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Flanders' theorem for many matrices under commutativity assumptions
No full textWe analyze the relationship between the Jordan canonical form of products, in different orders, of square matrices . Our results extend some classical results by H. Flanders. Motivated by a generalization of Fiedler matrices, we study permuted products of under the assumption that the graph of non-commutativity relations of is a forest. Under this condition, we show that the Jordan structure of all nonzero eigenvalues is the same for all permuted products. For the eigenvalue zero, we obtain an upper bound on the dierence between the sizes of Jordan blocks for any two permuted products, and we show that this bound is attainable. For we show that, moreover, the bound is exhaustive
Performance predictions of multilevel communication optimal LU and QR factorizations on hierarchical platforms
No full textIn this paper we study the performance of two classical dense linear algebra algorithms, the LU and the QR factorizations, on multi- level hierarchical platforms. We note that we focus on multilevel QR factorization, and give a brief description of the multilevel LU factoriza- tion. We first introduce a performance model called Hierarchical Cluster Platform (HCP), encapsulating the characteristics of such platforms. The focus is set on reducing the communication requirements of studied al- gorithms at each level of the hierarchy. Lower bounds on communication are therefore extended with respect to the Hcp model. We then present a multilevel QR factorization algorithm tailored for those platforms, and provide a detailed performance analysis. We also provide a set of perfor- mance predictions showing the need for such hierarchical algorithms on large platforms
A community-driven global reconstruction of human metabolism
Get PDFMultiple models of human metabolism have been reconstructed, but each represents only a subset of our knowledge. Here we describe Recon 2, a community-driven, consensus 'metabolic reconstruction', which is the most comprehensive representation of human metabolism that is applicable to computational modeling. Compared with its predecessors, the reconstruction has improved topological and functional features, including ~2� more reactions and ~1.7� more unique metabolites. Using Recon 2 we predicted changes in metabolite biomarkers for 49 inborn errors of metabolism with 77% accuracy when compared to experimental data. Mapping metabolomic data and drug information onto Recon 2 demonstrates its potential for integrating and analyzing diverse data types. Using protein expression data, we automatically generated a compendium of 65 cell type�specific models, providing a basis for manual curation or investigation of cell-specific metabolic properties. Recon 2 will facilitate many future biomedical studies and is freely available at http://humanmetabolism.org/
Hankel Determinant Structure of the Rational Solutions for Fifth Painlevé Equation
Get PDFIn this paper, we construct the Hankel determinant representation of the rational solutions for the fifth Painlev\'{e} equation through the Umemura polynomials. Our construction gives an explicit form of the Umemura polynomials for in terms of the
Hankel Determinant formula. Besides, We compute the generating function of the entries in terms of logarithmic derivative of the Heun Confluent Function
The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential
Get PDFA new matrix function corresponding to the scalar unwinding number of
Corless, Hare, and Jeffrey is introduced.
This matrix unwinding function, , is shown to
be a valuable tool for deriving
identities involving the matrix logarithm and
fractional matrix powers,
revealing, for example, the
precise relation between and .
The unwinding function is also shown to be closely connected with the matrix
sign function.
An algorithm for computing the unwinding function based on the
Schur--Parlett method with a special reordering is proposed.
It is shown that matrix argument reduction using the function
,
which has eigenvalues with imaginary parts in the interval
and for which \e^A = \e^{\mathrm{mod}(A)},
can give significant computational savings in the evaluation
of the exponential by scaling and squaring algorithms
Efficient high-order rational integration and deferred correction with equispaced data
Get PDFStable high-order linear interpolation schemes are well suited for the accurate approximation of antiderivatives and the construction of efficient quadrature rules. In this paper we utilize for this purpose the family of linear barycentric rational interpolants by Floater and Hormann, which are particularly useful for interpolation with equispaced nodes. We analyze the convergence of integrals of these interpolants to those of analytic functions as well as functions with a finite number of continuous derivatives. As a by-product, our convergence analysis leads to an extrapolation scheme for rational quadrature at equispaced nodes. Furthermore, as a main application of our analysis, we present and investigate a new iterated deferred correction method for the solution of initial value problems, which allows to work efficiently even with large numbers of equispaced data. This so-called rational deferred correction (RDC) method turns out to be highly competitive with other methods relying on more involved implementations or non-equispaced node distributions. Extensive numerical experiments are carried out, comparing the RDC method to the well established spectral deferred correction method by Dutt, Greengard and Rokhlin
The local Langlands correspondence for inner forms of SL_n
No full textThis is a beamer presentation of MIMS eprint 2013.38 of the same title (joint authors are A-M. Aubert, P. Baum, R. Plymen, M. Solleveld).
Let be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group . It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for enhanced with an irreducible representation of an S-group and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of . An analogous result is shown in the archimedean case.
To settle the case where has positive characteristic, we employ the method of close fields. We prove that this method is compatible with the local Langlands correspondence for inner forms of , when the fields are close enough compared to the depth of the representations
Real option analysis in resilient energy networks
Get PDFThe resilience of future power systems are being challenged in three fronts: (i) decarbonising energy supply will alter supply mix; (ii) shift of previous non-electric demand onto the energy network will require the system to work at higher capacity; and (iii) expected changes in climate will alter demand and performance of electrical network components. This thesis quantitatively assesses the impact of future climate change on the resilience of a power system, in secure and hazardous conditions. This is done through the use of reliability indices and probabilistic security assessment. Dynamical thermal ratings of circuits are used throughout this thesis given their potential for increased capacity over the standard static ratings. The first finding is that the predicted future climate scenarios will result in components with lower thermal ratings then if used currently. Due to this, it is found that the reliability of the system decreases under further climate scenarios. In order to keep a satisfactory level of reliability in the system, a method of temporary overloaded circuits is introduced which doesn't result in a higher risk of component failure. The temporary overload method allows for the rating constraint to be violated provided the temperature constraint isn't. Applying this to the system, and assessing the results under various climate scenarios, it is found that the method is beneficial in terms of economical cost and system reliability. When applied to hazardous conditions, it is found the method has a higher potential to strengthen the reliability of the system in comparison to when used on the 'safe' system.
An approach is taken to aid the system operator in decision making under uncertain conditions. A scenario is devised in which an operator wants to plan the power dispatch for a future time period. This is done through the use of stochastic optimisation, where the uncertainty is encapsulated by the conductor ratings which are calculated using dynamical thermal ratings in which the weather parameters are stochastic. This is developed for a one and two period model, in which the two period model has the first and second period coupled through the addition of a ramp rate constraint in the optimisation. System adequacy indices and probabilistic security indices are added as constraints so the system operator can control the reliability of his system
Efficient and stable Arnoldi restarts for matrix functions based on quadrature
Get PDFWhen using the Arnoldi method for approximating f(A)b, the action of a matrix function on a vector, the maximum number of iterations that can be performed is often limited by the storage requirements of the full Arnoldi basis. As a remedy, different restarting algorithms have been proposed in the literature, none of which was universally applicable, efficient, and stable at the same time. We utilize an integral representation for the error of the iterates in the Arnoldi method which then allows us to develop an efficient quadrature-based restarting algorithm suitable for a large class of functions, including the so-called Stieltjes functions and the exponential function. Our method is applicable for functions of Hermitian and non-Hermitian matrices, requires no a-priori spectral information, and runs with essentially constant computational work per restart cycle. We comment on the relation of this new restarting approach to other existing algorithms and illustrate its efficiency and numerical stability by various numerical experiments
Covariance Structure Regularization via Entropy Loss Function
No full textThe need to estimate structured covariance matrices arises in a variety of applications and the problem is widely studied in statistics. A new method is proposed for regularizing the covariance structure of a given
covariance matrix whose underlying structure has been blurred by random noise, particularly when the dimension of the covariance matrix is high. The regularization is made by choosing an optimal structure from an available class of covariance structures in terms of minimizing the discrepancy, defined via the entropy loss function, between the given matrix and the class. A range of potential candidate structures comprising tridiagonal Toeplitz, compound symmetry, AR(1), and banded Toeplitz is considered.
It is shown that for the first three structures local or global minimizers of the discrepancy can be computed by one-dimensional optimization, while for the fourth structure Newton's method enables efficient computation of the global minimizer. Simulation studies are conducted, showing that the proposed new approach provides a reliable way to regularize covariance structures. The approach is also applied to real data analysis, demonstrating the usefulness of the proposed approach in practice