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

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

diffusion in deterministic fractals a model for polymer electrolytes abstract we present simulation results for random walks on a family of sierpinski carpets under a periodically varying bias the diffusivity obtained from these results shows a power law variation of the bias with frequency the exponent depends on details of the carpet generator we discuss applicability of these results to real materials with fractal structure such as polymer electrolytes with different salt concentrations

scaling exponents for random walks on sierpinski carpets and number of distinct sites visited a new algorithm for infinite fractal lattices the scaling exponent for the mean square distance covered in a random walk dw and the average number of distinct sites visited dn are determined for a family of sierpinski carpet patterns we suggest a new random walk algorithm to generate walks on an effectively infinite deterministic fractal lattice the algorithm is applied to several sierpinski carpet patterns with the same hausdorff dimension we show that the systems have a quite different scaling exponent dw and further that the generally accepted result dn ds does not hold for all of these where ds is the spectral dimension

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

the pore structure of sierpinski carpets in this paper a new method is developed to investigate the pore structure of finitely and even infinitely ramified sierpinski carpets the holes in every iteration stage of the carpet are described by a hole counting polynomial this polynomial can be computed iteratively for all carpet stages and contains information about the distribution of holes with different areas and perimeters from which dimensions governing the scaling of these quantities can be determined whereas the hole area is known to be two dimensional the dimension of the hole perimeter may be related to the random walk dimension

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

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

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

embedded trees estimation of gaussian processes on graphs with cycles graphical models provide a powerful general framework for encoding the structure of large scale estimation problems however the graphs describing typical real world phenomena contain many cycles making direct estimation procedures prohibitively costly in this paper we develop an iterative inference algorithm for general gaussian graphical models it operates by exactly solving a series of modified estimation problems on spanning trees embedded within the original cyclic graph when these subproblems are suitably chosen the algorithm converges to the correct conditional means moreover and in contrast to many other iterative methods the tree based procedures we propose can also be used to calculate exact error variances although the conditional mean iteration is effective for quite densely connected graphical models the error variance computation is most efficient for sparser graphs in this context we present a modeling example suggesting that very sparsely connected graphs with cycles may provide significant advantages relative to their tree structured counterparts thanks both to the expressive power of these models and to the efficient inference algorithms developed herein the convergence properties of the proposed tree based iterations are characterized both analytically and experimentally in addition by using the basic tree based iteration to precondition the conjugate gradient method we develop an alternative accelerated iteration that is finitely convergent simulation results are presented that demonstrate this algorithm s effectiveness on several inference problems including a prototype distributed sensing application

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

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

some notes on multisplltting methods and m step preconditioners for linear systems abstract to solve the real nonsingular linear system ax b on parallel and vector machines multisplitting methods and m step preconditioners are considered in this work in particular 1 we extend adams and ong s idea and theory of the m step additive preconditioners and ruggiero and galligani s method of the arithmetic mean and we generalize the corresponding convergence results 2 we determine suitable relaxed m step preconditioners and solve the problem of minimizing the related condition number with respect to the relaxation extrapolation parameter involved 3 finally we complete the theoretical solution of the problem of determining the optimum sor additive iterative method for 2 cyclic positive definite matrices

an updating algorithm for subspace tracking in certain signal processing applications it is required to compute the null space of a matrix whose rows are samples of a signal with p components the usual tool for doing this is the singular value decomposition however the singular value decomposition has the drawback that it requires o p sup 3 operations to recompute when a new sample arrives it is shown that a different decomposition called the urv decomposition is equally effective in exhibiting the null space and can be updated in o p sup 2 time the updating technique can be run on a linear array of p processors in o p time

accelerated landweber iterations for the solution of ill posed equations in this paper the potentials of so calledlinear semiiterative methods are considered for the approximate solution of linear ill posed problems and ill conditioned matrix equations several efficient two step methods are presented most of which have been introduced earlier in the literature stipulating certain conditions concerning the smoothness of the solution a notion of optimal speed of convergence may be formulated various direct and converse results are derived to illustrate the properties of this concept if the problem s right hand side data are contaminated by noise semiiterative methods may be used asregularization methods assuming optimal rate of convergence of the iteration for the unperturbed problem the regularized approximations will be of order optimal accuracy to derive these results specific properties of polynomials are used in connection with the basic theory of solving ill posed problems rather recent results onfast decreasing polynomials are applied to answer an open question of brakhage numerical examples are given including a comparison to the method of conjugate gradients

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

on validity of m step multisplitting preconditioners for linear systems let ax b be a linear system where a is a symmetric positive definite spd matrix m step multisplitting preconditioners which include the preconditioners based on multisplittings obtained by incomplete cholesky factorizations r bru c corral a martinez j mas siam j matrix anal appl 16 1995 1210 1222 for the conjugate gradient method are studied the validity of the proposed m step multisplitting preconditioners when a is an spd matrix is proved our results improve and extend previous ones

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

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

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

a convergence theorem for chaotic asynchronous relaxation abstract we present a convergence result for chaotic asynchronous relaxation that is a modification of the result of chazan and miranker the modification is a restriction to the case of global memory or fast communication the extra restriction is that each update is based on a prior state of the system rather than on prior substates

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

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

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

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

a unified approach to solving ill conditioned matrix problems a unified approach to solving ill conditioned matrix problems is presented concentrating on those problems arising in engineering where measurement inaccuracy is the major source of error by applying the singular value decomposition it is shown that regularization with a parameter can be used without squaring the condition number of the matrix numerical examples demonstrate that a smaller relative error is attainable using the unified approach over the more standard technique of discarding singular values lastly a simple method is presented for locating the optimal solution using a computed residual

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

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

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

bi asymptotic fractals fractals between lower and upper bounds extending the idea of rigaut s asymptotic fractals issuing a turnover from a constant to a power law behaviour towards smaller scales we extend this idea to asymptotic fractals with both lower and upper turnover points i e the fractal region is terminated towards larger scales by another constant value this behaviour is typical for natural fractals such biological cell boundaries here we present a new analytic function describing this bi asymptotic behaviour the introduced parameters can be directly interpreted we show the advantage of this function in fitting processed images of natural and mathematical fractals in comparison with standard procedures the determined dimension is significantly more accurate a program package with this new function is available

the fast fourier transform method and ill conditioned matrices we study the problem of finding numerical solutions of the linear algebraic equation a x b where a denotes an nxn ill conditioned coefficient matrix it is well known that gaussian elimination methods coupled with pivoting strategies are ineffective in this setting due to round off error we propose a new and simple application of the fast fourier transform fft method other viable methods such as the qr method qrm or the singular value decomposition method svdm have been proposed in the literature the goal of this paper is to investigate the performance of the proposed method and compare it to other popular methods the comparison is illustrated by computer simulation results using matlab

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

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

anomalous diffusion on random fractal composites stochastic fractals generated from combinations of deterministic fractals have the advantage of being tractable to some extent but also being closer to real materials since they are partially disordered in the present work we focus our attention on the remarkable nonlinear mixing behavior exhibited by fractals generated as random combinations of two different sierpinski carpet generators when patterns with different anomalous diffusion exponents and the same or different fractal dimensions are combined together the effective diffusion exponent cannot in general be expressed as a linear weighted average of the diffusion exponents of the constituents the effective exponent may show a maximum or minimum for certain compositions an explanation of this interesting phenomenon is offered on the basis of details of the carpet generator particularly on the number and position of connection points which determine the connectivity of the fractal composite

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

modelling porous structures by repeated sierpinski carpets abstract porous materials such as sedimentary rocks often show a fractal character at certain length scales deterministic fractal generators iterated upto several stages and then repeated periodically provide a realistic model for such systems on the fractal diffusion is anomalous and obeys the law r 2 t 2 d w where r 2 is the mean square distance covered in time t and d w 2 is the random walk dimension the question is how is the macroscopic diffusivity related to the characteristics of the small scale fractal structure which is hidden in the large scale homogeneous material in particular do structures with same d w necessarily lead to the same diffusion coefficient at the same iteration stage the present paper tries to shed some light on these questions

polynomial preconditioners based on factorized sparse approximate inverses let ax b be a linear system where a is a symmetric positive definite matrix an additive polynomial preconditioner for the conjugate gradient method based on multisplittings is proposed the multisplittings are obtained by computing some factorized sparse approximate inverses of the coefficient matrix namely splittings of the form a email protected email protected t 1 n email protected email protected t a 1 induced by the ainv and fsai factorized approximate inverse preconditioners applied to diagonal blocks of a are used the applicability of this preconditioner is studied moreover the results of the numerical experiments obtained on a cray t3e for a representative set of matrices are presented specifically structural analysis and transport diffusion problems are considered the effect of the reverse cuthill mckee rcm and multiple minimum degree mmd orderings is evaluated

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

on multisplitting methods for band matrices we present new theoretical results on two classes of multisplitting methods for solving linear systems iteratively these classes are based on overlapping blocks of the underlying coefficient matrix a which is assumed to be a band matrix we show that under suitable conditions the spectral radius rho h of the iteration matrix h does not depend on the weights of the method even if these weights are allowed to be negative for a certain class of splittings we prove an optimality result for rho h with respect to the weights provided that a is an m matrix this result is based on the fact that the multisplitting method can be represented by a single splitting a m n which in our situation surprisingly turns out to be a regular splitting furthermore we show by numerical examples that weighting factors alpha not in 0 1 may considerably improve the convergence

application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media we consider the application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media it is found that answers to interesting physics questions can be expressed in terms of several new fractal dimensions in addition to the fractal dimensiond f 1 d the fractal dimension of the backbone arises in connection with electric current flow 2 d red the fractal dimension of the singly connected bonds in the backbone arises in connection with its equivalence to the thermal scaling power 3 d e the fractal dimension of the of the elastic backbone 4 d u the fractal dimension of the unscreened perimeter arises in connection with the viscosity singularity at the gelation threshold 5 d min the fractal dimension of the minimum path or chemical distance between two sites arises in co nnection with the aharony stauffer conjecture 6 dw the fractal dimension of a random walk 7 d g the fractal dimension of growth sites that arise as a random walk creates a cluster relations among these fractal dimensions are discussed some of which can be proved and others of which are conjectures whose validity has been established only in certain limiting cases

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