computing ata btb ltdl using generalized hyperbolic transformations abstract hyperbolic transformations permit the computation of the ltdl factorization of the matrix y ata btbwithout computing the individual components ata and btb this paper describes generalized hyperbolic householder and givens transformations based on hyperbolic imaginary numbers the use of imaginary numbers provides a simple conceptual framework for developing hyperbolic algorithms and produces algorithms which operate under more general conditions than current methods

a general orthogonalization technique with applications to time series analysis and signal processing a new orthogonalization technique is presented for computing the qr factorization of a general n x p matrix of full rank p n 2 p the method is based on the use of projections to solve increasingly larger subproblems recursively and has an o np2 operation count for general matrices the technique is readily adaptable to solving linear least squares problems if the initial matrix has a circulant structure the algorithm simplifies significantly and gives the so called lattice algorithm for solving linear prediction problems from this point of view it is seen that the lattice algorithm is really an efficient way of solving specially structured least squares problems by orthogonalization as opposed to solving the normal equations by fast toeplitz algorithms

element free galerkin method convergence of the continuous and discontinuous shape functions abstract we consider numerical solutions of second order elliptic partial differential equations such as laplace s equation or linear elasticity in two dimensional non convex domains by the element free galerkin method efg this is a meshless method in which the shape functions are constructed by using weight functions of compact support for non convex domains two approaches to the determination of whether a node affects approximation at a particular point are used a contained path criterion and the visibility criterion we show that for non convex domains the visibility criterion leads to discontinuous weight functions and discontinuous shape functions the resulting approximation is no longer conforming and its convergence must be established by inspection of the so called consistency term we show that the variant of the element free galerkin method which uses the discontinuous shape functions is convergent and that in the practically important case of linear shape functions the convergence rate is not affected by the discontinuities the convergence of the discontinuous approximation is first established by the classical and generalized patch test as these tests do not provide an estimate of the convergence rate the rate of convergence in the energy norm is examined for both the continuous and discontinuous efg shape functions and for smooth and non smooth solutions by a direct inspection of the error terms

fast block toeplitz orthogonalization algorithms are presented which compute theqr factorization of a block toeplitz matrix ino n 2 block operations wheren is the block order of the matrix and a block operation is a multiplication inversion or a set of householder operations involving one or two blocks the algorithms are in general analogous to those presented in the scalar toeplitz case in a previous paper but the basic operation is the householder transform rather than the givens transform and the computation of the householder coefficients and other working matrices requires special treatment two algorithms are presented the first computes onlyr explicitly and the second computes bothq andr

a meshless method with enriched weight functions for fatigue crack growth the meshless method is particularly appropriate to solve crack propagation problems in this paper the fatigue growth of cracks in two dimensional bodies is considered the analysis is based upon paris equation new enriched weight functions are introduced in the meshless method formulation to capture the singularity at the crack tip simple problems show the accuracy and efficiency of this method then it is applied to fatigue analysis of single and multi cracked bodies under mixed mode conditions copyright 2004 john wiley sons ltd

an analysis of the hr algorithm for computing the eigenvalues of a matrix abstract the hr algorithm a method of computing the eigenvalues of a matrix is presented it is based on the fact that almost every complex square matrix a can be decomposed into a product a hr of a so called pseudo hermitian matrix h and an upper triangular matrix r this algorithm is easily seen to be a generalization of the well known qr algorithm it is shown how it is related to the power method and inverse iteration and for special matrices the connection between the lr and hr algorithms is indicated

fast toeplitz orthogonalization algorithms are presented which compute theqr factorization of an order n toeplitz matrix ino n 2 operations the first algorithm computes onlyr explicitly and the second computes bothq andr the algorithms are derived from a well known procedure for performing the rank 1 update ofqr factors using the shift invariance property of the toeplitz matrix the algorithms can be used to solve the toeplitz least squares problem and can be modified to solve toeplitz systems ino n space

algebraic gauss manin systems and brieskorn modules we study the algebraic gauss manin system and the algebraic brieskorn module associated to a polynomial mapping with isolated singularities since the algebraic gauss manin system does not contain any information on the cohomology of singular fibers we first construct a non quasi coherent sheaf which gives the cohomology of every fiber then we study the algebraic brieskorn module and show that its position in the the algebraic gauss manin system is determined by a natural map to quotients of local analytic gauss manin systems and its pole part by the vanishing cycles at infinity comparing it with the deligne extension this implies for example a formula for the determinant of periods in the two dimensional case we can describe the global structure of the algebraic gauss manin system rather explicitly

systolic implementation of the lattice algorithm for least squares linear prediction problems abstract we present a linear systolic array of o n cells that solves the least squares linear prediction problem in time o n via an algorithm based on the so called lattice algorithm the total storage required is o n words i e only a constant number of words are needed at each cell

structure preserving γqr and γ lanczos algorithms for bethe salpeter eigenvalue problems abstract to solve the bethe salpeter eigenvalue problem with distinct sizes two efficient methods called qr algorithm and lanczos algorithm are proposed in this paper both algorithms preserve the special structure of the initial matrix a b b a resulting the computed eigenvalues and the associated eigenvectors still hold the properties similar to those of theorems are given to demonstrate the validity of the proposed two algorithms in theory numerical results are presented to illustrate the superiorities of our methods

fast orthogonalization algorithm and parallel architecture for ar spectral estimation based on forward backward linear prediction truncated qr methods have been shown to be comparable to svd based methods for the sinusoidal frequency estimation based on the forward backward linear prediction fblp model however without exploiting the special structure of the fblp matrix the qr decomposition qrd of the fblp matrix has a computational complexity on the order of 2 6m n n sup 2 3 o n sup 2 for a 2m n fblp matrix a fast algorithm to perform the qrd of the fblp matrix by exploiting its special toeplitz hankel form is proposed the computational complexity is reduced to 10n sup 2 4mn o n the fast algorithm can also be easily implemented by a linear systolic array reducing the number of time steps required to 2m 5n 4

center conditions rigidity of logarithmic differential equations abstract in this paper we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again this is a generalization of ilyashenko s result on hamiltonian differential equations the main tools are picard lefschetz theory of a polynomial with complex coefficients in two variables specially the gusein zade a campo s theorem on calculating the dynkin diagram of the polynomial and the action of gauss manin connection on the so called brieskorn lattice petrov module of the polynomial we will also generalize j p francoise recursion formula and condition for a polynomial which is a product of lines in a general position some applications on the cyclicity of cycles and the bautin ideals will be given

parallel solution of symmetric positive definite systems with hyperbolic rotations abstract an algorithm based on hyperbolic rotations is presented for the solution of linear systems of equations ax b with symmetric positive definite coefficient matrix a forward elimination and backsubstitution are replaced by matrix vector multiplications rendering the method amenable to implementation on a variety of parallel and vector machines this method can be simplified and formulated without square roots if a is also toeplitz a systolic vlsi architecture implementing the resulting recurrence equations is more efficient than previously proposed pipelined toeplitz system solvers the hardware count becomes independent of the matrix size if its inverse is banded

block rls using row householder reflections abstract householder reflections applied from the left are generally used to zero a contiguous sequence of entries in a column of a matrix a our purpose in this paper is to introduce new row householder and row hyperbolic householder reflections which are also applied from the left but now zero a contiguous sequence of entries in a row of a we then show how these row householder reflections can be used to design efficient sliding data window recursive least squares rls covariance algorithms which are based upon rank k modifications to the inverse cholesky factor r 1 of the covariance matrix the algorithms are rich in matrix matrix blas 3 computations making them efficient on vector and parallel architectures preliminary numerical experiments are reported comparing these row householder based rank k modification schemes with k applications of the classical updating and downdating covariance schemes which use givens and hyperbolic rotations

element free galerkin method for wave propagation and dynamic fracture abstract element free galerkin method efg is extended to dynamic problems efg method which is based on moving least square interpolants mls requires only nodal data no element connectivity is needed this makes the method particularly attractive for moving dynamic crack problems since remeshing can be avoided in contrast to the earlier formulation for static problems by authors the weak form of kinematic boundary conditions for dynamic problems is introduced in the implementation to enforce the kinematic boundary conditions with this formulation the stiffness matrix is symmetric and positive semi definite and hence the consistency conergence and stability analyses of time integration remain the same as those in finite element method numerical examples are presented to illustrate the performance of this method the relationship between the element free galerkin method and the smooth particle hydrodynamics sph method is also discussed in this paper results are presented for some one dimensional problems and two dimensional problems with static and moving cracks

perturbation analysis for block downdating of a cholesky decomposition let the cholesky decomposition of x rm t x be x rm t x r rm t r where r is upper triangular the cholesky block downdating problem is to determine tilde r such that tilde r rm t tilde r r rm t r z rm t z where z is a block of rows from the data matrix x we analyze the sensitivity of this block downdating problem of the cholesky factorization a measure of conditioning for the cholesky block downdating is presented and compared to the single row downdating case

hyperbolic householder transformations a class of transformation matrices analogous to the householder matrices is developed with a nonorthogonal property designed to permit the efficient deletion of data from least squares problems these matrices which we term hyperbolic householder are shown to effect deletion or simultaneous addition and deletion of data with much less sensitivity to rounding errors than for techniques based on normal equations when the addition deletion sets are large this numerical robustness is obtained at the expense of only a modest increase in computations and when only a relatively small fraction of the data set is modified there is a decrease in required computations two applications to signal processing problems are considered first these transformations are used to obtain a square root algorithm for windowed recursive least squares filtering second the transformations are employed to implement the rejection of spurious data from the weight vector estimator in an adaptive array

a meshless method with enriched weight functions for three dimensional crack propagation the propagation of cracks in three dimensions is analysed by a meshless method the cracks are modelled by a set of triangles that are added when the propagation occurs since the method is meshless no remeshing of the domain is necessary during the propagation to avoid using a large number of degrees of freedom the stress singularity along the front of the cracks is taken into account by an enrichment of the shape functions of the meshless method by means of appropriate weight functions this enrichment technique is an extension of the technique that proved to be successful in two dimensions in a previous paper several algorithms for efficiently implementing the meshless method in three dimensions are detailed the accuracy of the enrichment is first assessed on simple examples and some results of non planar propagation of multiple cracks are then presented copyright 2005 john wiley sons ltd

the weak and strong stability of algorithms in numerical linear algebra the stability of algorithms in numerical linear algebra is discussed the concept of stability is extended to notions of weak stability and strong stability justifications are given for these extensions and the implications of error analyses in terms of these definitions are discussed the concept of weak stability helps to clarify some of the controversy which has arisen concerning the stability of algorithms for toeplitz systems

computing the minimum phase filter using the ql factorization we investigate the ql factorization of a time invariant convolutive filtering matrix and show that this factorization not only provides the finite length equivalent to the minimum phase filter but also gives the associated all pass filter the convergence properties are analyzed and we derive the exact convergence rate and an upper bound for a simple single input single output system with filter length finally this upper bound is used to derive an approximation of the convergence rate for systems of arbitrary length implementation wise the method has the advantage of being numerically stable and straight forward to extend to the multiple input multiple output case furthermore due to the existence of fast ql factorization methods it is possible to compute the filters efficiently

an inverse factorization algorithm for linear prediction abstract a new inverse factorization technique is presented for solving linear prediction problems arising in signal processing the algorithm is similar to a scheme of qiao in that is uses the rectangular toeplitz structure of the data to recursively compute the prediction error and to solve the problem when the optimum filter order has been found the novelty of the scheme presented here is the use of an inverse factorization scheme due to pan and plemmons for solving the linear prediction problem with low computational complexity and without the need for solving triangular systems we also provide a linear systolic array for solving these problems

element free galerkin methods for static and dynamic fracture abstract element free galerkin efg methods are presented and applied to static and dynamic fracture problems efg methods which are based on moving least square mls interpolants require only nodal data no element connectivity is needed the description of the geometry and numerical model of the problem consists only of a set of nodes and a description of exterior boundaries and interior boundaries from any cracks this makes the method particularly attractive for growing crack problems since only minimal remeshing is needed to follow crack growth in moving least square interpolants the dependent variable at any point is obtained by minimizing a function in terms of the nodal values of the dependent variable in the domain of influence of the point numerical examples involving fatigue crack growth and dynamic crack propagation are presented to illustrate the performance and potential of this method

a least squares approach to reduce stable discrete linear systems preserving their stability abstract a new stability preserving model reduction algorithm for discrete linear siso systems based on a least squares approach is proposed similar to the pade approximation an equation system for the markov parameters involving a high dimensional hankel matrix is considered it is proved that approximate solutions computed via the moore penrose pseudo inverse give rise to a stability preserving reduction scheme furthermore the proposed algorithm is compared to the balanced truncation method showing comparable performance of the reduced systems

dynamic fracture using element free galerkin methods the element free galerkin method for dynamic crack propagation is described and applied to several problems this method is a gridless method which facilitates the modelling of growing crack problems because it does not require remeshing the growth of the crack is modelled by extending its surfaces the essential feature of the method is the use of moving least squares interpolants for the trial and test functions in these interpolants the dependent variable is obtained at any point by minimizing a weighted quadratic form involving the nodal variables within a small domain surrounding the point the discrete equations are obtained by a galerkin method the procedures for modelling dynamic crack propagation based on dynamic stress intensity factors are also described

accurate fracture modelling using meshless methods the visibility criterion and level sets formulation and 2d modelling fracture modelling using numerical methods is well advanced in 2d using techniques such as the extended finite element method xfem the use of meshless methods for these problems lags somewhat behind but the potential benefits of no meshing particularly in 3d prompt continued research into their development in methods where the crack face is not explicitly modelled as the edge of an element for instance two procedures are instead used to associate the displacement jump with the crack surface the visibility criterion and the diffraction method the visibility criterion is simple to implement and efficient to compute especially with the help of level set coordinates however spurious discontinuities have been reported around crack tips using the visibility criterion whereas implementing the diffraction method in 3d is much more complicated than the visibility criterion in this paper a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved the formulation is based on the use of level set coordinates and the element free galerkin method and is generally applicable for single or multiple crack problems in 2d or 3d the paper explains the formulation and provides verification of the method against a number of 2d crack problems copyright 2011 john wiley sons ltd

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

schur type methods for solving least squares problems with toeplitz structure we give an overview of fast algorithms for solving least squares problems with toeplitz structure based on generalization of the classical schur algorithm and discuss their stability properties in order to obtain more accurate triangular factors of a toeplitz matrix as well as accurate solutions for the least squares problems methods based on corrected seminormal equations csne can be used we show that the applicability of the generalized schur algorithm is considerably enhanced when the algorithm is used in conjunction with csne several numerical tests are reported where different variants of the generalized schur algorithm and csne are compared for their accuracy and speed

from bareiss algorithm to the stable computation of partial correlations abstract this paper presents a new algorithm for the stable computation of sample partial correlation coefficients we start by generalizing bareiss algorithm for the solution of linear systems of equations with non symmetric toeplitz coefficient matrix to matrices that are not toeplitz and show that it computes their lu and ul factorizations for symmetric positive definite matrices b the normalized version of bareiss algorithm is the hyperbolic cholesky algorithm which computes the upper and lower triangular cholesky factors u and l of b by means of 2 2 hyperbolic rotations if b is a convariance matrix certain partial correlation coefficients may be obtained as the geometric means of the multiplier pairs in bareiss algorithm or as the hyperbolic tangents of the hyperbolic rotations the partial correlation coefficients for a given data matrix a may suffer from serious loss of numerical accuracy when computed from the covariance matrix b a t a while the data flow graph of bareiss algorithm suggests a method due to cybenko for computing partial correlation coefficients directly from the columns of a the graph of hyperbolic cholesky suggests an alternative that requires fewer operations nd exhibits a higher degree of parallelism in that method one computes the qr decomposition a qu of the data matrix and applies orthogonal rotations to transform u to l the sines of the rotation angles constitute a set of partial correlation coefficients we extend this approach to determine arbitrary partial correlation coefficients at little extra computational cost

multiple crack weight for solution of multiple interacting cracks by meshless numerical methods summary 7 we devise a multiple crack weight mcw method for the accurate and effective solution of strongly interacting cracks by meshless numerical methods the mcw method constructs weight functions 9 around cracks so that they simultaneously characterize all the cracks present in the single nodal domain of influence this approach reduces the number of nodes necessary to achieve sufficient 11 accuracy and consequently it decreases the computational effort numerical examples demonstrate that the method allows an accurate solution of multiple cracks problems convergence of the method is 13 analysed and discussed copyright 2006 john wiley sons ltd

analysis of a recursive least squares hyperbolic rotation algorithm for signal processing abstract the application of hyperbolic plane rotations to the least squares downdating problem arising in windowed recursive least squares signal processing is studied a forward error analysis is given to show that this algorithm can be expected to perform well in the presence of rounding errors provided that the problem is not too ill conditioned the hyperbolic rotation algorithm is shown to be forward weakly stable and in fact comparable to an orthogonal downdating method shown to be backward stable by stewart it is shown in detail how the method s accuracy depends upon the conditioning numerical comparisons are made with the usual method based upon orthogonal rotations as implemented in linpack both methods have the important advantage over the classical normal equations appraoch that they can be effectively implemented on special purpose signal processing devices requiring shorter word lengths however the hyperbolic rotation requires n 2 fewer multiplications and additions for each downdating step than the orthogonal method where n is the number of least squares filter coefficients in addition it is more amenable to implementation on a variety of vector and parallel machines in many signal processing applications n is not large and if n processors are available then the downdating process can be accomplished in 2 n time steps by the hyperbolic rotation method

an enriched meshless method for non linear fracture mechanics this paper presents an enriched meshless method for fracture analysis of cracks in homogeneous isotropic non linear elastic two dimensional solids subject to mode i loading conditions the method involves an element free galerkin formulation and two new enriched basis functions types i and ii to capture the hutchinson rice rosengren singularity field in non linear fracture mechanics the type i enriched basis function can be viewed as a generalized enriched basis function which degenerates to the linear elastic basis function when the material hardening exponent is unity the type ii enriched basis function entails further improvements of the type i basis function by adding trigonometric functions four numerical examples are presented to illustrate the proposed method the boundary layer analysis indicates that the crack tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered whether for the near tip asymptotic field or for the far tip elastic field numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the j integral for the applied load intensities and material properties considered also the crack mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results furthermore the meshless results show excellent agreement with the experimental measurements indicating that the new basis functions are also capable of capturing elastic plastic deformations at a stress concentration effectively copyright 2003 john wiley sons ltd

eigenvalues of ax λbx for real symmetric matrices a and b computed by reduction to a pseudosymmetric form and the hr process abstract the paper presents a method for solving the eigenvalue problem ax bx where a and b are real symmetric but not necessarily positive definite matrices and b is nonsingular the method reduces the general case into a form cz z where c is a pseudosymmetric matrix a further reduction of c produces a tridiagonal pseudosymmetric form to which the iterative hr process is applied the tridiagonal pseudosymmetric form is invariant under the hr transformations the amount of computation is significantly less than in treating the problem by a general method

a fast least square deconvolution algorithm for vocal tract cross section estimation an efficient computation algorithm based on unitary jacobi type rotations is developed for fast least square deconvolution specifically this algorithm is able to solve a positive definite covariance matrix with a special toeplitz structure in o n sup 2 operations instead of o n sup 3 operations this algorithm which takes advantage of the inherent structure of the underlying deconvolution problem is guaranteed to be numerically stable since only unitary transformation is used it is implemented on an experimental system for the estimation of human vocal tract cross section a significant gain in processing is observed

fast recursive least squares transversal filters for adaptive filtering fast transversal filter ftf implementations of recursive least squares rls adaptive filtering algorithms are presented in this paper substantial improvements in transient behavior in comparison to stochastic gradient or lms adaptive algorithms are efficiently achieved by the presented algorithms the true not approximate solution of the rls problem is always obtained by the ftf algorithms even during the critical initialization period first n iterations of the adaptive filter this true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic gradient algorithms factor of 1 6 to 3 5 depending upon application additionally the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing fast rls algorithms such as the fast kalman algorithms of morf ljung and falconer 1976 and the fast ladder lattice algorithms of morf and lee 1977 1981 they are further shown to attain steady state unnormalized or improve upon first n initialization steps the very low computational requirements of the efficient rls solutions of carayannis manolakis and kalouptsidis 1983 finally several efficient procedures are presented by which to ensure the numerical stability of the transversal filter algorithms including the incorporation of soft constraints into the performance criteria internal bounding and rescuing procedures and dynamic range increasing square root normalized variations of the transversal filters

qr factorization of toeplitz matrices this paper presents a new algorithm for computing theqr factorization of anm n toeplitz matrix ino mn operations the algorithm exploits the procedure for the rank 1 modification and the fact that both principal m 1 n 1 submatrices of the toeplitz matrix are identical an efficient parallel implementation of the algorithm is possible

a superfast method for solving toeplitz linear least squares problems in this paper we develop a superfast o m n log 2 m n complexity algorithm to solve a linear least squares problem with an m n toeplitz coefficient matrix the algorithm is based on the augmented matrix approach the augmented matrix is further extended to a block circulant matrix and dft is applied this leads to an equivalent tangential interpolation problem where the nodes are roots of unity this interpolation problem can be solved by a divide and conquer strategy in a superfast way to avoid breakdowns and to stabilize the algorithm pivoting is used and a technique is applied that selects difficult points and treats them separately the effectiveness of the approach is demonstrated by several numerical examples 2003 elsevier science inc all rights reserved

relative cohomology with respect to a lefschetz pencil let m be a complex projective manifold of dimension n 1 and f a meromorphic function on m obtained by a generic pencil of hyperplane sections of m the n th cohomology vector bundle of f0 f m r where r is the set of indeterminacy points of f is defined on the set of regular values of f0 and we have the usual gauss manin connection on it following brieskorn s methods in br we extend the n th cohomology vector bundle of f0 and the associated gauss manin connection to p1 by means of differential forms the new connection turns out to be meromorphic on the critical values of f0 we prove that the meromorphic global sections of the vector bundle with poles of arbitrary order at p1 is isomorphic to the brieskorn module of f in a natural way and so the brieskorn module in this case is a free c t module of rank n where c t is the ring of polynomials in t and n is the dimension of n th cohomology group of a regular fiber of f0

a perturbation analysis of the problem of downdating a cholesky factorization abstract the rank one modification of a cholesky factorization r t r zz t d t d where r and d are upper triangular matrices and z is a column vector is called the downdating problem there are many articles devoted to this problem due to its broad range of applications and numerical difficulty this paper serves as a first order parametrized perturbation analysis of this problem

stability analysis and fast algorithms for triangularization of toeplitz matrices a new o mn algorithm for triangularizing an m times n toeplitz matrix is presented the algorithm is based on the previously developed recursive algorithms that exploit the toeplitz structure and compute each row of the triangular factor via updating and downdating steps a forward error analysis for this existing recursive algorithm is presented which allows us to monitor the conditioning of the problem and use the method of corrected semi normal equations to obtain higher accuracy for certain ill conditioned matrices numerical experiments show that the new algorithm improves the accuracy significantly while the computational complexity stays in o mn

smoothing and accelerated computations in the element free galerkin method two topics in the formulation and implementation of meshless methods are considered the smoothing of the approximating functions at concave boundaries and the speedup of the calculation of the approximating functions and their derivatives these techniques are described in the context of the element free galerkin method but they are applicable to other meshless methods results are presented for some elastostatic problems which show a moderate improvement in the accuracy of the smoothed interpolant the speedup in calculating the shape functions is about a factor of two

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 symmetric structure preserving γqr algorithm for linear response eigenvalue problems abstract in this paper we present an efficient qr algorithm for solving the linear response eigenvalue problem h x x where h is symmetric with respect to 0 diag i n i n based on newly introduced orthogonal transformations the qr algorithm preserves the symmetric structure of h throughout the whole process and thus guarantees the computed eigenvalues to appear pairwise as they should with the help of a newly established implicit orthogonality theorem we incorporate the implicit multi shift technique to accelerate the convergence of the qr algorithm numerical experiments are given to show the effectiveness of the algorithm

hybrid algorithm for fast toeplitz orthogonalization new techniques for fast toeplitz qr decomposition are presented the methods are based on the shift invariance property of a toeplitz matrix the numerical properties of the algorithms are discussed and some comparisons are made with two other fast toeplitz orthogonalization methods

complex variable approach to the bem for multiple crack problems a boundary element method for straight multiple center and edge crack problems is developed in this paper the method is constructed upon the systematic use of the elastic singularity solutions in complex variables the crack opening is represented by the continuous distribution of dislocation dipoles and the effect of the non crack boundary by the continuous distributions of point forces and dislocation dipoles the crack tip singularity is embedded into the interpolation using orthogonal polynomials i e chebyshev and jacobi and their associated singular weight functions the proposed analytical integration procedure of the cauchy type integrals defined over the crack eliminates the need for the quadrature formulae for numerical integration streamlines and enhances the accuracy of the traditional singular integral equation method for crack problems the stress intensity factors for the fifteen problems analyzed in this paper have been accurate enough to substitute those given in stress intensity factor handbooks since non crack boundary of arbitrary shape can be introduced at will the method is expected to give accurate stress intensity factors for complex real life problems