instance, [3], [4], [8], [9], [19]). ...
In recent years there has been a growing interest in the dynamics of matrix differential systems on a smooth manifold. Research effort extends to both theory and numerical methods, particularly on the manifolds of orthogonal and symplectic matrices. This paper concerns dynamical systems on the manifold ${\cal OB}(n)$ of square oblique rotation matrices, a constraint appearing in some minimization problems and in multivariate data analysis. Background and theoretical results on differential equations on ${\cal OB}(n)$ are provided. Moreover, numerical procedures preserving the structure of the solution are found among known quadratic invariant preserving methods. Numerical tests and simulations on the oblique Procrustes problem are also reported.
... Thus, here comes the sudden understanding that the classical Toda lattice does have an objective function in mind. The componentwise scaled Toda flow [14], ˙ ...
It is known that there is a close relationship between matrix groups and linear transformations. The purpose of this exposition is to explore that relationship and to bridge it to the area of applied mathematics. Some known connections between discrete algorithms and differential systems will be used to motivate a larger framework that allows embracing more general matrix groups. Different types of group actions will also be considered. The inherited topological structure of the Lie groups makes it possible to design various flows to approximate or to effect desired canonical forms of linear transformations. While the group action on a fixed matrix usually preserves a certain internal properties or structures along the orbit, the action alone often is not sufficient to direct the orbit to the desired canonical form. Various means to further control these actions will be introduced. These controlled group actions on linear transformations often can be characterized by a certain dynamical systems on a certain matrix manifolds. Wide range of applications starting from eigenvalue computation to structured low rank approximation, and to some inverse problems are demonstrated. A number of open problems are identified.
... where L + (L − ) is the strictly upper (lower) triangular part of L (see [3]). Table 6 reports the global and unitary errors, at the end of the time interval, for the studied methods applied with h = 1/64. ...
In recent years some numerical methods have been developed to integrate matrix differential systems whose solutions are unitary matrices. In this paper we propose a new approach that transforms the original problem into a skew-Hermitian differential system by means of the Cayley transform. The new methods are semi-explicit, that is, no iteration is required but the solution of a certain number of linear matrix systems at each step is needed. Several numerical comparisons with known unitary integrators are reported.
... We have already mentioned (see Section 2.1.2) the work of Brockett [Bro91,Bro89], who has shown how one can use matrix differential equations to perform computation often thought of as being intrinsically discrete, and Bloch [Blo85,Blo90] who has shown how Hamiltonian systems may be used to solve principal component and linear programming problems. Chu [Chu95] has studied the Toda flow as a continuous-time analog of the QR algorithm. ...
Thesis (Ph. D. in Electrical Engineering and Compter Sciences)--University of California, Berkeley. Includes bibliographical references (leaves 312-322).
We present a pedagogical discussion of Similarity Renormalization Group (SRG) methods, in particular the In-Medium SRG (IMSRG) approach for solving the nuclear many-body problem. These methods use continuous unitary transformations to evolve the nuclear Hamiltonian to a desired shape. The IMSRG, in particular, is used to decouple the ground state from all excitations and solve the many-body Schrödinger equation. We discuss the IMSRG formalism as well as its numerical implementation, and use the method to study the pairing model and infinite neutron matter. We compare our results with those of Coupled cluster theory (Chap. 8), Configuration-Interaction Monte Carlo (Chap. 9), and the Self-Consistent Green’s Function approach discussed in Chap. 11 The chapter concludes with an expanded overview of current research directions, and a look ahead at upcoming developments.
The basic goal of an inverse eigenvalue problem is to reconstruct the physical parameters of a certain system from the knowledge or desire of its dynamical behavior. Depending on the application, inverse eigenvalue problems appear in many different forms. This book discusses the fundamental questions, some known results, many applications, mathematical properties, a variety of numerical techniques, as well as several open problems.
We work in the space Rn×n of n-by-n real matrices. We say that a linear transformation τ on this space is Toda-like if it maps symmetric matrices to skew-symmetric matrices. With such a transformation we associate a bilinear operation α defined by α(X, Y) ≔ [X, τY] + [Y, τX] where [U, V] ≔ UV − VU. Then Rn×n together with this operation is a (usually not associative) algebra. We call any subalgebra of such an algebra a Toda-like algebra. We classify these algebras, which arise in connection with eigenvalue computations. We determine the Toda-like algebras which are of primary interest for this application. We accomplish this classification by studying certain difference-weighted graphs which we associate with the algebras.
We show how to obtain polar decomposition as well as inversion of fixed and time-varying matrices using a class of nonlinear continuous-time dynamical systems. First we construct a dynamical system that causes an initial approximation of the inverse of a time-varying matrix to flow exponentially toward the true time-varying inverse. Using a time-parametrized homotopy from the identity matrix to a fixed matrix with unknown inverse, and applying our result on the inversion of time-varying matrices, we show how any positive definite fixed matrix may be dynamically inverted by a prescribed time without an initial guess at the inverse. We then construct a dynamical system that solves for the polar decomposition factors of a time-varying matrix given an initial approximation for the inverse of the positive definite symmetric part of the polar decomposition. As a by-product, this method gives another method of inverting time-varying matrices. Finally, using homotopy again, we show how dynamic polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix by a prescribed time.
The problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure. This paper describes a general procedure in using the projected gradient method. It is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicity. This gives rise to the construction of a descent flow that can be followed numerically. The explicit form also facilitates the computation of the second-order optimality conditions. Examples of applications are discussed. With slight modifications, the procedure can be extended to solve least squares problems for general matrices subject to singular-value constraints.
1. The field of values 2. Stable matrices and inertia 3. Singular value inequalities 4. Matrix equations and Kronecker products 5. Hadamard products 6. Matrices and functions 7. Totally positive matrices.
A general framework for constructing isospectral flows in the space $\operatorname{gl}(n)$ of n by n matrices is proposed. Depending upon how $\operatorname{gl}(n)$ is split, this framework gives rise to different types of abstract matrix factorizations. When sampled at integer times, these flows naturally define special iterative processes, and each flow is associated with the sequence generated by the corresponding abstract factorization. The proposed theory unifies as special cases the well-known matrix decomposition techniques used in numerical linear algebra and is likely to offer a broader approach to the general matrix factorization problem.
This paper discusses two dynamical systems on the unit sphere $S^{n - 1} $ in $\mathbb{R}^n $ space, each defined in terms of a real square matrix M. The solutions of these systems are found to converge to points which provide essential information about eigenvalues of the matrix M. It is shown, in particular, how the dynamics of the second flow is analogous to that of the Rayleigh quotient iterations.
A continuous version of the classical QR algorithm, known as the Toda flow, is generalized to complex-valued, full and nonsymmetric matrices. It is shown that this generalized Toda flow, when sampled at integer times, gives the same sequence of matrices as the OR algorithm applied to the matrix exp $(G(X_0 ))$. When $G(X) = X$, global convergence is deduced for the case of distinct real eigenvalues. This convergence property can also be understood locally by the center manifold theory. It is shown that the manifold of upper triangular matrices with decreasing main diagonal entries is the stable center manifold for the Toda flow. One interesting example is given to demonstrate geometrically the dynamical behavior of this flow.
It is trite but true to say that research on the symmetric eigenvalue problem has flourished since the first edition of this book appeared in 1980. I had dreamed of including the significant new material in an expanded second edition, but my own research obsessions diverted me from reading, digesting, and then regurgitating all that work. So it was with some relief that I accepted Gene Golub's suggestion that I let SIAM republish the original book in its Classics in Applied Mathematics series.
Naturally I could not resist the temptation to remove blemishes in the original, to sharpen some results, and to include some slicker proofs. In the first of these activities I was greatly helped by three people who read all, or nearly all, of the first edition with close attention. They are Thomas Ericsson, Zhaojun Bai, and Richard O. Hill, Jr.
In this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations. New algorithms are suggested and old algorithms, including $QR$, are interpreted.
The Toda flow is a dynamical system whose dependent variables may be viewed as the entries of a symmetric tridiagonal matrix. The spectrum of the matrix is invariant in time and, as $t \to \infty $, the off-diagonal entries tend to zero, exposing the eigenvalues on the main diagonal. Recently a wave of interest in the Toda and related isospectral flows was sparked in the numerical analysis community by the suggestion that to integrate the Toda flow numerically might be a cost effective way to calculate the eigenvalues of large symmetric tridiagonal matrices. A second source of interest was the recently discovered connection between the Toda flow and the $QR$ algorithm. The present paper has two principal aims. The first is to point out that the Toda and related flows are intimately connected with the power method. This connection clarifies completely the convergence properties of the flows. The relationship of the Toda flow to both the $QR$ algorithm and the power method is based on a connection with the $QR$ matrix factorization. The second aim of this paper is to introduce and discuss families of isospectral flows associated with two other well-known matrix factorizations, the $LU$ factorization and the Cholesky factorization.
Many important mathematical problems are solved by iterative methods. Many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems. This paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods.
A large class of matching problems can be thought of in the following way. One is given a group G which acts on a Riemannian manifold and one is given two points x1 and x2 in the manifold. The problem is to find gϵG such that d(g(x1),x2) is as small as possible. If the group has finite cardinality, this is a combinatorial optimization problem whereas if G is a Lie group, one can use the calculus to obtain necessary conditions. Our main results establish circumstances under which one can relax the combinatorial problem to an appropriate Lie group problem and then solve it by the method of steepest descent. An interesting class of applications involves embedding the permutation group on n letters in the orthogonal group O(n) with the latter acting on Rn(n+1)/2 via a symmetricized tensor product. The convergence properties of the descent equation which arises in this way are analyzed in some detail.
We establish a number of properties associated with the dynamical system , where H and N are symmetric n by n matrices and [A, B] = AB − BA. The most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices. We are especially interested in the role of this equation as an analog computer. For example, we show how to map the data associated with a linear programming problem into H(0) and N in such a way as to have evolve to a solution of the linear programming problem. This result can be applied to find systems which solve a variety of genetic combinatorial optimization problems, and it even provides an algorithm for diagonalizing symmetric matrices.
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Matrix differential equations: A continuous realization process for linear algebra problems, Nonlinear Anal., to appear. P. Deift, T. Nanda, and C. Tomei, Differential equations for the symmetric eigenvalue problem, SIAM .I Numec Anal
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