randomized qlp decomposition abstract we present the randomized qlp rqlp algorithm and its enhanced version erqlp for computing a low rank approximation to a matrix a of size m n such that a q l p t where l is a rank k lower triangular matrix q and p are column orthogonal matrices the implementation of rqlp and erqlp needs o m n k flops which are mainly applied in blas 3 operations we derive the lower bounds on the l values which can track the singular values of a approximately our claims are supported by numerical experiments

structural instability analyses based on generalised path following this paper describes how quasi static conservative instability problems can be completely described using generalised path following procedures for augmented equilibrium formulations in particul

continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

filtering and frequency estimation using perturbation formulas a new method for updating the svd is introduced based on perturbation formulas the complexity of the method is o n sup 2 applications are made to frequency estimation and filtering

new studies of randomized augmentation and additive preprocessing abstract a standard gaussian random matrix hereafter referred to just as gaussian matrix has full rank with probability 1 and is well conditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from the square shape and becomes more rectangular if we append sufficiently many gaussian rows or columns to any normalized and possibly rank deficient or ill conditioned matrix then the augmented matrix has full rank with probability 1 and is well conditioned with a probability close to 1 we specify and prove these properties of augmentation and extend them to additive preprocessing that is to adding a product of two rectangular gaussian matrices by applying our randomization techniques to a matrix that has numerical rank r we accelerate the known algorithms for the approximation of its trailing singular spaces associated with all its positive singular values except for the r largest values our algorithms use much fewer random parameters and run much faster when various random sparse and structured preprocessors replace gaussian empirically the outputs of the resulting algorithms are as accurate as the outputs under gaussian preprocessing our novel duality techniques provides formal support so far missing for these empirical observations and opens door to de randomization of our preprocessing and to further acceleration and simplification of our algorithms by using more efficient sparse and structured preprocessors our techniques and our progress can be applied to various other fundamental matrix computations such as the celebrated low rank approximation of a matrix by means of random sampling

a systolic implementation of the chan foster rrqr algorithm the rank revealing qr factorization is a useful tool in many signal processing applications since it explicitly yields all the necessary information to solve rank deficient least squares problems and subset selection problems to compute signal and noise subspaces etc the authors present a systolic algorithm for computing a rank revealing qr factorization and consider the performance of the algorithm

deflated decomposition of solutions of nearly singular systems when solving the linear system ax b where a may be nearly singular and b is not consistent with a one is often interested in computing a deflated solution i e a unique solution to a nearby singular but consistent system a s x d tilde b kelley 14 15 has considered deflated solutions with a s corresponding to setting a small pivot of the lu factorization of a to zero stewart 22 proposed an iterative algorithm for computing a deflated solution with a s corresponding to setting the smallest singular value of a to zero kelley s approach explicitly uses submatrices of the lu factors whereas stewart s approach is implicit in that it only involves solving systems with a in this paper we generalize the concept of a deflated solution to that of a deflated decomposition which expresses the solution x in terms of x d and the null vectors of a s we treat such decompositions in a uniform framework that includes the approaches of kelley and stewart and introduce some new deflated

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

continuum percolation with unreliable and spread out connections we derive percolation results in the continuum plane that lead to what appears to be a general tendency of many stochastic network models namely when the selection mechanism according to which nodes are connected to each other is sufficiently spread out then a lower density of nodes or on average fewer connections per node are sufficient to obtain an unbounded connected component we look at two different transformations that spread out connections and decrease the critical percolation density while preserving the average node degree our results indicate that real networks can exploit the presence of spread out and unreliable connections to achieve connectivity more easily provided they can maintain the average number of functioning connections per node

improved bound for rank revealing lu factorizations abstract in many applications it is necessary to determine the rank or numerical rank of a matrix many of these situations involve matrices that are very large order or that are sparse or that may undergo some form of modification rank k update row or column appended or removed in these cases the singular value decomposition s cost may be prohibitively high or the decomposition may not be computationally feasible especially for large sparse problems we thus examine the theoretical merits of rank revealing lu rrlu factorizations we find that in those cases where the nullity is small and the gap is well defined an rrlu factorization could be a very useful tool

maximizing bilinear forms subject to linear constraints abstract we present methods for maximizing a bilinear form x t ay x y or its absolute value subject to linear constraints the constraints are either of the form c x d g y h or of the form c x e r d g y e r h where r d and r h represent the column spaces of matrices d and h we also show how to compute the angle between two subspaces of r m subject to these types of linear constraints and how to compute linearly constrained canonical correlations and the corresponding canonical vectors

matrix pencil methodologies for computing the greatest common divisor of polynomials hybrid algorithms and their performance the computation of the greatest common divisor gcd of several polynomials is a problem that emerges in many fields of applications the gcd computation has a non generic nature and thus its numerical computation is a hard problem in this paper we examine the family of matrix pencil methods for gcd computation and investigate their performance as far as their complexity error analysis and their effectiveness for evaluating approximate solutions the relative merits of the various variants of such methods are examined for the different cases of sets of polynomials with varying number of elements and degree the developed algorithms combine symbolical and numerical programming and this is what we define here as hybrid computations the combination of numerical operations with symbolical programming can improve the nature of the methods and guarantees the stability of the algorithm furthermore it emphasizes the significance of hybrid computations in complex problems such as the computation of gcd all m

rounding errors in solving block hessenberg systems a rounding error analysis is presented for a divide and conquer algorithm to solve linear systems with block hessenberg matrices conditions are derived under which the algorithm computes a stable solution the algorithm is shown to be stable for block diagonally dominant matrices and for m matrices

a critical exponent for shortest path scaling in continuum percolation we carry out monte carlo experiments to study the scaling behavior of shortest path lengths in continuum percolation these studies suggest that the critical exponent governing this scaling is the same for both continuum and lattice percolation we use splitting a technique that has not yet been fully exploited in the physics literature to increase the speed of our simulations this technique can also be applied to other models where clusters are grown sequentially

an inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems we discuss an inverse free highly parallel spectral divide and conquer algorithm it can compute either an invariant subspace of a nonsymmetric matrix a or a pair of left and right deflating subspaces of a regular matrix pencil a lambda b this algorithm is based on earlier ones of bulgakov godunov and malyshev but improves on them in several ways this algorithm only uses easily parallelizable linear algebra building blocks matrix multiplication and qr decomposition but not matrix inversion similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function which requires matrix inversion and is faster but can be less stable than the new algorithm

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

so 2 1 lie algebra the jacobi matrix and the scattering states of the morse oscillator work exploring the connection between so 2 1 lie algebra and the jacobi matrix method for scattering is extended to the morse potential regular and irregular solutions of the three term recursion associated with the jacobi matrix are specified by their asymptotic behaviour and closed form expressions in terms of generalised hypergeometric functions are obtained this example further illustrates the connection between scattering theory and orthogonal polynomials

fast linear algebra is stable in demmel et al numer math 106 2 199 224 2007 we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense and that in fact if multiplication of n by n matrices can be done by any algorithm in o n operations for any 0 then it can be done stably in o n operations for any 0 here we extend this result to show that essentially all standard linear algebra operations including lu decomposition qr decomposition linear equation solving matrix inversion solving least squares problems generalized eigenvalue problems and the singular value decomposition can also be done stably in a normwise sense in o n operations

analytical approximation of the percolation threshold for overlapping ellipsoids of revolution analytic approximations for percolation points in two dimensional and three dimensional particulate arrays have been reported for only a very few simple particle geometries here an analytical approach is presented to determine the percolative properties i e statistical cluster properties of permeable ellipsoids of revolution we generalize a series expansion technique previously used by other authors to study arrays of spheres and cubes our analytic solutions are compared with monte carlo simulation results and show good agreement at low particle aspect ratio at higher aspect ratios the analytic approximation becomes even more computationally intensive than direct simulation of a number of realizations additional simulation results and simplified closed form bounding expressions for percolation thresholds are also presented limitations and applications of the asymptotic expressions are discussed in the context of materials design and design of sensor arrays

on the existence and computation of rank revealing lu factorizations abstract by exploring properties of schur complements this paper presents bounds on the existence of rank revealing lu factorizations that are comparable with those of rank revealing qr factorizations the new bounds provide substantial improvement over previously derived bounds this paper also proposes two algorithms using gaussian elimination with a block pivoting strategy to select a subset of columns from a given matrix which has a guaranteed relatively large smallest singular value each of these two algorithms is faster than its orthogonal counterpart for dense matrices if implemented appropriately these algorithms are faster than the corresponding rank revealing qr methods even when the orthogonal matrices are not explicitly updated based on these two algorithms an algorithm using only gaussian elimination for computing rank revealing lu factorizations is introduced

perturbation theory for the rank deficient equality constrained least squares problem this paper presents perturbation theory for the equality constrained least squares problem min f in s kf g 2 with s f lf h 2 min y in c n ly h 2 lse in which l and begin array 20 c l k end array may not be of full rank the analysis generalizes the results of elden 4 perturbation theorems for the lse problem are proved numerical experiments are also given to verify the error bounds

a krylov multisplitting algorithm for solving linear systems of equations abstract we consider the practical implementation of krylov subspace methods conjugate gradients gmres etc for parallel computers in the case where the preconditioning matrix arises from a multisplitting we show that the algorithm can be efficiently implemented by dividing the work into tasks that generate search directions and a single task that minimizes over the resulting subspace each task is assigned to a subset of processors it is not necessary for the minimization task to send frequent information to the direction generating tasks and this leads to high utilization with a minimum of synchronization we study the convergence properties of various forms of the algorithm and present results of numerical examples on a sequential computer

the 2 norm of random matrices abstract numerical experiments show that it is possible to derive simple estimates for the expected 2 norm of random matrices a with elements from a normal distribution with zero mean and standard deviation and from a poisson distribution with mean value these estimates are max m n 2 max m n and e 2 mn respectively where m and n are the dimensions of a

computation of coprime factorizations of rational matrices abstract we propose numerically reliable state space algorithms for computing several coprime factorizations of rational matrices 1 factorizations with factors having poles in a given stability domain 2 factorizations with proper stable factors 3 factorizations with inner and j inner denominators the new algorithms are based on a recursive generalized schur algorithm for pole dislocation they are generally applicable whether the underlying descriptor state space representation is minimal or not and whether it is stabilizable detectable or not the proposed algorithms are useful in solving various computational problems for both standard and descriptor system representations

a fast algorithm for index of annihilation computations in this paper a fast algorithm for computing the index of annihilation of the associated pencil of a given matrix is presented knowledge of this index leads us to the specification of the elementary divisors of the matrix and thus we can specify its canonical forms it is shown that the new algorithm which is based on the rrqr decomposition is faster than the existing svd approach the algorithm can be also applied to matrix pencils and thus specify the structure of their elementary divisors

a parallel butterfly algorithm the butterfly algorithm is a fast algorithm which approximately evaluates a discrete analogue of the integral transform int mathbb r d k x y g y dy at large numbers of target points when the kernel k x y is approximately low rank when restricted to subdomains satisfying a certain simple geometric condition in d dimensions with o n d quasi uniformly distributed source and target points when each appropriate submatrix of k is approximately rank r the running time of the algorithm is at most o r 2 n d log n a parallelization of the butterfly algorithm is introduced which assuming a message latency of alpha and per process inverse bandwidth of beta executes in at most o r 2 frac n d p log n beta r frac n d p alpha log p time using p processes this parallel algorithm was then instantiated in the form of the open source textttdistbutterfly library for the special case where k x y exp i phi x y where phi x y is a black box sufficiently smooth

efficient low rank approximation of matrices based on randomized pivoted decomposition given a matrix bf a with numerical rank k the two sided orthogonal decomposition tsod computes a factorization bf a bf udv t where bf u and bf v are orthogonal and bf d is upper lower triangular tsod is rank revealing as the middle factor bf d reveals the rank of bf a the computation of tsod however is demanding in this paper we present an algorithm called randomized pivoted tsod rp tsod where the middle factor is lower triangular key in our work is the exploitation of randomization and rp tsod is primarily devised to efficiently construct an approximation to a low rank matrix we provide three different types of bounds for rp tsod i we furnish upper bounds on the error of the low rank approximation ii we bound the k approximate principal singular values and iii we derive bounds for the canonical angles between the approximate and the exact singular subspaces our bounds describe the characteristics and behavior of our proposed algorithm through numerical tests we show the effectiveness of the devised bounds as well as our proposed algorithm

tracking poles representing hankel operators and the nehari problem we present an algorithm which locates the poles and zeros of a rational function given the values at the roots of unity so long as enough values are specified to make the problem well posed the algorithm is robust in a strong sense if the sample values are perturbed slightly it will identify the correct number of candidate poles and zeros and they will be close to the correct poles and zeros the algorithm proceeds by first calculating the discrete fourier transform from the given sample values and then examining the singular value decompositions of truncations of four hankel matrices formed from them we then show how given such sample values for a rational function o we may exploit the poles and zeros to construct the szego bases for ho the hankel operator with symbol o and hence solve the nehari problem the algorithms are shown to be robust and they are very accurate the results improve considerably on those of helton spain and young

simple and efficient nonparametric method for estimating the number of signals without eigendecomposition inspired by the computational simplicity and numerical stability of qr decomposition a nonparametric method for estimating the number of signals without eigendecomposition mense is proposed for the coherent narrowband signals impinging on a uniform linear array ula by exploiting the array geometry and its shift invariance property to decorrelate the coherency of signals through subarray averaging the number of signals is revealed in the rank of the qr upper trapezoidal factor of the autoproduct of a combined hankel matrix formed from the cross correlations between some sensor data since the infection of additive noise is defused signal detection capability is improved a new detection criterion is then formulated in terms of the row elements of the qr upper triangular factor when finite array data are available and the number of signals is determined as a value of the running index for which this ratio criterion is maximized where the qr decomposition with column pivoting is also used to improve detection performance the statistical analysis clarifies that the mense detection criterion is asymptotically consistent furthermore the proposed mense algorithm is robust against the array uncertainties including sensor gain and phase errors and mutual coupling and against the deviations from the spatial homogeneity of noise model the effectiveness of the mense is verified through numerical examples and the simulation results show that the mense is superior in detecting closely spaced signals with a small number of snapshots and or at relatively low signal to noise ratio snr

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

group theoretical foundation of the j matrix theory of scattering group theoretical analysis shows that so 2 1 is an underlying dynamical symmetry for all hamiltonians that are compatible with the jacobi matrix j matrix formalism the class of central potentials with this property is obtained including but not limited to the oscillator coulomb and morse potentials the l2 bases and j matrix elements for these potentials are found so 2 1 invariant transformation of the solutions of the recursion relation for one potential gives those for another potential in the class phase shift and resonance calculations for a single channel potential are carried out in the oscillator basis to illustrate the use of our results

j factorizations of a general discrete time system abstract the problems of extending the j lossless conjugation the j j spectral factorization and the j j lossless factorization to a completely general discrete time system are considered existence conditions together with a numerically sound prototype algorithm for computing the factors are provided for a system having any type of singularity including arbitrary normal rank poles and zeros at infinity at zero or on the unit circle

numerical solution of the discrete time periodic riccati equation in this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic riccati equation this method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the euler lagrange difference equations in doing so it is possible to extract a basis for the stable deflating subspace of the extended pencil from which the riccati solution is obtained this algorithm is an extension of the standard qz algorithm and retains its attractive features such as quadratic convergence and small relative backward error a method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with

computing rank deficiency of rectangular matrix pencils abstract rectangular matrix pencils arise in many contexts in linear control theory for example the pbh test for controllability and the computation of transmission zeros in this paper we examine a method to find the distance from an arbitrary given pencil to a nearby rank deficient one in light of the fact that rectangular pencils are generically full rank we propose an experimental computational method that exhibits quadratic convergence to a local minimum of this distance function this partially answers the question of the existence of a rank deficient pencil in a neighborhood of a given pencil we use the algorithm to illustrate some limitations of previous algorithms to measure this distance

the jacobi matrix method in parabolic coordinates expansion of coulomb functions in parabolic sturmians previous analysis of the jacobi matrix method based on the underlying so 2 1 lie algebra is extended to the coulomb hamiltonian in parabolic coordinates the general solution of the generic so 2 1 eigenvalue equation is constructed and special cases which furnish expansions of the coulomb functions k r in a complete set of parabolic sturmians are discussed

microstructure functions for a model of statistically inhomogeneous random media we propose a model for statistically inhomogeneous two phase random media including functionally graded materials consisting of inhomogeneous fully penetrable poisson distributed spheres this model can be constructed for any specified variation of volume fraction and permits one to represent and evaluate certain n point correlation functions that statistically characterize the microstructure for this model unlike the case of statistically homogeneous media the microstructure functions depend upon the absolute positions of their arguments however as with homogeneous random media this microstructural information will be useful in obtaining rigorous estimates of the effective properties of such systems copyright ital 1998 ital the american physical society

computing the singular value decomposition with high relative accuracy we analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy this means that each computed singular value is guaranteed to have some correct digits even if the singular values have widely varying magnitudes this is in contrast to the absolute accuracy provided by conventional backward stable algorithms which in general only guarantee correct digits in the singular values with large enough magniturds it is of interest to compute the tiniest singular values with several correct digits because in some cases such as finite element problems and quantum mechanics it is the smallest singular values that have physical meaning and should be determined accurately by the data many recent papers have identified special classes of matrices where high relative accuracy is possible since it is not possible in general the perturbation theory and algorithms for these matrix classes have been quite different motivating us to seek a common perturbation theory and common algorithm we provide these in this paper and show that high relative accuracy is possible in many new cases as well the briefest way to describe our results is that we can compute the svd to high relative accuracy provided we can compute a high accuracy pivoted ldu decomposition we provide many examples of matrix classes permitting such an ldu decomposition

on stabilization methods of descriptor systems abstract numerically reliable computational methods are proposed for the stabilization of a linear descriptor system with or without simultaneous elimination of its impulsive behavior two basic stabilization approaches are discussed the first approach relies on methods which represent generalizations of direct stabilization techniques for standard state space systems the second approach is based on a recursive generalized schur algorithm for pole assignment both approaches are based exclusively on numerically reliable procedures and can serve for robust software implementations

investigations in the numerical behavior of the adaptive rank revealing qr factorization we present a tracking procedure based on the rank revealing qr rrqr factorization and investigate its numerical properties by applying it to the direction of arrival problem we address numerical issues raised by the related work proposed earlier by prasad et al 1991 and we compare the performance of the proposed algorithm to that obtained using an evd based technique

unitary esprit how to obtain increased estimation accuracy with a reduced computational burden esprit is a high resolution signal parameter estimation technique based on the translational invariance structure of a sensor array previous esprit algorithms do not use the fact that the operator representing the phase delays between the two subarrays is unitary the authors present a simple and efficient method to constrain the estimated phase factors to the unit circle if centro symmetric array configurations are used unitary esprit the resulting closed form algorithm has an esprit like structure except for the fact that it is formulated in terms of real valued computations throughout since the dimension of the matrices is not increased this completely real valued algorithm achieves a substantial reduction of the computational complexity furthermore unitary esprit incorporates forward backward averaging leading to an improved performance compared to the standard esprit algorithm especially for correlated source signals like standard esprit unitary esprit offers an inexpensive possibility to reconstruct the impinging wavefronts signal copy these signal estimates are more accurate since unitary esprit improves the underlying signal subspace estimates simulations confirm that even for uncorrelated signals the standard esprit algorithm needs twice the number of snapshots to achieve a precision comparable to that of unitary esprit thus unitary esprit provides increased estimation accuracy with a reduced computational burden

on the error estimate for the projection of a point onto a linear manifold abstract suppose that the linear system a h x b is consistent the problem of finding the solution x to min a h x b 1 2 x p 2 2 is studied where a c n m b c m x p c n and a b p are given when a b and p are perturbed an error bound for the solution x is obtained roundoff error bounds for the singular value decomposition method the column pivoting method and the rank revealing method are obtained the results in this paper extend those of arioli laratta and menchi to the case when a is not of full rank and either m n or m n

on computing normalized coprime factorizations of general rational matrices presents a numerically reliable approach for computing the normalized coprime factorization of an arbitrary possibly improper rational matrix we consider both the cases in which the factorization is coprime and normalized with respect to either the left half plane or the unit disk corresponding to systems with continuous or discrete time evolutions the proposed algorithm remedies the numerical drawbacks of alternative methods available thus far in the literature

direction of arrival estimation using rank revealing qr factorization the authors describe a novel technique for direction of arrival estimation based on computing a permutation matrix e and a qr factorization re hb of the permuted covariance matrix r such that a possible rank deficiency of r is revealed in the triangular factor b having a minimum norm lower right block a subset of the columns of the orthogonal matrix h is shown to be orthogonal to the direction vectors of sources and hence can be used to estimate their bearings the cost of this algorithm is only slightly more than that of one qr factorization but is much lower than that of an eigen decomposition simulation results are included to show that the proposed method performs nearly as well as music in terms of signal resolution bias and variance of the estimated bearings

analysis of facial motion patterns during speech using a matrix factorization algorithm this paper presents an analysis of facial motion during speech to identify linearly independent kinematic regions the data consists of three dimensional displacement records of a set of markers located on a subject s face while producing speech a qr factorization with column pivoting algorithm selects a subset of markers with independent motion patterns the subset is used as a basis to fit the motion of the other facial markers which determines facial regions of influence of each of the linearly independent markers those regions constitute kinematic eigenregions whose combined motion produces the total motion of the face facial animations may be generated by driving the independent markers with collected displacement records

some new properties of the equality constrained and weighted least squares problem abstract in this paper some new properties of the equality constrained and weighted least squares problem wlse min w 1 2 kx g 2 subject to lx h are obtained we derive a perturbation bound based on an unconstrained least squares problem and deduce some equivalent formulae for the projectors of this unconstrained ls problem we also present a new way to compute the minimum norm solution x wlse of the wlse problem by using the qr decomposition of the corresponding matrices and propose an algorithm to compute x wlse using the qr factorizations some numerical examples are provided to compare different methods for solving the wlse problem

universality and cluster structures in continuum models of percolation with two different radius distributions the percolation thresholds and the fractal cluster structures for continuum models of percolation with uniform cm1 and variable radius cm2 distributions of discs and spheres are investigated and compared with the results of ordinary lattice percolation configurations of up to 250000 discs 2 dimensions and 100000 spheres 3 dimensions are numerically simulated in two dimensions the authors find distinctly different percolation concentrationss for models cm1 and cm2 in the three dimensional systems the percolation concentration for both models cannot be distinguished within their limits of accuracy the fractal dimensions of the cluster hull surface and volume are the same as in the corresponding lattice models the harris criterion for the continuum percolation problem is confirmed by their simulation

improving cur matrix decomposition and the nystrom approximation via adaptive sampling the cur matrix decomposition and the nystrom approximation are two important low rank matrix approximation techniques the nystrom method approximates a symmetric positive semidefinite matrix in terms of a small number of its columns while cur approximates an arbitrary data matrix by a small number of its columns and rows thus cur decomposition can be regarded as an extension of the nystrom approximation in this paper we establish a more general error bound for the adaptive column row sampling algorithm based on which we propose more accurate cur and nystrom algorithms with expected relative error bounds the proposed cur and nystrom algorithms also have low time complexity and can avoid maintaining the whole data matrix in ram in addition we give theoretical analysis for the lower error bounds of the standard nystrom method and the ensemble nystrom method the main theoretical results established in this paper are novel and our analysis makes no special assumption on the data matrices

perturbation of the least squares problem consider the least squares solutions to ax b ax b where a a a and b b b are perturbed from a b respectively in an earlier paper the author deduced the perturbation bounds for the solution when b r a in this paper we extend the results to cover more general cases when b r a and a is not of full rank

one and two dimensional coulomb green s function matrices in parabolic sturmians basis one and two dimensional operators which originate from the asymptotic form of the three body coulomb wave equation in parabolic coordinates are treated within the context of a square integrable basis set the matrix representations of green s functions corresponding to these operators are obtained

stable subspace tracking algorithm based on a signed urv decomposition subspace estimation and tracking are of fundamental importance in many signal processing algorithms the class of schur subspace estimators provides a complete parametrization of all principal subspace estimates defined as the column spans of corresponding low rank matrix approximants that lie within a specified 2 norm distance of a given matrix the parametrization is found in terms of a two sided hyperbolic decomposition hyperbolic urv or hurv which can be computed using hyperbolic rotations unfortunately such rotations are commonly associated with numerical instabilities in this paper we present a numerically stable non iterative algorithm to compute the hurv called the signed urv surv algorithm we show that this algorithm implicitly imposes certain constraints on the hurv such that important norm bounds that guarantee stability are satisfied the constraints also restrict the parametrization of the subspace estimate such that it becomes close to the principal subspace provided by the svd which is a special case within this class the complexity of the algorithm is of the same order as that of a qr update updating and downdating are of the same complexity and are both numerically stable surv is proven to provide rank estimates consistent with the svd with the same rank threshold it can replace an svd where only subspace estimation is needed typical applications would e g be the detection of the number of signals in array signal processing and subspace estimation for source separation and interference mitigation such as the first step in music and esprit type algorithms simulation results demonstrate the numerical stability and confirm that this algorithm provides exact rank estimates and good principal subspace estimates as compared to the svd

percolation thresholds on two dimensional voronoi networks and delaunay triangulations the site percolation threshold for the random voronoi network is determined numerically with the result pc 0 714 10 0 000 02 using monte carlo simulation on periodic systems of up to 40 000 sites the result is very close to the recent theoretical estimate pc approximately 0 7151 of neher for the bond threshold on the voronoi network we find pc 0 666 931 0 000 005 implying that for its dual the delaunay triangulation pc 0 333 069 0 000 005 these results rule out the conjecture by hsu and huang that the bond thresholds are 2 3 and 1 3 respectively but support the conjecture of wierman that for fully triangulated lattices other than the regular triangular lattice the bond threshold is less than 2 sin pi 18 approximately 0 3473

the qlp approximation to the singular value decomposition in this paper we introduce a new decomposition called the pivoted qlp decomposition it is computed by applying pivoted orthogonal triangularization to the columns of the matrix x in question to get an upper triangular factor r and then applying the same procedure to the rows of r to get a lower triangular matrix l the diagonal elements of r are called the r values of x those of l are called the l values numerical examples show that the l values track the singular values of x with considerable fidelity far better than the r values at a gap in the l values the decomposition provides orthonormal bases of analogues of row column and null spaces provided of x the decomposition requires no more than twice the work required for a pivoted qr decomposition the computation of r and l can be interleaved so that the computation can be terminated at any suitable point which makes the decomposition especially suitable for low rank determination problems the interleaved algorithm also suggests a new efficient 2 norm estimator

numerical methods in control we study classical control problems like pole assignment stabilization linear quadratic control and h control from a numerical analysis point of view we present several examples that show the difficulties with classical approaches and suggest reformulations of the problems in a more general framework we also discuss some new algorithmic approaches

generation of percolation cluster perimeters by a random walk a type of self avoiding random walk whish generates the perimeter of two dimensional lattice percolation clusters is given the algorithm has been simulated on a computer yielding the mean perimeter length as a function of occupation probability

on numerical methods for discrete least squares approximation by trigonometric polynomials fast efficient and reliable algorithms for discrete least squares approximation of a real valued function given at arbitrary distinct nodes in 0 2 by trigonometric polynomials are presented the algorithms are based on schemes for the solution of inverse unitary eigenproblems and require only o mn arithmetic operations as compared to o mn 2 operations needed for algorithms that ignore the structure of the problem an algorithm which solves this problem with real valued data and real valued solution using only real arithmetic is given numerical examples are presented that show that the proposed algorithms produce consistently accurate results that are often better than those obtained by general qr decomposition methods for the least squares problem

estimation of percolation thresholds via percolation in inhomogeneous media this paper mathematically justifies techniques used to estimate the percolation thresholds of fully penetrable disks or boolean models of planar disks generalizations to systems of other particles in two or more dimensions are also discussed

general linearization formulae for products of continuous hypergeometric type polynomials the linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials hps of a continuous variable which consists of the expansion of the product of two arbitrary hps in series of an orthogonal hp set here this problem is algebraically solved directly in terms of the coefficients of the second order differential equations satisfied by the involved polynomials general expressions for the expansion coefficients are given in integral form and they are applied to derive the connection formulae relating the three classical families of hypergeometric polynomials orthogonal on the real axis hermite laguerre and jacobi as well as several generalized linearization formulae involving these families the connection and linearization coefficients are generally expressed as finite sums of terminating hypergeometric functions which often reduce to a single function of the same type when possible these functions are evaluated in closed form in some cases sign properties of the coefficients such as positivity or non negativity conditions are derived as a by product from their resulting explicit representations

concentration gradient approach to continuum percolation in two dimensions the author has investigated percolation properties of a system of overlapping discs randomly distributed with a gradient of concentration fractal properties and critical exponents of this system appear to be identical to their counterparts on a lattice this is in agreement with the universality of critical exponents of percolation the concentration gradient approach permits a precise calculation of the percolation threshold corresponding to a critical area fraction of 0 6766 or 0 0005 he has investigated the part of the external perimeter of the percolation cluster which is accessible to discs of various sizes the fractal dimension of this accessible perimeter is found to decrease abruptly as a function of the radius of invading discs from 1 75 or 0 02 to 1 35 or 0 02

subspace iteration randomization and singular value problems a classical problem in matrix computations is the efficient and reliable approximation of a given matrix by a matrix of lower rank the truncated singular value decomposition svd is known to provide the best such approximation for any given fixed rank however the svd is also known to be very costly to compute among the different approaches in the literature for computing low rank approximations randomized algorithms have attracted researchers attention recently due to their surprising reliability and computational efficiency in different application areas typically such algorithms are shown to compute with very high probability low rank approximations that are within a constant factor from optimal and are known to perform even better in many practical situations in this paper we present a novel error analysis that considers randomized algorithms within the subspace iteration framework and show with very high probability that highly accurate low rank approximations as well as singular values ca

on the existence and computation of lu factorizations with small pivots let a be an n by n matrix which may be singular with a one dimensional null space and consider the lu factorization of a when a is exactly singular we show conditions under which a pivoting strategy will produce a zero n th pivot when a is not singular we show conditions under which a pivoting strategy will produce an nth pivot that is o g or o k a where is the smallest singular value of a and k a is the condition number of a these conditions are expressed in terms of the elements of a in general but reduce to conditions on the elements of the singular vectors corresponding to when a is nearly or exactly singular they can be used to build a 2 pass factorization algorithm which is guaranteed to produce a small n th pivot for nearly singular matrices as an example we exhibit an lu factorization of the n by n upper triangular matrix

computation of structural invariants of generalized state space systems in this paper we develop an algorithm for computing the zeros of a generalized state space model described by the matrix 5 tuple e a b c d where e may be a singular matrix but det a he 0 the characterization of these zeros is based on the system matrix of the corresponding 5 tuple both the characterization and the computational algorithm are extensions of equivalent results for state space models described by the 4 tuples a b c d we also extend these results to the computation of infinite zeros and left and right minimal indices of the system matrix several non trivial numerical examples are included to illustrate the proposed results

asymmetry in the percolation thresholds of fully penetrable disks with two different radii we perform simulations of continuum gradient percolation to measure the percolation threshold of systems of fully penetrable disks of two different radii as a function of volumetric proportion v the percolation threshold phic v lambda is approximately symmetric about v 0 5 for a fixed ratio lambda of the radii however the difference from symmetry is strongly statistically significant we also improve by an order of magnitude the measurement of the percolation threshold for disks of equal radius phic 0 676 347 5 6

on updating signal subspaces the authors develop an algorithm for adaptively estimating the noise subspace of a data matrix as is required in signal processing applications employing the signal subspace approach the noise subspace is estimated using a rank revealing qr factorization instead of the more expensive singular value or eigenvalue decompositions using incremental condition estimation to monitor the smallest singular values of triangular matrices the authors can update the rank revealing triangular factorization inexpensively when new rows are added and old rows are deleted experiments demonstrate that the new approach usually requires o n sup 2 work to update an n n matrix and that it accurately tracks the noise subspace

a new method for computing the stable invariant subspace of a real hamiltonian matrix abstract a new backward stable structure preserving method of complexity o n3 is presented for computing the stable invariant subspace of a real hamiltonian matrix and the stabilizing solution of the continuous time algebraic riccati equation the new method is based on the relationship between the invariant subspaces of the hamiltonian matrix h and the extended matrix 0 h h 0 and makes use of the symplectic urv like decomposition that was recently introduced by the authors

efficient measurement of the percolation threshold for fully penetrable discs we study the percolation threshold for fully penetrable discs by measuring the average location of the frontier for a statistically inhomogeneous distribution of fully penetrable discs we use two different algorithms to efficiently simulate the frontier including the continuum analogue of an algorithm previously used for gradient percolation on a square lattice we find that c 0 676 339 0 000 004 thus providing an extra significant digit of accuracy to this constant some figures in this article appear in colour in the electronic version see www iop org

a newton squaring algorithm for computing the negative invariant subspace of a matrix by combining newton s method for the matrix sign function with a squaring procedure a basis for the negative invariant subspace of a matrix can be computed efficiently the algorithm presented is a variant of multiplication rich schemes for computing the matrix sign function such as the well known inversion free schulz method which requires two matrix multiplications per step however by avoiding a complete computation of the matrix sign and instead concentrating only on the negative invariant subspace the final newton steps can be replaced by steps which require only one matrix squaring each this efficiency is attained without sacrificing the quadratic convergence of newton s method

rank and null space calculations using matrix decomposition without column interchanges abstract the most widely used stable methods for numerical determination of the rank of a matrix a are the singular value decomposition and the qr algorithm with column interchanges here two algorithms are presented which determine rank and nullity in a numerically stable manner without using column interchanges one algorithm makes use of the condition estimator of cline moler stewart and wilkinson and relative to alternative stable algorithms is particularly efficient for sparse matrices the second algorithm is important in the case that one wishes to test for rank and nullity while sequentially adding columns to a matrix

determination of the bond percolation threshold for the kagome lattice the hull gradient method is used to determine the critical threshold for bond percolation on the two dimensional kagome lattice and its dual the dice lattice for this system the hull walk is represented as a self avoiding trail or mirror model trajectory on the 3 4 6 4 archimedean tiling lattice the result one standard deviation of error is not consistent with previously conjectured values

componentwise analysis of direct factorization of real symmetric and hermitian matrices abstract we derive componentwise error bound for the factorization h gjg t where h is a real symmetric matrix g has full column rank and j is diagonal with 1 s on the diagonal we also derive a componentwise forward error bound that is we bound the difference between the exact and the computed factor g in the cases where such a bound is possible we extend these results to the hermitian case and to the well known bunch parlett factorization finally we prove bounds for the scaled condition of the matrix g and show that the factorization can have the rank revealing property

accuracy of tsvd solutions computed from rank revealing decompositions rank revealing decompositions are favorable alternatives to the singular value decomposition svd because they are faster to compute and easier to update although they do not yield all the information that the svd does they yield enough information to solve various problems because they provide accurate bases for the relevant subspaces in this paper we consider rank revealing decompositions in computing estimates of the truncated svd tsvd solution to an overdetermined system of linear equations a x approx b where a is numerically rank deficient we derive analytical bounds which show how the accuracy of the solution is intimately connected to the quality of the subspaces

rank revealing qr factorizations and the singular value decomposition t chan has noted that even when the singular value decomposition of a matrix a is known it is still not obvious how to find a rank revealing qr factorization rrqr of a if a has numerical rank deficiency this paper offers a constructive proof of the existence of the rrqr factorization of any matrix a of size m x n with numerical rank r the bounds derived in this paper that guarantee the existence of rrqr are all of order f i in comparison with chan s 0 2 it has been known for some time that if a is only numerically rank one deficient then the column permutation l7 of a that guarantees a small rnn in the qr factorization of a n can be obtained by inspecting the size of the elements of the right singular vector of a corresponding to the smallest singular value of a to some extent our paper generalizes this well known result we consider the interplay between two important matrix decompositions the singular value decomposition and the qr factorization of a matrix a in particular we are interested in the case when a is singular or nearly singular it is well known that for any a e r m x n a real matrix with rn rows and n columns where without loss of generality we assume rn n there are orthogonal matrices u and v such that where c is a diagonal matrix with nonnegative diagonal elements we assume that a 2 a2 2 2 on 2 0 the decomposition 0 1 is the singular value decomposition svd of a and the ai are the singular values of a the columns of v are the right singular vectors of a and the columns of u are the left singular vectors of a mathematically in terms of the singular values received december 1 1990 revised february 8 199 1 199 1 mathematics subject classification primary 65f30 15a23 15a42 15a15