Showing papers on "Matrix (mathematics) published in 2000"

Random Matrix Theory and ζ(1/2+it)

Abstract: We study the characteristic polynomials Z(U, θ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of |Z| and Z/Z*, and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N→∞. In the limit, we show that these two distributions are independent and Gaussian. Costin and Lebowitz [15] previously found the Gaussian limit distribution for Im log Z using a different approach, and our result for the cumulants proves a conjecture made by them in this case. We also calculate the leading order N→∞ asymptotics of the moments of |Z| and Z/Z*. These CUE results are then compared with what is known about the Riemann zeta function ζ (s) on its critical line Re s= 1/2, assuming the Riemann hypothesis. Equating the mean density of the non-trivial zeros of the zeta function at a height T up the critical line with the mean density of the matrix eigenvalues gives a connection between N and T. Invoking this connection, our CUE results coincide with a theorem of Selberg for the value distribution of log ζ(1/2+iT) in the limit T→∞. They are also in close agreement with numerical data computed by Odlyzko [29] for large but finite T. This leads us to a conjecture for the moments of |ζ(1/2+it) |. Finally, we generalize our random matrix results to the Circular Orthogonal (COE) and Circular Symplectic (CSE) Ensembles.

Multireference perturbation theory for large restricted and selected active space reference wave functions

Abstract: A multireference second-order perturbation theory (MRPT2) has been developed which allows the use of reference wave functions with large active spaces and arbitrary configuration selection. Internally contracted configurations are used as a basis for all configuration subspaces of the first-order wave function for which the overlap matrix depends only on the second-order density matrix of the reference function. Some other subspaces which would require the third- or fourth-order density matrices are left uncontracted. This removes bottlenecks of the complete active space second order pertubation theory (CASPT2) caused by the need to construct and diagonalize large overlap matrices. Preliminary applications of the new method for 1,2-dihydronaphthalene (DHN) and free base porphin are presented in which the effect of applying occupancy restrictions in the reference wave function (restricted active space second-order perturbation theory, RASPT2) and reference configuration selection (general MRPT2) on electro...

Matrix iterative analysis (2nd edn), by Richard S. Varga. Springer Series in Computational Mathematics 27. Pp. 358. £55. 2000. ISBN 3 540 66321 5 (Springer Verlag).

Abstract: a 'sleeper', receiving only modest attention for 50 years before emerging as a cornerstone of communications engineering in more recent times. Roughly one in six of Walsh's 281 publications are included, photographically reproduced. Reproduction is excellent except for one paper from 1918. The book also reproduces three brief papers about Walsh and his work, by W. E. Sewell, D. V. Widder and Morris Marden. The first two were written for a special issue of the SIAM Journal celebrating Walsh's 70th birthday; the third is an obituary.

Size-specific sensitivity: applying a new structured population model

Abstract: Matrix population models require the population to be divided into discrete stage classes. In many cases, especially when classes are defined by a continuous variable, such as length or mass, there are no natural breakpoints, and the division is artificial. The authors introduce the integral projection model, which eliminates the need for division into discrete classes, without requiring any additional biological assumptions. Like a traditional matrix model, the integral projection model provides estimates of the asymptotic growth rate, stable size distribution, reproductive values, and sensitivities of the growth rate to changes in vital rates. However, where the matrix model represents the size distributions, reproductive value, and sensitivities as step functions (constant within a stage class), the integral projection model yields smooth curves for each of these as a function of individual size. The authors describe a method for fitting the model to data, and they apply this method to data on an endangered plant species, northern monkshood (Aconitum noveboracense), with individuals classified by stem diameter. The matrix and integral models yield similar estimates of the asymptotic growth rate, but the reproductive values and sensitivities in the matrix model are sensitive to the choice of stage classes. The integral projection modelmore » avoids this problem and yields size-specific sensitivities that are not affected by stage duration. These general properties of the integral projection model will make it advantageous for other populations where there is no natural division of individuals into stage classes.« less

Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations

Abstract: Madych and Nelson [1] proved multiquadric (MQ) mesh-independent radial basis functions (RBFs) enjoy exponential convergence. The primary disadvantage of the MQ scheme is that it is global, hence, the coefficient matrices obtained from this discretization scheme are full. Full matrices tend to become progressively more ill-conditioned as the rank increases. In this paper, we explore several techniques, each of which improves the conditioning of the coefficient matrix and the solution accuracy. The methods that were investigated are 1. (1) replacement of global solvers by block partitioning, LU decomposition schemes, 2. (2) matrix preconditioners, 3. (3) variable MQ shape parameters based upon the local radius of curvature of the function being solved, 4. (4) a truncated MQ basis function having a finite, rather than a full band-width, 5. (5) multizone methods for large simulation problems, and 6. (6) knot adaptivity that minimizes the total number of knots required in a simulation problem. The hybrid combination of these methods contribute to very accurate solutions. Even though FEM gives rise to sparse coefficient matrices, these matrices in practice can become very ill-conditioned. We recommend using what has been learned from the FEM practitioners and combining their methods with what has been learned in RBF simulations to form a flexible, hybrid approach to solve complex multidimensional problems.

Integral equation solution of Maxwell's equations from zero frequency to microwave frequencies

Abstract: We develop a new method to precondition the matrix equation resulting from applying the method of moments (MoM) to the electric field integral equation (EFIE). This preconditioning method is based on first applying the loop-tree or loop-star decomposition of the currents to arrive at a Helmholtz decomposition of the unknown currents. However, the MoM matrix thus obtained still cannot be solved efficiently by iterative solvers due to the large number of iterations required. We propose a permutation of the loop-tree or loop-star currents by a connection matrix, to arrive at a current basis that yields a MoM matrix that can be solved efficiently by iterative solvers. Consequently, dramatic reduction in iteration count has been observed. The various steps can be regarded as a rearrangement of the basis functions to arrive at the MoM matrix. Therefore, they are related to the original MoM matrix by matrix transformation, where the transformation requires the inverse of the connection matrix. We have also developed a fast method to invert the connection matrix so that the complexity of the preconditioning procedure is of O(N) and, hence, can be used in fast solvers such as the low-frequency multilevel fast multipole algorithm (LP-MLFMA). This procedure also makes viable the use of fast solvers such as MLFMA to seek the iterative solutions of Maxwell's equations from zero frequency to microwave frequencies.

Advances in linear matrix inequality methods in control

Abstract: Preface Notation Part I. Introduction. Robust Decision Problems in Engineering: A linear matrix inequality approach L. El Ghaoui and S.-I. Niculescu Part II. Algorithms and Software: Mixed Semidefinite-Quadratic-Linear Programs J.-P. A. Haeberly, M. V. Nayakkankuppam and M. L. Overton Nonsmooth algorithms to solve semidefinite programs C. Lemarechal and F. Oustry sdpsol: A Parser/Solver for Semidefinite Programs with Matrix Structure S.-P. Wu and S. Boyd Part III. Analysis: Parametric Lyapunov Functions for Uncertain Systems: The Multiplier Approach M. Fu and S. Dasgupta Optimization of Integral Quadratic Constraints U. Jonsson and A. Rantzer Linear Matrix Inequality Methods for Robust H2 Analysis: A Survey with Comparisons F. Paganini and E. Feron Part IV. Synthesis. Robust H2 Control K. Y. Yang, S. R. Hall and E. Feron Linear Matrix Inequality Approach to the Design of Robust H2 Filters C. E. de Souza and A. Trofino Robust Mixed Control and Linear Parameter-Varying Control with Full Block Scalings C. W. Scherer Advanced Gain-Scheduling Techniques for Uncertain Systems P. Apkarian and R. J. Adams Control Synthesis for Well-Posedness of Feedback Systems T. Iwasaki Part V. Nonconvex Problems. Alternating Projection Algorithms for Linear Matrix Inequalities Problems with Rank Constraints K. M. Grigoriadis and E. B. Beran Bilinearity and Complementarity in Robust Control M. Mesbahi, M. G. Safonov and G. P. Papavassilopoulos Part VI. Applications:Linear Controller Design for the NEC Laser Bonder via Linear Matrix Inequality Optimization J. Oishi and V. Balakrishnan Multiobjective Robust Control Toolbox for LMI-Based Control S. Dussy Multiobjective Control for Robot Telemanipulators J. P. Folcher and C. Andriot Bibliography Index.

A sparse H -matrix arithmetic. Part II: application to multi-dimensional problems

Abstract: The preceding Part I of this paper has introduced a class of matrices (ℋ-matrices) which are data-sparse and allow an approximate matrix arithmetic of almost linear complexity. The matrices discussed in Part I are able to approximate discrete integral operators in the case of one spatial dimension. In the present Part II, the construction of ℋ-matrices is explained for FEM and BEM applications in two and three spatial dimensions. The orders of complexity of the various matrix operations are exactly the same as in Part I. In particular, it is shown that the applicability of ℋ-matrices does not require a regular mesh. We discuss quasi-uniform unstructured meshes and the case of composed surfaces as well.

Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs. II

Abstract: In part I of this paper [S. Chen, X. Li, and X. Zhou, SIAM J. Control Optim., 36 (1998), pp. 1685--1702], an optimization model of stochastic linear quadratic regulators (LQRs) with indefinite control cost weighting matrices is proposed and studied. In this sequel, the problem of solving LQR models with system diffusions dependent on both state and control variables, which is left open in part I, is tackled. First, the solvability of the associated stochastic Riccati equations (SREs) is studied in the normal case (namely, all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite, with at least one positive definite), which in turn leads to an optimal state feedback control of the LQR problem. In the general indefinite case, the problem is decomposed into two optimal LQR problems, one with a forward dynamics and the other with a backward dynamics. The well-posedness and solvability of the original LQR problem are then obtained by solving these two subproblems, and an optimal control is explicitly constructed. Examples are presented to illustrate the results.

Special Functions: A Unified Theory Based on Singularities

Abstract: Preface 1. Linear Second-order ODE with Polynomial Coefficients 2. The Hypergeometric Class of Equations 3. The Heun Class of Equations 4. Application to Physical Sciences 5. The Painleve Class of Equations A. Gamma-Function and Related Functions B. CTCPs for Heun Equations in General Form C. Multipole Matrix Elements D. SFTools - Database of the Special Functions

Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework

Abstract: A critical disadvantage of primal-dual interior-point methods compared to dual interior-point methods for large scale semidefinite programs (SDPs) has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semidefinite matrix completion, this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage. Our method is used in two ways. One is a conversion of a sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior-point method for SDPs employing a standard block-diagonal matrix data structure. The other way is an incorporation of our method into primal-dual interior-point methods which we can apply directly to a given SDP. In Part II of this article, we will investigate an implementation of such a primal-dual interior-point method based on positive definite matrix completion, and report some numerical results.

Constraint Preconditioning for Indefinite Linear Systems

Abstract: The problem of finding good preconditioners for the numerical solution of indefinite linear systems is considered. Special emphasis is put on preconditioners that have a 2 × 2 block structure and that incorporate the (1,2) and (2,1) blocks of the original matrix. Results concerning the spectrum and form of the eigenvectors of the preconditioned matrix and its minimum polynomial are given. The consequences of these results are considered for a variety of Krylov subspace methods. Numerical experiments validate these conclusions.

Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation.

Abstract: General equations are derived for computing the amplitude matrix for a nonspherical particle in an arbitrary orientation and for arbitrary illumination and scattering directions with respect to the laboratory reference frame, provided that the scattering problem can be solved with respect to the particle reference frame. These equations are used along with the T-matrix method to provide benchmark results for homogeneous, dielectric, rotationally symmetric particles. The computer code is publicly available on the World-Wide Web at http://www.giss.nasa.gov/~crmim.

On Matrix–Valued Herglotz Functions

Abstract: We provide a comprehensive analysis of matrix–valued Herglotz functions and illustrate their applications in the spectral theory of self–adjoint Hamiltonian systems including matrix–valued Schrodinger and Dirac–type operators. Special emphasis is devoted to appropriate matrix–valued extensions of the well–known Aronszajn–Donoghue theory concerning support properties of measures in their Nevanlinna–Riesz–Herglotz representation. In particular, we study a class of linear fractional transformations MA(z) of a given n × n Herglotz matrix M(z) and prove that the minimal support of the absolutely continuous part of the measure associated to MA(z) is invariant under these linear fractional transformations. Additional applications discussed in detail include self–adjoint finite–rank perturbations of self–adjoint operators, self–adjoint extensions of densely defined symmetric linear operators (especially, Friedrichs and Krein extensions), model operators for these two cases, and associated realization theorems for certain classes of Herglotz matrices.

Comparison of permutation methods for the partial correlation and partial mantel tests

Abstract: This study compares empirical type I error and power of different permutation techniques that can be used for partial correlation analysis involving three data vectors and for partial Mantel tests. The partial Mantel test is a form of first-order partial correlation analysis involving three distance matrices which is widely used in such fields as population genetics, ecology, anthropology, psychometry and sociology. The methods compared are the following: (1) permute the objects in one of the vectors (or matrices); (2) permute the residuals of a null model; (3) correlate residualized vector 1 (or matrix A) to residualized vector 2 (or matrix B); permute one of the residualized vectors (or matrices); (4) permute the residuals of a full model. In the partial correlation study, the results were compared to those of the parametric t-test which provides a reference under normality. Simulations were carried out to measure the type I error and power of these permutatio methods, using normal and non-normal data, ...

On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix

Abstract: We consider bipartite matching algorithms for computing permutations of a sparse matrix so that the diagonal of the permuted matrix has entries of large absolute value. We discuss various strategies for this and consider their implementation as computer codes. We also consider scaling techniques to further increase the relative values of the diagonal entries. Numerical experiments show the effect of the reorderings and the scaling on the solution of sparse equations by a direct method and by preconditioned iterative techniques.

Algorithms for matrix canonical forms

Abstract: Computing canonical forms of matrices over rings is a classical math¬ ematical problem with many applications to computational linear alge¬ bra. These forms include the Frobenius form over a field, the Hermite form over a principal ideal domain and the Howell and Smith form over a principal ideal ring. Generic algorithms are presented for computing each of these forms together with associated unimodular transformation matrices. The algorithms are analysed, with respect to the worst case, in terms of number of required operations from the ring. All algorithms are deterministic. For a square input matrix, the algorithms recover each of these forms in about the same number of operations as required for matrix multiplication. Special emphasis is placed on the efficient computation of transforms for the Hermite and Smith form in the case of rectangular input matrices. Here we analyse the running time of our algorithms in terms of three parameters: the row dimension, the column dimension and the number of nonzero rows in the output matrix. The generic algorithms are applied to the problem of computing the Hermite and Smith form of an integer matrix. Here the complexity anal¬ ysis is in terms of number of bit operations. Some additional techniques are developed to avoid intermediate expression swell. New algorithms are demonstrated to construct transformation matrices which have good bounds on the size of entries. These algorithms recover transforms in essentially the same time as required by our algorithms to compute only the form itself. Kurzfassung Kanonischen Formen von Matrizen über Ringen zu berechnen, ist ein klassisches mathematisches Problem mit vielen Anwendungen zur konstruktiven linearen Algebra. Diese Formen umfassen die Frobenius Form über einem Körper und die Hermite-, Howellund Smith-Form über einem Hauptidealring. Wir studieren die Berechnung dieser Formen aus der Sicht von sequentiellen deterministischen Komplexitätsschranken im schlimmsten Fall. Wir präsentieren Algorithmen für das Berechnen aller dieser Formen sowie der dazugehörigen unimodularen Transforma¬ tionsmatrizen samt Analyse der Anzahl benötigten Ringoperationen. Die Howell-, HermiteSmithund Frobenius-Form einer quadratischen Matrix kann mit ungefähr gleich vielen Operationen wie die Matrixmul¬ tiplikation berechnet werden. Ein Schwerpunkt liegt hier bei der effizienten Berechnung der Hermiteund Smith-Form sowie der dazugehörigen Transformationsmatrizen im Falle einer nichtquadratischen Eingabematrix. In diesem Fall analysieren wir die Laufzeit unserer Algorithmen abhänhig von drei Parametern: die Anzahl der Zeilen, die Anzahl der Spalten und die Anzahl der Zeilen in der berechneten Form, die mindestens ein Element ungleich Null enthal¬ ten. Die generische Algorithmen werden auf das Problem des Aufsteilens der Hermiteund Smith-Form einer ganzzahligen Matrix angewendet. Hier wird die Komplizität des Verfahren in der Anzahl der benötigten Bitoperationen ausgedrückt. Einige zusätzliche Techniken wurden en¬ twickelt, um das übermässige Wachsen von Zwischenergebnissen zu ver¬ meiden. Neue Verfahren zur Konstruktion von Transformationsmatrizen für die Hermiteund Smith-Form einer ganzzahligen Matrix wurden en¬ twickelt. Ziel der Bemühungen bei der Entwicklung dieser Verfahren war im Wesentlichen das erreichen der gleichen obere Schranke für die Laufzeit, die unsere Algorithmen benötigen, um nur die Form selbst zu berechnen.

Uncorrelated modes of the non-linear power spectrum

Abstract: Non-linear evolution causes the galaxy power spectrum to become broadly correlated over different wavenumbers. It is shown that pre-whitening the power spectrum – transforming the power spectrum in such a way that the noise covariance becomes proportional to the unit matrix – greatly narrows the covariance of power. The eigenfunctions of the covariance of the pre-whitened non-linear power spectrum provide a set of almost uncorrelated non-linear modes somewhat analogous to the Fourier modes of the power spectrum itself in the linear, Gaussian regime. These almost uncorrelated modes make it possible to construct a near-minimum variance estimator and Fisher matrix of the pre-whitened non-linear power spectrum analogous to the Feldman–Kaiser–Peacock (FKP) estimator of the linear power spectrum. The paper concludes with summary recipes, in gourmet, fine and fastfood versions, of how to measure the pre-whitened non-linear power spectrum from a galaxy survey in the FKP approximation. An appendix presents FFTLog, a code for taking the fast Fourier or Hankel transform of a periodic sequence of logarithmically spaced points, which proves useful in some of the manipulations.

All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication

Abstract: We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms.The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value in O(n2+μ) time, where μ satisfies the equation ω(1, μ, 1) = 1 + 2μ and ω(1, μ, 1) is the exponent of the multiplication of an n × nμ matrix by an nμ × n matrix. Currently, the best available bounds on ω(1, μ, 1), obtained by Coppersmith, imply that μ 0 is an error parameter and W is the largest edge weight in the graph, after the edge weights are scaled so that the smallest non-zero edge weight in the graph is 1. It returns estimates of all the distances in the graph with a stretch of at most 1 + ϵ. Corresponding paths can also be found efficiently.

Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II—A framework for volume mesh optimization and the condition number of the Jacobian matrix

Abstract: Three-dimensional unstructured tetrahedral and hexahedral finite element mesh optimization is studied from a theoretical perspective and by computer experiments to determine what objective functions are most effective in attaining valid, high quality meshes. The approach uses matrices and matrix norms to extend the work in Part I to build suitable 3D objective functions. Because certain matrix norm identities which hold for 2 x 2 matrices do not hold for 3 x 3 matrices. significant differences arise between surface and volume mesh optimization objective functions. It is shown, for example, that the equivalence in two-dimensions of the Smoothness and Condition Number of the Jacobian matrix objective functions does not extend to three dimensions and further. that the equivalence of the Oddy and Condition Number of the Metric Tensor objective functions in two-dimensions also fails to extend to three-dimensions. Matrix norm identities are used to systematically construct dimensionally homogeneous groups of objective functions. The concept of an ideal minimizing matrix is introduced for both hexahedral and tetrahedral elements. Non-dimensional objective functions having barriers are emphasized as the most logical choice for mesh optimization. The performance of a number of objective functions in improving mesh quality was assessed on a suite of realistic test problems, focusing particularly on all-hexahedral ''whisker-weaved'' meshes. Performance is investigated on both structured and unstructured meshes and on both hexahedral and tetrahedral meshes. Although several objective functions are competitive, the condition number objective function is particularly attractive. The objective functions are closely related to mesh quality measures. To illustrate, it is shown that the condition number metric can be viewed as a new tetrahedral element quality measure.

Incomplete cross approximation in the mosaic-skeleton method

Abstract: The mosaic-skeleton method was bred in a simple observation that rather large blocks in very large matrices coming from integral formulations can be approximated accurately by a sum of just few rank-one matrices (skeletons). These blocks might correspond to a region where the kernel is smooth enough, and anyway it can be a region where the kernel is approximated by a short sum of separable functions (functional skeletons). Since the effect of approximations is like that of having small-rank matrices, we find it pertinent to say about mosaic ranks of a matrix which turn out to be pretty small for many nonsingular matrices. On the first stage, the method builds up an appropriate mosaic partitioning using the concept of a tree of clusters and some extra information rather than the matrix entries (related to the mesh). On the second stage, it approximates every allowed block by skeletons using the entries of some rather small cross which is chosen by an adaptive procedure. We focus chiefly on some aspects of practical implementation and numerical examples on which the approximation time was found to grow almost linearly in the matrix size.

Time-Explicit Representation of Relative Motion Between Elliptical Orbits

Abstract: Theclassictreatmentofrendezvousmechanicsandotherproblemsinvolvingtherelativemotionoftwospacecraft assumesa circularreferenceorbit, allowing a simpleclosed-form description ofthemotion. Asolution isdeveloped using an elliptical reference orbit, expanding the state transition matrix in powers of eccentricity, while retaining the explicit time dependence of the three-dimensional motion. The solution includes separate matrix elements for e rst-and second-order terms in eccentricity and for both Cartesian and cylindrical coordinates. Assessment of the maximum errors in position and velocity components over one complete revolution of the reference satellite shows that the solution is accurate for practical purposes with eccentricities in the range 0 ‐0.3. An example application is given for the proposed laser interferometer space antenna gravity wave experiment.

Breakdown of universality in multi-cut matrix models

Abstract: We solve the puzzle of the disagreement between orthogonal polynomials methods and mean-field calculations for random N×N matrices with a disconnected eigenvalue support. We show that the difference does not stem from a 2 symmetry breaking, but from the discreteness of the number of eigenvalues. This leads to additional terms (quasiperiodic in N) which must be added to the naive mean-field expressions. Our result invalidates the existence of a smooth topological large-N expansion and some postulated universality properties of correlators. We derive the large-N expansion of the free energy for the general two-cut case. From it we rederive by a direct and easy mean-field-like method the two-point correlators and the asymptotic orthogonal polynomials. We extend our results to any number of cuts and to non-real potentials.