Search

The inverse scaling and squaring algorithm computes the logarithm of a square matrix $A$ by evaluating a rational approximant to the logarithm at the matrix $B:=A^{2^{-s}}$ for a suitable choice of $s$. We introduce a dual approach and approximate the logarithm of $B$ by solving the rational equation $r(X)=B$, where $r$ is a diagonal Padé approximant to the matrix exponential at $0$. This equation is solved by a substitution technique in the style of those developed by Fasi & Iannazzo (2020, Substitution algorithms for rational matrix equations. Elect. Trans. Num. Anal., 53, 500–521). The new method is tailored to the special structure of the diagonal Padé approximants to the exponential and in terms of computational cost is more efficient than the state-of-the-art inverse scaling and squaring algorithm.

This paper proposes a new model for the dynamics of correlation matrices, where the dynamics are driven by the likelihood score with respect to the matrix logarithm of the correlation matrix. In analogy to the exponential GARCH model for volatility, this transformation ensures that the correlation matrices remain positive definite, even in high dimensions. For the conditional distribution of returns, we assume a student-t copula to explain the dependence structure and univariate student-t for the marginals with potentially different degrees of freedom. The separation into volatility and correlation parts allows for a two-step estimation, which facilitates estimation in high dimensions. We derive estimation theory for one-step and two-step estimation. In an application to a set of six asset indices including financial and alternative assets we show that the model performs well in terms of diagnostics, specification tests, and out-of-sample forecasting.

The asymptotic behavior of exponential sums ∑ k = 1 N exp ⁡ ( 2 π i n k α ) \sum _{k=1}^N \exp ( 2\pi i n_k \alpha ) for Hadamard lacunary ( n k ) (n_k) is well known, but for general ( n k ) (n_k) very few precise results exist, due to number theoretic difficulties. It is therefore natural to consider random ( n k ) (n_k) , and in this paper we prove the law of the iterated logarithm for ∑ k = 1 N exp ⁡ ( 2 π i n k α ) \sum _{k=1}^N \exp (2\pi i n_k \alpha ) if the gaps n k + 1 − n k n_{k+1}-n_k are independent, identically distributed random variables. As a comparison, we give a lower bound for the discrepancy of { n k α } \{n_k \alpha \} under the same random model, exhibiting a completely different behavior.

[EN] The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Pade approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson-Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison., This research was funded by the European Regional Development Fund (ERDF) and the Spanish Ministerio de Economia y Competitividad Grant TIN2017-89314-P.

[EN] Recently, two general methods for evaluating matrix polynomials requiring one matrix product less than the Paterson-Stockmeyer method were proposed, where the cost of evaluating a matrix polynomial is given asymptotically by the total number of matrix product evaluations. An analysis of the stability of those methods was given and the methods have been applied to Taylor-based implementations for computing the exponential, the cosine and the hyperbolic tangent matrix functions. Moreover, a particular example for the evaluation of the matrix exponential Taylor approximation of degree 15 requiring four matrix products was given, whereas the maximum polynomial degree available using Paterson-Stockmeyer method with four matrix products is 9. Based on this example, a new family of methods for evaluating matrix polynomials more efficiently than the Paterson-Stockmeyer method was proposed, having the potential to achieve a much higher efficiency, i.e., requiring less matrix products for evaluating a matrix polynomial of certain degree, or increasing the available degree for the same cost. However, the difficulty of these family of methods lies in the calculation of the coefficients involved for the evaluation of general matrix polynomials and approximations. In this paper, we provide a general matrix polynomial evaluation method for evaluating matrix polynomials requiring two matrix products less than the Paterson-Stockmeyer method for degrees higher than 30. Moreover, we provide general methods for evaluating matrix polynomial approximations of degrees 15 and 21 with four and five matrix product evaluations, respectively, whereas the maximum available degrees for the same cost with the Paterson-Stockmeyer method are 9 and 12, respectively. Finally, practical examples for evaluating Taylor approximations of the matrix cosine and the matrix logarithm accurately and efficiently with these new methods are given., This research was partially funded by the European Regional Development Fund (ERDF) and the Spanish Ministerio de Economia y Competitividad grant TIN2017-89314-P, and by the Programa de Apoyo a la Investigacion y Desarrollo 2018 of the Universitat Politecnica de Valencia grant PAID-06-18-SP20180016.

This paper shows the practical necessity to equip the adaptive controller with a module capable of converting transfer function of the object of control from a discrete form to continuous and vice verse and to recalculating its parameters to another sampling period with the high accuracy. High accuracy is required by the modal regulator and observer for their normal operation. To meet this need, the authors designed a program in C++, which uses zero-order holder method which, in turn, includes calculation of matrix logarithm and matrix exponential. This program has minimal dependencies and minimal system requirements and was successfully tested on single-board minicomputer CubieBoard3.

In this work, we proposed a fast texture image retrieval method by using the learning Copula model of multiple Dual-tree complex wavelet transforms (DTCWTs). Compared with the discrete wavelet transform, DTCWT provides multiple-directions and multiple-scales decomposition to image and also has the fast calculation capability. In the proposed method Multiple DTCWTs is incorporated to get more texture features; compare to Gabor wavelet, DTCWT has less computational cost of decomposition. In DTCWT domains, we developed a Learning Copula Model (called LCMoMD) to describe the dependence between the subbands of multiple DTCWTs. For improving the retrieval performance, LCMoMD is first embedded in the linear space by utilizing matrix logarithm and Kernel Principal Component Analysis (KPCA) is used to calculate the features from the embedding Copula model in the linear space. Experiments demonstrate that our method has fast and robust performance of texture extraction compared to the state-of-the-art methods.

Randomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of $$x^T Bx$$ x T B x for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

The exponential of a matrix plays an important role in the theory of Lie groups. The main purpose of this paper is to examine matrix groups over the split quaternions and the exponential map from their Lie algebras into the groups. Since the set of split quaternions is a noncommutative algebra, the way of computing the exponential of a matrix over the split quaternions is more difficult than calculating the exponential of a matrix over the real or complex numbers. Therefore, we give a method of finding exponential of a split quaternion matrix by its complex adjoint matrix.

Inferring the latent structure of complex nonlinear dynamical systems in a data driven setting is a challenging mathematical problem with an ever increasing spectrum of applications in sciences and engineering. Koopman operator-based linearization provides a powerful framework that is suitable for identification of nonlinear systems in various scenarios. A recently proposed method by Mauroy and Goncalves is based on lifting the data snapshots into a suitable finite dimensional function space and identification of the infinitesimal generator of the Koopman semigroup. This elegant and mathematically appealing approach has good analytical (convergence) properties, but numerical experiments show that software implementation of the method has certain limitations. More precisely, with the increased dimension that guarantees theoretically better approximation and ultimate convergence, the numerical implementation may become unstable and it may even break down. The main sources of numerical difficulties are the computations of the matrix representation of the compressed Koopman operator and its logarithm. This paper addresses the subtle numerical details and proposes a new implementation algorithm that alleviates these problems.

It is well known that when the dimension of the data becomes very large, the sample covariance matrix S will not be a good estimator of the population covariance matrix Σ . Using such estimator, one typical consequence is that the estimated eigenvalues from S will be distorted. Many existing methods tried to solve the problem, and examples of which include regularizing Σ by thresholding or banding. In this paper, we estimate Σ by maximizing the likelihood using a new penalization on the matrix logarithm of Σ (denoted by A ) of the form: ‖ A − m I ‖ F 2 = ∑ i ( log ( d i ) − m ) 2 , where d i is the i th eigenvalue of Σ . This penalty aims at shrinking the estimated eigenvalues of A toward the mean eigenvalue m . The merits of our method are that it guarantees Σ to be non-negative definite and is computational efficient. The simulation study and applications on portfolio optimization and classification of genomic data show that the proposed method outperforms existing methods.

Let 𝔖 n be a semigroup of mappings of a set X with n elements into itself, A be some fixed subset of the set N of natural numbers, and V n (A) be a set of mappings from 𝔖 n , with lengths of cycles belonging to A. The mappings from V n (A) are called A-mappings. We suppose that the set A has an asymptotic density ϱ > 0, and that |k : k ≤ n, k ∈ A, m − k ∈ A|/n → ϱ 2 as n → ∞ uniformly over m ∈ [n, Cn] for each constant C > 1. A number M(α) of different elements in a set {α, α 2, α 3, …} is called an order of mapping α ∈ 𝔖 n . Consider a random mapping σ = σ n (A) having uniform distribution on V n (A). In the present paper it is shown that random variable ln M(σ n (A)) is asymptotically normal with mean l ( n ) = ∑ k ∈ A ( n ) ln ⁡ ( k ) / k $l(n)=\sum_{k\in A(\sqrt{n})}\ln(k)/{k}$ and variance ϱln3(n)/24, where A(t) = {k : k ∈ A, k ≤ t}, t > 0. For the case A = N this result was proved by B. Harris in 1973.

This paper outlines the matrix exponential description of radiative transfer. The eigendecomposition method which serves as a basis for computing the matrix exponential and for representing the solution in a discrete ordinate setting is considered. The mathematical equivalence of the discrete ordinate method, the matrix operator method, and the matrix Riccati equations method is proved rigorously by means of the matrix exponential formalism. For optically thin layers, approximate solution methods relying on the Padé and Taylor series approximations to the matrix exponential, as well as on the matrix Riccati equations, are presented. For optically thick layers, the asymptotic theory with higher-order corrections is derived, and parameterizations of the asymptotic functions and constants for a water-cloud model with a Gamma size distribution are obtained.

In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory.

Deciding whether a Markov matrix is embeddable (i.e. can be written as the exponential of a rate matrix) is an open problem even for $4\times 4$ matrices. We study the embedding problem and rate identifiability for the K80 model of nucleotide substitution. For these $4\times 4$ matrices, we fully characterize the set of embeddable K80 Markov matrices and the set of embeddable matrices for which rates are identifiable. In particular, we describe an open subset of embeddable matrices with non-identifiable rates. This set contains matrices with positive eigenvalues and also diagonal largest in column matrices, which might lead to consequences in parameter estimation in phylogenetics. Finally, we compute the relative volumes of embeddable K80 matrices and of embeddable matrices with identifiable rates. This study concludes the embedding problem for the more general model K81 and its submodels, which had been initiated by the last two authors in a separate work., Comment: 20 pages; 10 figures

We consider the computation of the matrix logarithm by using numerical quadrature. The efficiency of numerical quadrature depends on the integrand and the choice of quadrature formula. The Gauss–Legendre quadrature has been conventionally employed; however, the convergence could be slow for ill-conditioned matrices. This effect may stem from the rapid change of the integrand values. To avoid such situations, we focus on the double exponential formula, which has been developed to address integrands with endpoint singularity. In order to utilize the double exponential formula, we must determine a suitable finite integration interval, which provides the required accuracy and efficiency. In this paper, we present a method for selecting a suitable finite interval based on an error analysis as well as two algorithms, and one of these algorithms is an adaptive quadrature algorithm.

Noferini, Vanni/0000-0002-1775-041X; Tisseur, Francoise/0000-0002-1011-2570 WOS:000473026800016 We study the structured condition number of differentiable maps between smooth matrix manifolds, extending previous results to maps that are only R-differentiable for complex manifolds. We present algorithms to compute the structured condition number. As special cases of smooth manifolds, we analyze automorphism groups, and Lie and Jordan algebras associated with a scalar product. For such manifolds, we derive a lower bound on the structured condition number that is cheaper to compute than the structured condition number. We provide numerical comparisons between the structured and unstructured condition numbers for the principal matrix logarithm and principal matrix square root of matrices in automorphism groups as well as for the map between matrices in automorphism groups and their polar decomposition. We show that our lower bound can be used as a good estimate for the structured condition number when the matrix argument is well conditioned. We show that the structured and unstructured condition numbers can differ by many orders of magnitude, thus motivating the development of algorithms preserving structure. Republic of Turkey Ministry of National EducationMinistry of National Education - Turkey; European Research Council Advanced grant MATFUN [267526]; Engineering and Physical Sciences Research CouncilUK Research & Innovation (UKRI)Engineering & Physical Sciences Research Council (EPSRC) [EP/I005293]; Royal Society-Wolfson Research Merit AwardRoyal Society of London The first author's research was supported by the Republic of Turkey Ministry of National Education. The second author's work was partially supported by European Research Council Advanced grant MATFUN (267526). The third author's research was supported by Engineering and Physical Sciences Research Council grant EP/I005293 and by a Royal Society-Wolfson Research Merit Award.

In this paper, we handle the problem of human action recognition by combining covariance matrices as local spatio-temporal (ST) descriptors and local ST features extracted densely from action video. Unlike traditional methods that separately utilizing gradient-based feature and optical flow-based feature, we use covariance matrix to fuse the two types of feature. Since covariance matrices are Symmetric Positive Definite (SPD) matrices, which form a special type of Riemannian manifold. To measure the distance of SPDs while avoid computing the geodesic distance between them, covariance features are transformed to log-Euclidean covariance matrices (LECM) by matrix logarithm operation. After encoding LECM by Locality-constrained Linear Coding method, in order to provide position information to ST-LECM features, spatial pyramid is used to partition the video frames, and the average-pooling-on-absolute-value function is implemented over each sub-frames. Finally, non-linear support vector machine is used as classifier. Experiments on public human action datasets show that the proposed method obtains great improvements in recognition accuracy, in comparison to several state-ofthe-art methods.

In this work we investigate in detail the underlying spin-dynamics associated with 1H-14N double CP experiments under fast MAS, recently demonstrated by Carnevale et al. We employ matrix logarithm and Floquet theory to compute numerically the effective Hamiltonian associated to the time-dependent problem. Certain common features related to construction of effective Hamiltonians by both approaches are discussed. The main observations related to 1H-14N CPMAS/double CP transfer are: (a) various spin terms of the effective Hamiltonian strongly depend on the crystallite orientation; (b) significant CP transfer occurs only when the magnitudes of the effective 1H and 14N RF strengths are comparable, and simultaneously all pure 14N terms in the effective Hamiltonian are small, except for the longitudinal and the RF terms; (c) the sign of 14N CPMAS signal follows the sign of 14N effective RF strength; (d) sign of the double CP signal is largely independent of crystallite orientation. We predict and verify matching conditions employing multiples of the spinning frequency or involving different 14N RF strengths. We provide an analytical proof for (d). The proof also provides an estimate for the ratio of 1H-14N and 14N-1H transfer amplitudes which is further substantiated through simulations. In addition, we find that double CP signals include contributions from several single-quantum coherences present after the first CP process. The uneven contribution from different coherences leads to a reversal of signal at very short contact times, a feature noted experimentally by Carnevale et al. The connection between CPMAS transfer and efficient spin-lock is discussed and illustrated. The factors affecting second-order quadrupolar lineshapes in double CP experiment are examined. With a linear ramp of 1H RF amplitude we have observed that significant CP transfer occurs for more crystallite orientations resulting in improved sensitivity.

Unitary matrices arise in many ways in physics, in particular as a time evolution operator. For a periodically driven system one frequently wishes to compute a Floquet Hamilonian that should be a Hermitian operator $H$ such that $e^{-iTH}=U(T)$ where $U(T)$ is the time evolution operator at time corresponding the period of the system. That is, we want $H$ to be equal to $-i$ times a matrix logarithm of $U(T)$. If the system has a symmetry, such as time reversal symmetry, one can expect $H$ to have a symmetry beyond being Hermitian. We discuss here practical numerical algorithms on computing matrix logarithms that have certain symmetries which can be used to compute Floquet Hamiltonians that have appropriate symmetries. Along the way, we prove some results on how a symmetry in the Floquet operator $U(T)$ can lead to a symmetry in a basis of Floquet eigenstates., 25 pages, 13 figure, 14 ancillary files, primarily Matlab files. Updated to have correct spelling of author names of authors in the Arxiv metadata

Many quantum information measures can be written as an optimization of the quantum relative entropy between sets of states. For example, the relative entropy of entanglement of a state is the minimum relative entropy to the set of separable states. The various capacities of quantum channels can also be written in this way. We propose a unified framework to numerically compute these quantities using off-the-shelf semidefinite programming solvers, exploiting the approximation method proposed in [Fawzi, Saunderson, Parrilo, Semidefinite approximations of the matrix logarithm, arXiv:1705.00812]. As a notable application, this method allows us to provide numerical counterexamples for a proposed lower bound on the quantum conditional mutual information in terms of the relative entropy of recovery., Comment: 14 pages, v2: minor updates

Constitutive laws in terms of the logarithmic strain tensor log U , i.e. the principal matrix logarithm of the stretch tensor U = F T F corresponding to the deformation gradient F , have been a subject of interest in nonlinear elasticity theory for a long time. In particular, there have been multiple attempts to derive a viable constitutive law of nonlinear elasticity from an elastic energy potential which depends solely on the logarithmic strain measure ‖ log U ‖ 2 , i.e. an energy function W : GL + ( n ) → R of the form (1) W ( F ) = Ψ ( ‖ log U ‖ 2 ) with a suitable function Ψ : [ 0 , ∞ ) → R , where ‖ . ‖ denotes the Frobenius matrix norm and GL + ( n ) is the group of invertible matrices with positive determinant. However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly monotone function Ψ such that W of the form (1) is Legendre–Hadamard elliptic. Similarly, we consider the related isochoric strain measure ‖ dev n log U ‖ 2 , where dev n log U is the deviatoric part of log U . Although a polyconvex energy function in terms of this strain measure has recently been constructed in the planar case n = 2 , we show that for n ≥ 3 , no strictly monotone function Ψ : [ 0 , ∞ ) → R exists such that F ↦ Ψ ( ‖ dev n log U ‖ 2 ) is polyconvex or even rank-one convex. Moreover, a volumetric-isochorically decoupled energy of the form F ↦ Ψ ( ‖ dev n log U ‖ 2 ) + W vol ( det F ) cannot be rank-one convex for any function W vol : ( 0 , ∞ ) → R if Ψ is strictly monotone.

We consider some classes S of structured matrices endowed each one with a structure of Lie or Jordan algebra. We show that any S decomposes as the direct sum S = ⨁ s S s of well-described minimal ideals, being each one a class of structured matrices of the same type as S .

In the present paper, the micro-dilatation theory or void elasticity is extended to both large displacement and large dilatation. Firstly, the deformation gradient tensor has been freshly defined by means of the matrix exponential function. The newly defined relation for the deformation gradient has painstakingly investigated for the uniqueness, decomposition issues as well as objectivity and isotropy considerations. The relation of the displacement gradient and deformation gradient tensor is brought via the matrix logarithm function. The micro-dilatation theory constitutive laws are derived using the thermodynamic principles under the zero-centrosymmetric, weakly-centrosymmetric and fully-centrosymmetric cases. These cases have been derived and scrutinized by the numerical experiments. To achieve this assignment, the basic loadings are taken into account, e.g. the hydrostatic loading, simple traction and shear. Some conclusions and outlook pertaining to the above-mentioned cases and variable bulk density have thereafter discussed.

The paper reports interesting but unnoticed facts about irreducibility resp., reducibility of Flouqet factorisations and their harmonic implication in term of controllability in finite-dimensional linear continuous-time periodic FDLCP systems. Reducibility and irreducibility are attributed to matrix logarithm algorithms during computing Floquet factorisations in FDLCP systems, which are a pair of essential features but remain unnoticed in the Floquet theory so far. The study reveals that reducible Floquet factorisations may bring in harmonic waves variance into the Fourier analysis of FDLCP systems that in turn may alter our interpretation of controllability when the Floquet factors are used separately during controllability testing; namely, controllability interpretation discrepancy or simply, controllability discrepancy may occur and must be examined whenever reducible Floquet factorisations are involved. On the contrary, when irreducible Floquet factorisations are employed, controllability interpretation discrepancy can be avoided. Examples are included to illustrate such observations.

This study presents the vortex structure and numerical instability increase occurring when the level of elasticity is enhanced in inertial flows in planar contraction configuration for finitely extensible nonlinear elastic model by Peterlin (FENE-P) fluid . The re-entrant corner effect on corner vortices is also considered. The calculations are performed using extended matrix logarithm formulation described in a previous paper: A. Jafari et al. A new extended matrix logarithm formulation for the simulation of viscoelastic fluids by spectral elements. Computer & Fluids 2010; 39(9):1425-1438. In that reference, the proposed algorithm has been tested for simple geometry such as Poiseuille flow. In this study, we are interested in the capability of this algorithm for more complex geometry. This formulation helps to reach higher values of the Weissenberg number when compared with the classical one. Copyright (c) 2015John Wiley & Sons, Ltd.

It is well known that the eigenvalues of any unitary matrix lie on the unit circle. The purpose of this paper is to prove that the eigenvalues of any matrix polynomial, with unitary coefficients, lie inside the annulus A_{1/2,2) := {z ∈ C | 1/2 < |z| < 2}. The foundations of this result rely on an operator version of Rouche’s theorem and the intermediate value theorem.

In this article, we discuss the exponential and the logarithmic functions in the realm of matrices. These notions are very useful in the mathematical and the physical sciences [1,2]. We discuss some important results including the connections established between skew-symmetric and orthogonal matrices, etc., through the exponentialmap.

This paper presents a new type of circulant matrices. We call it the first and the last difference r-circulant matrix (FLDcircr matrix). We can verify that the linear operation, the matrix product and the inverse matrix of this type of matrices are still FLDcircr matrices. By constructing the basic FLDcircr matrix, we give the discriminance for FLDcircr matrices and the fast algorithm of the inverse and generalized inverse of the FLDcircr matrices.

There is a one to one mapping between a $p$ dimensional strictly positive definite covariance matrix $\Sigma$ and its matrix logarithm $L$. We exploit this relationship to study the structure induced on $\Sigma$ through a sparsity constraint on $L$. Consider $L$ as a random matrix generated through a basis expansion, with the support of the basis coefficients taken as a simple random sample of size $s=s^{*}$ from the index set $[p(p+1)/2]=\{1,\ldots,p(p+1)/2\}$. We find that the expected number of non-unit eigenvalues of $\Sigma$, denoted $\mathbb{E}[|\mathcal{A}|]$, is approximated with near perfect accuracy by the solution of the equation ¶ \[\frac{4p+p(p-1)}{2(p+1)}[\log (\frac{p}{p-d})-\frac{d}{2p(p-d)}]-s^{*}=0.\] Furthermore, the corresponding eigenvectors are shown to possess only ${p-|\mathcal{A}^{c}|}$ non-zero entries. We use this result to elucidate the precise structure induced on $\Sigma$ and $\Sigma^{-1}$. We demonstrate that a positive definite symmetric matrix whose matrix logarithm is sparse is significantly less sparse in the original domain. This finding has important implications in high dimensional statistics where it is important to exploit structure in order to construct consistent estimators in non-trivial norms. An estimator exploiting the structure of the proposed class is presented.

The matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix functions, many of which are of importance in quantum information theory. In this paper we show how to approximate the matrix logarithm with functions that preserve operator concavity and can be described using the feasible regions of semidefinite optimization problems of fairly small size. Such approximations allow us to use off-the-shelf semidefinite optimization solvers for convex optimization problems involving the matrix logarithm and related functions, such as the quantum relative entropy. The basic ingredients of our approach apply, beyond the matrix logarithm, to functions that are operator concave and operator monotone. As such, we introduce strategies for constructing semidefinite approximations that we expect will be useful, more generally, for studying the approximation power of functions with small semidefinite representations., 31 pages, Introduction rewritten and numerous small changes throughout

We consider the two logarithmic strain measures \(\omega _{\mathrm {iso}}= ||{{\mathrm{dev}}}_n \log U ||\) and \(\omega _{\mathrm {vol}}= |{{\mathrm{tr}}}(\log U) |\), which are isotropic invariants of the Hencky strain tensor \(\log U = \log (F^TF)\), and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group \({{\mathrm{GL}}}(n)\). Here, F is the deformation gradient, \(U=\sqrt{F^TF}\) is the right Biot-stretch tensor, \(\log \) denotes the principal matrix logarithm, \(||\,.\, ||\) is the Frobenius matrix norm, \({{\mathrm{tr}}}\) is the trace operator and Open image in new window is the n-dimensional deviator of \(X\in \mathbb {R}^{n\times n}\). This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor \(\varepsilon ={{\mathrm{sym}}}\nabla u\), which is the symmetric part of the displacement gradient \(\nabla u\), and reveals a close geometric relation between the classical quadratic isotropic energy potential in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor R, where \(F=RU\) is the polar decomposition of F.

We investigate a finite element formulation of the exponentiated Hencky-logarithmic model whose strain energy function is given by $$\begin{aligned} W_\mathrm {eH}(\varvec{F}) = \dfrac{\mu }{k}\, e^{\displaystyle k \left||\text{ dev }_n \log \varvec{U}\right||^2} + \dfrac{\kappa }{2 \hat{k}}\, e^{\displaystyle \hat{k} [\text{ tr } (\log \varvec{U})]^2 }, \end{aligned}$$ where $$\mu >0$$ is the (infinitesimal) shear modulus, $$\kappa >0$$ is the (infinitesimal) bulk modulus, k and $$\hat{k}$$ are additional dimensionless material parameters, $$\varvec{U}=\sqrt{\varvec{F}^T\varvec{F}}$$ is the right stretch tensor corresponding to the deformation gradient $$\varvec{F}$$ , $$\log $$ denotes the principal matrix logarithm on the set of positive definite symmetric matrices, $$\text{ dev }_n \varvec{X} = \varvec{X}-\frac{\text{ tr } \varvec{X}}{n}\varvec{1}$$ and $$||\varvec{X} ||= \sqrt{\text{ tr }\varvec{X}^T\varvec{X}}$$ are the deviatoric part and the Frobenius matrix norm of an $$n\times n$$ -matrix $$\varvec{X}$$ , respectively, and $$\text{ tr }$$ denotes the trace operator. To do so, the equivalent different forms of the constitutive equation are recast in terms of the principal logarithmic stretches by use of the spectral decomposition together with the undergoing properties. We show the capability of our approach with a number of relevant examples, including the challenging “eversion of elastic tubes” problem.

Bilinear pooling of Convolutional Neural Network (CNN) features [22, 23], and their compact variants [10], have been shown to be effective at fine-grained recognition, scene categorization, texture recognition, and visual question-answering tasks among others. The resulting representation captures second-order statistics of convolutional features in a translationally invariant manner. In this paper we investigate various ways of normalizing these statistics to improve their representation power. In particular we find that the matrix square-root normalization offers significant improvements and outperforms alternative schemes such as the matrix logarithm normalization when combined with elementwise square-root and l2 normalization. This improves the accuracy by 2-3% on a range of fine-grained recognition datasets leading to a new state of the art. We also investigate how the accuracy of matrix function computations effect network training and evaluation. In particular we compare against a technique for estimating matrix square-root gradients via solving a Lyapunov equation that is more numerically accurate than computing gradients via a Singular Value Decomposition (SVD). We find that while SVD gradients are numerically inaccurate the overall effect on the final accuracy is negligible once boundary cases are handled carefully. We present an alternative scheme for computing gradients that is faster and yet it offers improvements over the baseline model. Finally we show that the matrix square-root computed approximately using a few Newton iterations is just as accurate for the classification task but allows an order-of-magnitude faster GPU implementation compared to SVD decomposition.

We continue our investigation of the asymptotic behavior of dynamical systems described by matrices, which was started in last chapter, now moving to the continuous time case. This means that we investigate the asymptotic properties of the matrix exponential function.

We discuss a completely forgotten work of the geologist GF Becker on the ideal isotropic nonlinear stress–strain function ( Am J Sci 1893; 46: 337–356). Due to the fact that the mathematical modelling of elastic deformations has evolved greatly since the original publication we give a modern reinterpretation of Becker’s work, combining his approach with the current framework of the theory of nonlinear elasticity. Interestingly, Becker introduces a multiaxial constitutive law incorporating the logarithmic strain tensor, more than 35 years before the quadratic Hencky strain energy was introduced by Heinrich Hencky in 1929. Becker’s deduction is purely axiomatic in nature. He considers the finite strain response to applied shear stresses and spherical stresses, formulated in terms of the principal strains and stresses, and postulates a principle of superposition for principal forces which leads, in a straightforward way, to a unique invertible constitutive relation, which in today’s notation can be written as [Formula: see text] where TBiot is the Biot stress tensor, log( U) is the principal matrix logarithm of the right Biot stretch tensor [Formula: see text], [Formula: see text] denotes the trace and dev3 X = X − (1/3) tr ( X) · 11 denotes the deviatoric part of a matrix [Formula: see text]. Here, G is the shear modulus and K is the bulk modulus. For Poisson’s number ν = 0 the formulation is hyperelastic and the corresponding strain energy [Formula: see text] has the form of the maximum entropy function.

A limiting property of the matrix exponential is proven: if the (1,1)-block of a 2-by-2 block matrix becomes “arbitrarily small” in a limiting process, the matrix exponential of that matrix tends to zero in the (1,1)-, (1,2)-, and (2,1)-blocks. The limiting process is such that either the log norm of the (1,1)-block goes to negative infinity, or, for a certain polynomial dependency, the matrix associated with the largest power of the variable that tends to infinity is stable. The limiting property is useful for simplification of dynamic systems that exhibit modes with sufficiently different time scales. The obtained limit then implies the decoupling of the corresponding dynamics.

© 2014 SIAM.The Fréchet derivative Lf of a matrix function f: ℂn×n → ℂn×n controls the sensitivity of the function to small perturbations in the matrix. While much is known about the properties of Lf and how to compute it, little attention has been given to higher order Fréchet derivatives. We derive sufficient conditions for the kth Fréchet derivative to exist and be continuous in its arguments and we develop algorithms for computing the kth derivative and its Kronecker form. We analyze the level-2 absolute condition number of a matrix function ("the condition number of the condition number") and bound it in terms of the second Fréchet derivative. For normal matrices and the exponential we show that in the 2-norm the level-1 and level-2 absolute condition numbers are equal and that the relative condition numbers are within a small constant factor of each other. We also obtain an exact relationship between the level-1 and level-2 absolute condition numbers for the matrix inverse and arbitrary nonsingular matrices, as well as a weaker connection for Hermitian matrices for a class of functions that includes the logarithm and square root. Finally, the relation between the level-1 and level-2 condition numbers is investigated more generally through numerical experiments.

It poses the inverse problem that consists in finding the logarithm of a function. It shows that when the function is holomorphic in a simply connected domain , the solution at the inverse problem exists and is unique if a branch of the logarithm is fixed. In addition, it’s demonstrated that when the function is continuous in a domain , where is Hausdorff space and connected by paths. The solution of the problem exists and is unique if a branch of the logarithm is fixed and is stable; for what in this case, the inverse problem turns out to be well-posed.

To approximate $f(A){b}$---the action of a matrix function on a vector---by a Krylov subspace method, restarts may become mandatory due to storage requirements for the Arnoldi basis or due to the growing computational complexity of evaluating $f$ on a Hessenberg matrix of growing size. A number of restarting methods have been proposed in the literature in recent years and there has been substantial algorithmic advancement concerning their stability and computational efficiency. However, the question under which circumstances convergence of these methods can be guaranteed has remained largely unanswered. In this paper we consider the class of Stieltjes functions and a related class, which contain important functions like the (inverse) square root and the matrix logarithm. For these classes of functions we present new theoretical results which prove convergence for Hermitian positive definite matrices $A$ and arbitrary restart lengths. We also propose a modification of the Arnoldi approximation which guaran...

The matrix exponential eA t plays a central role in linear system and control theory. This paper develops a method to compute the accurate solution for the matrix exponential eA t with the assumption that the matrix A has an eigenvalue s1=0. The examples show the effectiveness of the proposed method.

The mathematical model of pseudo-random number generator is given in the paper. The problems of discrete logarithm tasks solving and the concept of «hard bits» for discrete logarithm are considered in the paper. Constraints are imposed related to the absence of logarithm which can compute the discrete logarithm of y = g x mod p , where x ≤ 2 c for polynomial time. The constraint is called the assumption on discrete logarithm with short с bit exponents ( с – DLSE ). As an example, the Sundaram- Patel’s generator is given, qualitative and quantitative characteristics of the generator resistance to the main types of attacks are proposed. The paper gives the analysis of algorithms for generating pseudo-random numbers, such as the algorithm of Blum-Blum- Shub algorithm, Blum-Micali, Fortuna and Yarrow. Based on specified criteria, evaluation of algorithms is given, conclusions on the advantages and disadvantages of each algorithm are made.

We describe a method for evaluating both the Fibonacci-Horner and the polynomial decomposition of the principal matrix logarithm, with a view to solve the lifting problem of its explicit computation. The Binet formula for linear recursive sequences serves as a triggering factor for giving the exact formula. We supply some illustrative examples.

For statistical inferences that involve covariance matrices, it is desirable to obtain an accurate covariance matrix estimate with a well-structured eigen-system. We propose to estimate the covariance matrix through its matrix logarithm based on an approximate log-likelihood function. We develop a generalization of the Leonard and Hsu log-likelihood approximation that no longer requires a nonsingular sample covariance matrix. The matrix log-transformation provides the ability to impose a convex penalty on the transformed likelihood such that the largest and smallest eigenvalues of the covariance matrix estimate can be regularized simultaneously. The proposed method transforms the problem of estimating the covariance matrix into the problem of estimating a symmetric matrix, which can be solved efficiently by an iterative quadratic programming algorithm. The merits of the proposed method are illustrated by a simulation study and two real applications in classification and portfolio optimization. Supplementa...

Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present and investigate a novel strategy for the automated parameter selection when the function to be approximated is of Cauchy-Stieltjes (or Markov) type, such as the matrix square root or the logarithm. The performance of this approach is demonstrated by numerical examples involving symmetric and nonsymmetric matrices. These examples suggest that our black-box method performs at least as well, and typically better, as the standard rational Arnoldi method with parameters being manually optimized for a given matrix.

Suppose that the moving average process is based on a doubly infinite sequence of identically distributed and dependent random variables with zero mean and finite variance and that the sequence of coefficients is absolutely summable. Under suitable conditions of dependence, we show the precise rates in the law of logarithm of a kind of weighted infinite series for the first moment of the partial sums of the moving average process. This generalizes the common law of logarithm with the square root of logarithm to that with any positive power of logarithm. Moreover, we provide another law of logarithm as a supplement.