representations of the quantum group suq 2 and the little q jacobi polynomials abstract in this paper we study the finite dimensional unitary representations of the quantum group suq 2 then we obtain the peter weyl theorem for suq 2 and the matrix elements of these unitary representations are explicitly expressed in terms of the little q jacobi polynomials which are known as q analogues of orthogonal polynomials using these expressions the orthogonality relations of these polynomials are obtained in terms of the haar measure on the quantum group suq 2

acoustic propagation in a refracting ocean waveguide with an irregular interface abstract a method is derived for computing the acoustic field in a cylindrically symmetric ocean with an irregular water sediment interface and a general depth dependent sound speed profile the approach is based on a stepwise coupled mode solution which was originally developed for a homogeneous water column the extension to a general sound speed profile is carried out with the aid of galerkin s method numerical examples are given for a shallow water waveguide with an upward refracting sound speed profile the first example shows that propagation loss is significantly greater in a rough bottomed waveguide than in a smooth bottomed waveguide in the second example it is shown that upward refraction reduces the effect of bottom roughness on sound propagation by reducing the insonification of the rough water sediment interface

sum rules and perfect screening conditions for the one component plasma first we show from the bbgky hierarchy and under exponential clustering assumption that the three dimensional one component plasma obeys the compressibility sum rule which links the compressibility to the fourth moment of the two point correlation function then it is proved that the first equation of the hierarchy is equivalent to the known value of the correlation function of momentum current and density finally we are concerned with the energy density its definition and its correlation with the numerical density and with itself

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

on matrix differential equations and abstract fg algorithm abstract there is well known relationship between both the isospectral flows and the unitarily self equivalent flows and the abstract fg eigenvalue algorithm namely the values of the isospectral and self equivalent flows at integer times coincide with the iterates of fg algorithm we show that such a relation is characteristic of a whole class of matrix differential equations all these equations are built by means of various representations of an arbitrary lie algebra on k n n

understanding the qr algorithm the qr algorithm is currently the most popular method for finding all eigenvalues of a full matrix while qr is now well understood by specialists in eigenvalue computations this understanding is not being conveyed effectively to the mathematical public many accounts present wilkinson s 1965 convergence proof others establish some of the connections between the qr algorithm the power method and inverse iteration usually much emphasis is rightly placed on the refinements such as shifts of origin which are required to make the algorithm competitive but practically all accounts fail to explain the basic meaning of qr iterations as a consequence the qr algorithm is widely thought to be difficult to understand the purpose of this paper is to try to convince the reader that the opposite is true in fact the qr algorithm is neither more nor less than a clever implementation of simultaneous iteration which is itself a natural easily understood extension of the power method this point

spectral decomposition and matrix valued orthogonal polynomials the relation between the spectral decomposition of a self adjoint operator which is realizable as a higher order recurrence operator and matrix valued orthogonal polynomials is investigated a general construction of such operators from scalar valued orthogonal polynomials is presented two examples of matrix valued orthogonal poly nomials with explicit orthogonality relations and three term recurrence relation are pre sented which both can be considered as 2 2 matrix valued analogues of subfamilies of askey wilson polynomials

spectral analysis and the haar functional on the quantum su 2 group the haar functional on the quantumsu 2 group is the analogue of invariant integration on the groupsu 2 if restricted to a subalgebra generated by a self adjoint element the haar functional can be expressed as an integral with a continuous measure or with a discrete measure or by a combination of both these results by woronowicz and koornwinder have been proved by using the corepresentation theory of the quantumsu 2 group and schur s orthogonality relations for matrix elements of irreducible unitary corepresentations these results are proved here by using a spectral analysis of the generator of the subalgebra the spectral measures can be described in terms of the orthogonality measures of orthogonal polynomials by using the theory of jacobi matrices

markov chains with memory tensor formulation and the dynamics of power iteration a markov chain with memory is no different from the conventional markov chain on the product state space such a markovianization however increases the dimensionality exponentially instead markov chain with memory can naturally be represented as a tensor whence the transitions of the state distribution and the memory distribution can be characterized by specially defined tensor products in this context the progression of a markov chain can be interpreted as variants of power like iterations moving toward the limiting probability distributions what is not clear is the makeup of the second dominant eigenvalue that affects the convergence rate of the iteration if the method converges at all casting the power method as a fixed point iteration this paper examines the local behavior of the nonlinear map and identifies the cause of convergence or divergence as an application it is found that there exists an open set of irreducible and aperiodic transition probability tensors where the z eigenvector type power iteration fails to converge

the elr method for computing the eigenvalues of a general matrix this paper presents an algorithm for computing all the eigenvalues of a full nonsymmetric matrix a all the current algorithms begin with the reduction of a to a hessenberg matrix h it is widely believed that the qr algorithm is the best method for computing the eigenvalues of h in this paper we suggest an alternative method which based on the fact that the eigenvalues of a tridiagonal matrix can be computed much faster than those of a hessenberg matrix we first reduce h to a tridiagonal matrix t and then compute the eigenvalues of t the reduction of h to t is performed by the elimination method 3 while the eigenvalues of t are computed by the lr method although the above two methods have been well known for a long time their use in this way has traditionally been rejected because of the possibility of numerical instability we show here that under certain conditions the elimination method is quite stable furthermore we describe a strategy for the lr algorithm that again results in stability

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

classical chaos and fluctuation dissipation relations for nonlinear response the classical nonlinear optical response is expressed in a form that closely resembles the fluctuation dissipation theorem the nth order response is shown to depend on interferences among n closely lying trajectories the relevant dynamical information on the vicinity of a given trajectory can be recast using the stability matrix related to the lyapunov exponents no such interference exists in the linear response and the nonlinear response is consequently a much more sensitive probe for classical chaos sequences of multiple femtosecond pulses can be designed to directly probe the stability matrix textcopyright 1996 the american physical society

on the continuous realization of iterative processes 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 grassmann rayleigh quotient iteration for computing invariant subspaces the classical rayleigh quotient iteration rqi allows one to compute a one dimensional invariant subspace of a symmetric matrix a here we propose a generalization of the rqi which computes a p dimensional invariant subspace of a cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature

asymptotic analysis of toda lattice on diagonalizable matrices soit x x 0 g x x 0 x 0 x est une matrice a valeur complexe et g x une integrale de contour a valeur matricielle on etudie la dynamique de x t sous les hypotheses suivantes 1 x 0 x 0 r n n est diagonalisable 2 g z z 3 x 0 est une matrice de hessenberg superieure irreductible

rational qr iteration without inversion in this manuscript a new method will be presented for performing a qr iteration with a i a i 1 qr without explicit inversion of the factor a i 1 a qr method driven by a rational function is attractive since convergence can occur at both sides of the matrix each step of this new iteration consists of two substeps in the explicit version first an rq factorization of the initial matrix a i rq will be computed followed by a qr factorization of the matrix a i q h the factorization of a i q h can be computed in an intelligent manner exploiting properties of the already known rq factorization of a i the similarity transformation yielding the qr step is defined by the unitary factor q in the qr factorization of the transformed matrix a i q h examples will be given illustrating how to efficiently compute the factorization for some specific classes of matrices the novelties of this approach with respect to these matrix classes will be discussed

generalized nonlinear fluctuation dissipation relation for the one component plasma we present a novel procedure leading to compact and tractable cubic p 3 and quartic p 4 scalar fluctuation dissipation relations for the classical one component plasma subjected to a weak external coulombic perturbation there is no external magnetic field the ultimate goal here is to establish a generalized nonlinear fluctuation dissipation relation in which a single p 1 point dynamical structure function is expressed as a linear combination of pth order external density response functions when cast in this form such an architecture by definition is invariant with respect to rotation on the p 1 sided polygon formed by the wave vector frequency arguments of the dynamical structure function the application of this rotation symmetry to suppress unphysical isolated singularities is a basic ingredient of the derivation

two connections between the sr and hr eigenvalue algorithms abstract the sr and hr algorithms are members of the family of gr algorithms for calculating eigenvalues and invariant subspaces of matrices this paper makes two connections between the sr and hr algorithms 1 an iteration of the sr algorithm on a 2 n 2 n symplectic butterfly matrix using shifts i 1 i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts i 1 i i 1 k 2 an iteration of the sr algorithm on a 2 n 2 n j tridagonal hamiltonian matrix using shifts i i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts 2 i i 1 k

a quaternion qr algorithm this paper extends the francis qr algorithm to quaternion and antiquaternion matrices it calculates a quaternion version of the schur decomposition using quaternion unitary similarity transformations following a finite step reduction to a hessenberg like condensed form a sequence of implicit qr steps reduces the matrix to triangular form eigenvalues may be read off the diagonal eigenvectors may be obtained from simple back substitutions for serial computation the algorithm uses only half the work and storage of the unstructured francis qr iteration by preserving quaternion structure the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors

qr like algorithms for eigenvalue problems in the year 2000 the dominant method for solving matrix eigenvalue problems is still the qr algorithm this paper discusses the family of gr algorithms with emphasis on the qr algorithm included are historical remarks an outline of what gr algorithms are and why they work and descriptions of the latest highly parallelizable versions of the qr algorithm now that we know how to parallelize it the qr algorithm seems likely to retain its dominance for many years to come

self similar flows abstract large classes of self similar isospectral flows can be viewed as continuous analogues of certain matrix eigenvalue algorithms in particular there exist families of flows associated with the qr lr and cholesky eigenvalue algorithms this paper uses lie theory to develop a general theory of self similar flows which includes the qr lr and cholesky flows as special cases also included are new families of flows associated with the sr and hr eigenvalue algorithms the basic theory produces analogues of unshifted single step eigenvalue algorithms but it is also shown how the theory can be extended to include flows which are continuous analogues of shifted and multiple step eigenvalue algorithms

two sided grassmann rayleigh quotient iteration the two sided rayleigh quotient iteration proposed by ostrowski computes a pair of corresponding left right eigenvectors of a matrix c we propose a grassmannian version of this iteration i e its iterates are pairs of p dimensional subspaces instead of one dimensional subspaces in the classical case the new iteration generically converges locally cubically to the pairs of left right p dimensional invariant subspaces of c moreover grassmannian versions of the rayleigh quotient iteration are given for the generalized hermitian eigenproblem the hamiltonian eigenproblem and the skew hamiltonian eigenproblem

the continuous time rayleigh quotient flow on the sphere a continuous time differential equation inspired by the rayleigh quotient iteration for a symmetric matrix a is studied the set of all continuous solutions termed the rayleigh quotient flow is shown to be a time scaled version of the newton flow for rayleigh quotient cost functional the scaling factor ensures that the rate of variation of the rayleigh quotient is constant and positive along solutions this interpretation leads to a precise phase portrait for the rayleigh quotient flow it is shown that a complete non degenerate solution of the rayleigh quotient flow visits each of the eigenvectors of a in ascending order

the qr algorithm revisited the qr algorithm is still one of the most important methods for computing eigenvalues and eigenvectors of matrices most discussions of the qr algorithm begin with a very basic version and move by steps toward the versions of the algorithm that are actually used this paper outlines a pedagogical path that leads directly to the implicit multishift qr algorithms that are used in practice bypassing the basic qr algorithm completely

a quadratically convergent qr like method without shifts for the hermitian eigenvalue problem abstract we propose a new qr like algorithm symmetric squared qr ssqr method that can be readily parallelized using commonly available parallel computational primitives such as matrix matrix multiplication and qr decomposition the algorithm converges quadratically and the quadratic convergence is achieved through a squaring technique without utilizing any kind of shifts we provide a rigorous convergence analysis of ssqr and derive structures for several of the important quantities generated by the algorithm we also discuss various practical implementation issues such as stopping criteria and deflation techniques we demonstrate the convergence behavior of ssqr using several numerical examples

convergence of algorithms of decomposition type for the eigenvalue problem we develop the theory of convergence of a generic gr algorithm for the matrix eigenvalue problem that includes the qr lr sr and other algorithms as special cases our formulation allows for shifts of origin and multiple gr steps the convergence theory is based on the idea that the gr algorithm performs nested subspace iteration with a change of coordinate system at each step thus the convergence of the gr algorithm depends on the convergence of certain sequences of subspaces it also depends on the quality of the coordinate transformation matrices as measured by their condition numbers we show that with a certain obvious shifting strategy the gr algorithm typically has a quadratic asymptotic convergence rate for matrices possessing certain special types of structure cubic convergence can be achieved