solution of the large matrix equations which occur in response theory abstract the eigenvalue equation and the sets of linear equations that occur in linear and nonlinear response function calculations have a specific paired structure we have developed iterative algorithms which utilize this structure to efficiently solve the equations the algorithms have been designed so they do not require the matrices to be explicitly available which makes it possible to perform accurate calculations with a dimension of maybe 106 numerical tests show convergence behavior superior to previously suggested algorithms

riccati based preconditioner for computing invariant subspaces of large matrices this paper introduces and analyzes the convergence properties of a method that computes an approximation to the invariant subspace associated with a group of eigenvalues of a large not necessarily diagonalizable matrix the method belongs to the family of projection type methods at each step it refines the approximate invariant subspace using a linearized riccati s equation which turns out to be the block analogue of the correction used in the jacobi davidson method the analysis conducted in this paper shows that the method converges at a rate quasi quadratic provided that the approximate invariant subspace is close to the exact one the implementation of the method based on multigrid techniques is also discussed and numerical experiments are reported

uniform asymptotic expansions of a double integral coalescence of two stationary points consider the double integral i d g x y e i f x y d x d y where is a large positive variable and is an auxiliary parameter we consider the case in which the phase function f x y has two simple stationary points x y and x y in d which coalesce at a point x y as approaches a critical value the point x y can either be an interior point of d or a boundary point of d asymptotic expansions are derived in both cases which hold uniformly in a neighbourhood of our derivation is mathematically rigorous

inverse shifted inverse and rayleigh quotient iteration as newton s method the l 2 normalized inverse shifted inverse and rayleigh quotient iterations are classic algorithms for approximating an eigenvector of a symmetric matrix this work establishes rigorously that each iterate produced by one of these three algorithms can be viewed as a newton s method iterate followed by a normalization the equivalences given here are not meant to suggest changes to the implementations of the classic eigenvalue algorithms however they add further understanding to the formal structure of these iterations and they provide an explanation for their good behavior despite the possible need to solve systems with nearly singular coefficient matrices a historical development of these eigenvalue algorithms is presented using our equivalences and traditional newton s method theory helps to gain understanding as to why normalized newton s method inverse iteration and shifted inverse iteration are only linearly convergent and not quadratically convergent as would be expected and why a new l

method of robust stability analysis with highly structured uncertainties this paper presents a method of analyzing the stability of linear time invariant interconnected systems with uncertainties each subsystem is single input single output and its vector locus lies inside a polygon at each frequency it is shown that the convex closure of the image of the cartesian product of these polygons under the mapping phi det i fh agrees with the convex closure of the image of the vertices of these polygons under the mapping from this result the image can be estimated by a finite dumber of calculations a sufficient stability condition is obtained by applying this result to the multivariable nyquist stability criterion

the calculation of acoustic fields in solids for transient normal surface force sources of arbitrary geometry and apodization methods used to calculate the radiated acoustic field of transient normal surface force sources applied to elastic solids have previously included significant approximations or have constrained the surface force distribution in practically restrictive ways in this paper a theoretically exact quantitative numerical solution to the problem is described the only constraint on the applied suface force in this solution is that it is cophased nevertheless by efficient numerical construction the computational expense required is not inconveniently large theoretical results are presented and compared with results in the literature

extremal robustness properties of multilinear interval systems this paper deals with robustness properties of a control system subject to parameter uncertainty when the parameters appear affine multilinearly in the coefficients of the characteristic polynomial the robust hurwitz stability of such a multilinear interval polynomial family reduces to that of the stability of a set of manifolds derived from the kharitonov polynomials several extremal properties of these stability manifolds are derived these properties are useful in determining the worst case parametric h and nonlinear sector bounded stability margins for control systems containing interval parametric uncertainty they generalize previous results obtained by the authors for the linear case

a priori error bounds on invariant subspace approximations by block krylov subspaces abstract we derive a priori error bounds for the block krylov subspace methods in terms of the sine between the desired invariant subspace and the block krylov subspace the obtained results can be seen as the block analogue of the classical a priori estimates for standard projection methods

efficient solution of the simplified p n equations in this paper we show new solver strategies for the multigroup sp n equations for nuclear reactor analysis by forming the complete matrix over space moments and energy a robust set of solution strategies may be applied power iteration shifted power iteration rayleigh quotient iteration arnoldi s method and a generalized davidson method each using algebraic and physics based multigrid preconditioners have been compared on the c5g7 mox test problem as well as an operational pressurized water reactor model our results show that the most efficient approach is the generalized davidson method which is 30 40 times faster than traditional power iteration and 6 10 times faster than arnoldi s method

minres and minerr are better than symmlq in eigenpair computations the symmlq and minres methods for solving sparse indefinite symmetric systems of equations siam j numer anal 12 1975 pp 617 629 can be used inside the inverse or rayleigh quotient iterations for computing the eigenpairs of large sparse generalized eigenproblems a y lambda b y it is shown in this paper that the common opinion concerning the superiority of symmlq over minres is not true in this context several results taken from quantum chemistry acoustics and structural mechanics are presented that show the superiority of minres and the new minerr method over symmlq especially for ill conditioned eigenproblems where symmlq may even diverge theoretical explanations of the observed phenomena are given the use of minres or minerr instead of symmlq may thus lead to a significant increase in efficiency in large scale eigenvalue computations

chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems abstract the present paper deals with the problem of computing a few of the eigenvalues with largest or smallest real parts of a large sparse nonsymmetric matrix we present a general acceleration technique based on chebyshev polynomials and discuss its practical application to arnoldi s method and the subspace iteration method the resulting algorithms are compared with the classical ones in a few experiments which exhibit a sharp superiority of the arnoldi chebyshev approach

large scale complex eigenvalue problems abstract the nonsymmetric eigenvalue problem ax bx is discussed with special emphasis on linear algebra theory on algorithms implemented for solving large scale systems and on interpreting complex spectra obtained in applications from physics and engineering present day supercomputers make the required matrix computations feasible especially in view of the increase in memory

parameter partitioning via shaping conditions for the stability of families of polynomials it is shown that if parameter uncertainty enters a polynomial in a manner which allows an appropriate partitioning generated by certain shaping conditions then the q locus becomes at each frequency a convex parpolygon in the complex plane the results generalize some earlier work and expand the class of problems for which simple robust stability tests exist these results imply that when uncertainty affects polynomial coefficients in a linear manner the q locus is guaranteed to be a convex parpolygon at each frequency

easily testable sufficient conditions for the robust stability of systems with multilinear parameter dependence abstract a number of robust stability problems take the following form a polynomial has real coefficients which are multilinear in real parameters that are confined to a rectangular axis parallel box in parameter space an efficient method is required for checking the stability of this set of polynomials we present two sufficient conditions in this paper which can be applied in many cases they involve checking certain properties at the corners and edges of the parameter space box

time domain formulation for pulse propagation including nonlinear behavior at a caustic this paper derives a simple three dimensional time domain formulation for small angle propagation of finite amplitude acoustic pulses and weak shocks in a medium that can be range dependent the formulation is closely related to a full wave model and does not contain the artificial singularities of ray based models at caustics the simplicity of the formulation suggests that it may be of value for broadband linear as well as nonlinear propagation two derivations are given one heuristic and the other a formal series expansion from the fluid equations the result is a first order nonlinear progressive wave equation npe cast in a wave following frame of reference the npe is shown to be the nonlinear time domain counterpart of the frequency domain parabolic wave equation pe the npe gives a natural separation of terms governing refraction spreading and nonlinear steepening numerical calculations not involving fourier synthesis are presented using the npe to simulate the following cases a broadband linear pulse propagation in a waveguide b the development of an initially smooth nonlinear pulse into an n wave and c the behavior of a weak shock at a caustic cases a and b are compared with analytic solutions

robust stability of polynomials with multilinear parameter dependence abstract the problem is studied of testing for stability a class of real polynomials in which the coefficients depend on a number of variable parameters in a multilinear way we show that the testing for real unstable roots can be achieved by examining the stability of a finite number of corner polynomials obtained by setting parameters at their extreme values while checking for unstable complex roots normally involves examining the real solutions of up to m 1 simultaneous polynomial equations where m is the number of parameters when m 2 this is an especially simple task

a hybrid arnoldi faber iterative method for nonsymmetric systems of linear equations we present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations say of the formax b wherea n n witha nonsingular andb n are given this hybrid method begins with a limited number of steps of the arnoldi method to obtain some information on the location of the spectrum ofa and then switches to a richardson iterative method based on faber polynomials for a polygonal domain the faber polynomials can be constructed recursively from the parameters in the schwarz christoffel mapping function in four specific numerical examples of non normal matrices we show that this hybrid algorithm converges quite well and is approximately as fast or faster than the hybrid gmres or restarted versions of the gmres algorithm it is however sensitive as other hybrid methods also are to the amount of information on the spectrum ofa acquired during the first arnoldi phase of this procedure

a block arnoldi chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices in this paper we describe a block version of arnoldi s method for computing a few eigenvalues with largest or smallest real parts the method is accelerated via chebyshev iteration and a procedure is developed to identify the optimal ellipse which encloses the spectrum a parallel implementation of this method is investigated on the eight processor alliant fx 80 numerical results and comparisons with simultaneous iteration on some harwell boeing matrices are reported

a new iterative scheme for obtaining eigenvectors of large real symmetric matrices a rapidly convergent block iterative scheme for the computation of a few of the lowest eigenvalues and eigenvectors of a large dense real symmetric matrix is proposed the method is especially applicable to matrix eigenvalue problems that arise from the discretization of self adjoint partial differential equations one such application to certain symmetric matrices that arise in solid state band structure calculations is considered in detail the most timeconsuming parts of the present algorithm are a matrix multiplication and a gauss siedel relaxation step which are performed on each iteration these two parts can however be very efficiently implemented on a vector or parallel processing computer

time domain asymptotics and the method of stationary phase a technique is described for obtaining the asymptotic behaviour at large lambda of integrals having the form i lambda int g x p lambda f x dx where p is an arbitrary periodic function with mean zero it is based on the fact that the method of stationary phase may be applied directly to p lambda f x without decomposition of p into fourier components thus a simple minimum of f at x x 0 gives a term containing a fractional integral of order one half proportional to hat p y int y infty p x x y frac 1 2 dx evaluated at y lambda f x 0 a simple maximum gives a similar term in many physical problems f depends linearly on a parameter say t in such a way that i is periodic in t and the quantity of interest is q t di dt the theory of how the shape of q is determined by that of p when lambda is large but fixed is here called waveform asymptotics and its main features investigated using barber s pole integrals for example the singularity in q produced by a discontinuity in p is found explicitly as an inverse square root multiplied by a coefficient so need not be inferred from the tail of a fourier series more generally the effect on q of any rapid change in p may be obtained by the present method of stationary phase in the time domain without resolution into components since gibbs phenomenon is thereby avoided the method is suited to highly non sinusoidal wave problems an asymptotic representation of q by zeta functions is possible four extensions of the basic theory are analysed in detail coalescence of a maximum and minimum of f contributions from the endpoints of the range of integration collision of a maximum or minimum with an endpoint and the behaviour of integrals with no stationary points or end points the first and third of these lead to time domain airy functions and fresnel integrals respectively with singularity structures dual to the smooth patterns found in diffraction catastrophes the second recovers the original waveform and the fourth gives exponential asymptotics the theory is illustrated throughout by analysis and computation for functions p describing a square wave and an intermittent n wave and by diagrams of the resulting waveforms

new methods for calculations of the lowest eigenvalues of the real symmetric generalized eigenvalue problem a new iterative method based on a newton correction vector for extension of the krylov subspace its diagonal and band versions are proposed for calculation of selected lowest eigenvalues and corresponding eigenvectors of the generalized symmetric eigenvalue problem additionally diagonal and band jacobi davidson methods are introduced test calculations show that the new iterative method usually converges faster than quadratic near a solution the new iterative method along with its band version uses a smaller number of iterative steps to obtain a solution compared to the jacobi davidson band jacobi davidson method and generalized davidson method correspondingly the diagonal version of the new method preserves an advantage over the diagonal jacobi davidson and the davidson method

a rayleigh chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse hermitian matrices a procedure is presented for finding a number of the smallest eigenvalues and their associated eigenvectors of large sparse hermitian matrices the procedure a modification of an inverse subspace iteration procedure uses adaptively determined chebyshev polynomials to approximate the required application of the inverse operator on the subspace the method is robust converges with acceptable rapidity and can easily handle operators with eigenvalues of multiplicity greater than one numerical results are shown that demonstrate the utility of the procedure

generalized block lanczos methods for large unsymmetric eigenproblems generalized block lanczos methods for large unsymmetric eigenproblems are presented which contain the block arnoldi method and the block arnoldi algorithms are developed the convergence of this class of methods is analyzed when the matrix a is diagonalizable upper bounds for the distances between normalized eigenvectors and a block krylov subspace are derived and a priori theoretical error bounds for ritz elements are established compared with generalized lanczos methods which contain arnoldi s method the convergence analysis shows that the block versions have two advantages first they may be efficient for computing clustered eigenvalues second they are able to solve multiple eigenproblems however a deep analysis exposes that the approximate eigenvectors or ritz vectors obtained by general orthogonal projection methods including generalized block methods may fail to converge theoretically for a general unsymmetric matrix a even if corresponding approximate eigenvalues or ritz values do since the convergence of ritz vectors needs more sufficient conditions which may be impossible to satisfy theoretically than that of ritz values does the issues of how to restart and to solve multiple eigenproblems are addressed and some numerical examples are reported to confirm the theoretical analysis

time domain asymptotics ii application to propeller acoustics the method of time domain asymptotics of double integrals described in paper i this issue is applied to the thickness noise of a simple sector shaped propeller blade with supersonic tip speed and no axial motion the structure of the pressure field is determined by topological changes in the retarded blade shape corresponding to the presence of stationary points of a certain phase function the overall pressure waveform takes the form of a series of short bursts of high pressure fluctuation each burst corresponding to the crossing of a stationary point by the retarded blade shape asymptotic formulae are derived using the asymptotic regime of small chord length the complete near field structure is considered with detailed calculations of the acoustic field near the various transition surfaces in observer space the results are illustrated throughout with waveform graphs and contour plots for a blade with quadratic chord section

convergence properties of some block krylov subspace methods for multiple linear systems in the present paper we give some convergence results of the global minimal residual methods and the global orthogonal residual methods for multiple linear systems using the schur complement formulae and a new matrix product we give expressions of the approximate solutions and the corresponding residuals we also derive some useful relations between the norm of the residuals

block arnoldi and davidson methods for unsymmetric large eigenvalue problems we present two methods for computing the leading eigenpairs of large sparse unsymmetric matrices namely the block arnoldi method and an adaptation of the davidson method to unsymmetric matrices we give some theoretical results concerning the convergence and discuss implementation aspects of the two methods finally some results of numerical tests on a variety of matrices in which we compare these two methods are reported

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

on the radiation of ultrasound into an isotropic elastic half space via wavefront expansions of the impulse response the problem of propagation of pulses in the radiating near zone of a large circular normal transducer directly coupled to a homogeneous and isotropic elastic half space is re visited it is shown that for certain observation angles the impulse response approach is computationally inefficient a new method based on the so called wavefront expansions of the impulse response is developed instead the expansions are obtained by the analytical harmonic synthesis of the high frequency asymptotics of the transducer field unlike these asymptotics the wavefront expansions are expressed in terms of elementary functions only the direct p edge p and s waves as well as the transition regions penumbra and axial region are described the uniform asymptotic expansions applicable throughout the radiating near zone are derived as well the code based on the time convolution of the pressure input function with the wavefront expansions is compared to a direct numerical code it is thousands of times faster but practical

modification of the liu davidson method for obtaining one or simultaneously several eigensolutions of a large real symmetric matrix a new and efficient iteration method for obtaining simultaneously several eigensolutions and even for obtaining only one solution of a large real symmetric matrix is presented by modifying the simultaneous expansion method by davidson and liu the method is basically the ritz iteration method to correct trial vector s simultaneously using correction vectors however the number of the correction vectors determined in each iteration need not be the same as the number of the desired solutions it is advantageous for the former number to be smaller than the latter when many eigensolutions are sought in addition trial vectors which need not be obtained exactly nor be corrected are included so as to bound the desired solutions which are to be obtained exactly from above and or below when interior solutions are sought the correction vector space may be kept constant throughout the iterations when the correction vector space has room enough to include the trial vectors of the previous iterations the old trial vectors can serve as expansion vectors the performance of the algorithm is tested for five matrices and is compared with the original liu davidson algorithm the results demonstrate that the present algorithm attains efficient use of memory space and reduction of iteration cycles arithmetic operations and i o processings

time domain asymptotics i general theory for double integrals a technique is described for obtaining asymptotic formulae for the time domain waveforms associated with double integrals of rapidly varying isolated pulses the method is an extension of the theory of the time domain asymptotics of single integrals given by c j chapman proc r soc lond a 437 25 40 1992 and represents a generalization of the method of stationary phase to integrals with non sinusoidal integrands stationary points of the phase function in the domain of integration each give rise to a particular waveform shape in the time history of the solution the waveforms associated with interior extrema and saddle points are proportional to the original source function and the hilbert transform of the source function respectively boundary stationary points also give rise to distinctive waveforms proportional to fractional integral transforms of the source function the transitional case of the coalescence of two interior stationary points is considered in some detail an asymptotic formula describing the coalescence is found and the limiting behaviour of this formula after coalescence is calculated i e the residual waveform after the annihilation of the stationary points asymptotic and numerical results are compared for an example integral and good agreement is found even for moderate values of the asymptotic parameter

asymptotic evaluation of the pulse train radiated by an angled beam and fluid coupled rectangular ultrasonic transducer the near field underneath the ultrasonic probe fluid coupled to an isotropic solid is approximated in the frequency domain by a closed form asymptotic solution the approximation is based on the problem decomposition and uses the stationary phase method evaluating the response to an equivalent surface source this results in a sum of contributions each dominating in a specific geometrical region the main beam or a side lobe the transitional zones are also described the pulse trains are computed using the harmonic synthesis and compared with the results obtained by direct calculation of fourier integrals it is shown that the asymptotic approach permits us to elucidate the physics of problem and leads to a numerical algorithm which is about 104 times faster than the direct computations the near field underneath the ultrasonic probe fluid coupled to an isotropic solid is approximated in the frequency domain by a closed form asymptotic solution the approximation is based on the problem decomposition and uses the stationary phase method evaluating the response to an equivalent surface source this results in a sum of contributions each dominating in a specific geometrical region the main beam or a side lobe the transitional zones are also described the pulse trains are computed using the harmonic synthesis and compared with the results obtained by direct calculation of fourier integrals it is shown that the asymptotic approach permits us to elucidate the physics of problem and leads to a numerical algorithm which is about 104 times faster than the direct computations

deflated block krylov subspace methods for large scale eigenvalue problems we discuss a class of deflated block krylov subspace methods for solving large scale matrix eigenvalue problems the efficiency of an arnoldi type method is examined in computing partial or closely clustered eigenvalues of large matrices as an improvement we also propose a refined variant of the arnoldi type method comparisons show that the refined variant can further improve the arnoldi type method and both methods exhibit very regular convergence behavior

tchebychev acceleration technique for large scale nonsymmetric matrices the acceleration by tchebychev iteration for solving nonsymmetric eigenvalue problems is dicussed a simple algorithm is derived to obtain the optimal ellipse which passes through two eigenvalues in a complex plane relative to a reference complex eigenvalue new criteria are established to identify the optimal ellipse of the eigenspectrum the algorithm is fast reliable and does not require a search for all possible ellipses which enclose the spectrum the procedure is applicable to nonsymmetric linear systems as well

shape optimization towards stability in constrained hydrodynamic systems hydrodynamic stability plays a crucial role for many applications existing approaches focus on the dependence of the stability properties on control parameters such as the reynolds or the rayleigh number in this paper we propose a numerical method which aims at solving shape optimization problems in the context of hydrodynamic stability the considered approach allows to guarantee hydrodynamic stability by modifying parts of the underlying geometry within a certain flow regime this leads to a formulation of a shape optimization problem with constraints on the eigenvalues related to the linearized navier stokes equations in that context the eigenvalue problem is generally non symmetric and may involve complex eigenvalues to validate the proposed numerical approach we consider the flow around a body in a channel the shape of the body is parameterized and can be changed by means of a discrete number of design variables it is our aim to find a design which minimizes the drag force and ensures at the same time hydrodynamic stability while keeping the volume of the body constant the numerical results show that a transition from an unstable design to a stable one is attainable by considering an adequate change of the body shape the resulting bodies are long and flat which corresponds to common intuition

physically based parameterizations for designing adaptive algorithms abstract structural properties are examined of systems with physical component values as parameters both state variable realizations and transfer function descriptions are investigated the transfer functions in particular are shown to be the ratios of polynomials with coefficients multilinear in the parameters these structures prove useful in formulating adaptive algorithms

block newton method and block rayleigh quotient iteration for computing invariant subspaces of general complex matrices abstract we consider the block newton method and a modification of it the block rayleigh quotient iteration for approximating a simple p dimensional invariant subspace x im x and the corresponding eigenvalues collected in the projection l x h x 1 x h a x of an arbitrary complex matrix a c n n both methods generate a sequence of bases u k such that the subspaces im u k approximate the target subspace x the block newton method is newton s method applied to the extended system a x x l 0 w h x i with respect to x l where w is a normalization matrix we give a direct convergence analysis which in contrast to the approach by beyn kless thummler 2001 explicitly exploits the product form of the second derivative and therefore leads to new and essentially sharper error recursions for each of the sequences u k and k where u k and k are the k th x and l iterates resp then following losche schwetlick timmermann 1998 where the case of real symmetric a is considered we modify the block newton method as follows in step k we set w u k and use only the x update from the newton method to improve u k and then with the improved basis perform a rayleigh schur process and take the orthonormal basis and the generalized rayleigh quotient matrix obtained from it as new iterates so the method can be considered as a block rayleigh quotient iteration we show that for a sufficiently good initial approximation u 0 the method converges in that the sequence sin k with k im u k x converges q quadratically to zero provided that the invariant subspace x is simple i e that the eigenvalues corresponding to x which are the eigenvalues of the projection l x h x 1 x h a x of a onto x are separated from those corresponding to x which are given by a l the convergence results are directly proven without exploiting standard newton convergence results by using the angles between the corresponding subspaces instead of working with the norm differences of the bases as done by beyn kless thummler 2001 as a byproduct the method is shown to converge cubically if a a h as in the case p 1

projection methods for large lyapunov matrix equations abstract in the present paper we propose krylov subspace methods for solving large lyapunov matrix equations of the form ax xat bbt 0 where a and b are real n n and n s matrices respectively with s n such problems appear in many areas of control theory such as the computation of hankel singular values model reduction algorithms and others the proposed methods are based on the arnoldi process we show how to extract low rank approximate solutions to lyapunov matrix equations and give some theoretical results finally some numerical tests will be reported to illustrate the effectiveness of the proposed method

gmres with deflated restarting a modification is given of the gmres iterative method for nonsymmetric systems of linear equations the new method deflates eigenvalues using wu and simon s thick restarting approach siam j matrix anal appl 22 2000 pp 602 616 it has the efficiency of implicit restarting but is simpler and does not have the same numerical concerns the deflation of small eigenvalues can greatly improve the convergence of restarted gmres also it is demonstrated that using harmonic ritz vectors is important because then the whole subspace is a krylov subspace that contains certain important smaller subspaces

acceleration of inverse subspace iteration with newton s method this work is focused on the computation of the invariant subspace associated with a separated group of eigenvalues near a specified shift of a large sparse matrix first we consider the inverse subspace iteration with the preconditioned gmres method it guarantees a convergence to the desired invariant subspace but the rate of convergence is at best linear we propose to use it as a preprocessing for a newton scheme which necessitates at each iteration the solution of a sylvester type equation for which an iterative algorithm based on the preconditioned gmres method is specially devised this combination results in a fast and reliable method we discuss the implementation aspects and propose a theory of convergence numerical tests are given to illustrate our approach

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a refined iterative algorithm based on the block arnoldi process for large unsymmetric eigenproblems when the matrix in question is unsymmetric the approximate eigenvectors or ritz vectors obtained by orthogonal projection methods including arnoldi s method and the block arnoldi method cannot be guaranteed to converge in theory even if the corresponding approximate eigenvalues or ritz values do in order to circumvent this danger a new strategy has been proposed by the author for arnoldi s method the strategy used is generalized to the block arnoldi method in this paper it discards ritz vectors and instead computes refined approximate eigenvectors by small sized singular value decompositions it is proved that the new strategy can guarantee the convergence of refined approximate eigenvectors if the corresponding ritz values do the resulting refined iterative algorithm is realized by the block arnoldi process numerical experiments show that the refined algorithm is much more efficient than the iterative block arnoldi algorithm

smoothing iterative block methods for linear systems with multiple right hand sides in the present paper we present smoothing procedures for iterative block methods for solving nonsymmetric linear systems of equations with multiple right hand sides these procedures generalize those known when solving one right hand linear systems we give some properties of these new methods and then using these procedures we show connections between some known iterative block methods finally we give some numerical examples

an iterative block arnoldi algorithm with modified approximate eigenvectors for large unsymmetric eigenvalue problems a modified m step arnoldi algorithm proposed by jia and elsner for computing a few selected eigenpairs of large unsymmetric matrices is generalized to its block version in the new method we use new modified approximate eigenvectors obtained from a linear combination of ritz vectors and the wasted m 1 th block basis vector whose residual norms are satisfied with some p 1 dimensional minimization problems the resulting block arnoldi algorithm is not only better than the standard m step one both in theory and in practice but also cheaper than the standard m 1 step one the relationships among the residual norms of ritz pairs and those of new approximate eigenpairs are analyzed theoretical results show that the new method can overcome the drawback of nonconvergence which exists in the conventional one in some extent we carry out some numerical experiments which exhibit that the new algorithm is more efficient and works better than its counterparts

acoustic waveform singularities from supersonic rotating surface sources this paper discusses the analytical characteristics of waveforms produced by rotating surface sources with tip mach number 1 the analysis covers both monopole and quadrupole sources that have the greatest importance for prediction of high speed rotor noise by using idealized source distributions over the rectangular surfaces it is shown that the lighthiu curle analogy with time derivatives taken outside the integrals poses no additional difficulty attributable to numerical error the integrals themselves require careful analysis of critical points in the integrand so that an adaptive integration grid can be developed for each time step a mathematical analysis of the integrands arising from the idealized sources predicts a logarithmic singularity at critical points in the monopole waveform and a single pole singularity at such points in the quadrupole waveform the computation bears out these predictions

the asymptotic theory of rapidly rotating sound fields a rapidly rotating sound field such as that produced by a supersonic propeller may contain several types of diffraction pattern each in a different region of space this paper determines these patterns and their locations for the field around a propeller of simple type the method used is stationary phase analysis of a certain double integral and leads to asymptotic formulae valid when the number of blades is not too small or the high harmonics are being investigated physically the results describe propagation along rays each stationary phase point is a loud spot producing a ray which points directly at the observer most of the noise comes from these loud spots because extensive cancellation takes place everywhere else at most two interior and two boundary stationary points may be present the number and type depend on the position of the observer in relation to a cusped torus and two hyperboloids of one sheet as these surfaces are crossed the acoustic field changes in character for example when two stationary points coalesce and annihilate each other as they do at a caustic an airy function describes the transition from a loud zone of rapid oscillation to a quiet zone of exponential decay and when an interior stationary point crosses the boundary of the disc the transition region is described either by a fresnel integral or by a generalized airy function separate analyses are given for regions close to and well away from the transition surfaces and inner and outer limits are calculated for use in the method of matched asymptotic expansions in all cases an overlap region is found in which the leading terms agree the results of the paper determine completely the geometry of the acoustic field because the different regions have boundaries at known positions and cover the whole of space

convergence of polynomial restart krylov methods for eigenvalue computations krylov subspace methods have led to reliable and effective tools for resolving large scale non hermitian eigenvalue problems since practical considerations often limit the dimension of the approximating krylov subspace modern algorithms attempt to identify and condense significant components from the current subspace encode them into a polynomial filter and then restart the krylov process with a suitably refined starting vector in effect polynomial filters dynamically steer low dimensional krylov spaces toward a desired invariant subspace through their action on the starting vector the spectral complexity of nonnormal matrices makes convergence of these methods difficult to analyze and these effects are further complicated by the polynomial filter process the principal object of study in this paper is the angle an approximating krylov subspace forms with a desired invariant subspace convergence analysis is posed in a geometric framework that is robust to eigenvalue ill conditioning yet remains relatively uncluttered the bounds described here suggest that the sensitivity of desired eigenvalues exerts little influence on convergence provided the associated invariant subspace is well conditioned ill conditioning of unwanted eigenvalues plays an essential role this framework also gives insight into the design of effective polynomial filters numerical examples illustrate the subtleties that arise when restarting non hermitian iterations

robust stability under a class of nonlinear parametric perturbations this paper considers the robust stability verification of linear time invariant systems admitting a class of nonlinear parametric perturbations the general setting is one of determining the closed loop stability of systems whose open loop transfer functions consist of powers products and ratios of polytopes of polynomials apart from this general setting two special cases of independent interest are also considered the first special case concerns uncertainties in the open loop gain and real poles and zeros while the second special case treats uncertainties in the open loop gain and complex poles and zeros in light of the zero exclusion principle robust stability is equivalent to zero exclusion of the value sets of the system characteristic function a value set consists of the values of the characteristic functions at a fixed frequency the main results of this paper are as follows 1 the value set of the characteristic function at each fixed frequency is determined by the edges and some frequency dependent internal line segments 2 consequently hurwitz invariance verification simplifies to that of checking certain continuous scalar functions for avoidance of the negative real axis 3 for the case of real zero pole gain variations the critical lines are all frequency independent and therefore the determination of the robust stability is even simpler 4 for the case of complex zero pole gain variations the critical internal lines are shown to be either frequency independent or to be confined in certain two dimensional planes or three dimensional boxes

convergence properties of block gmres and matrix polynomials abstract this paper studies convergence properties of the block gmres algorithm when applied to nonsymmetric systems with multiple right hand sides a convergence theory is developed based on a representation of the method using matrix valued polynomials relations between the roots of the residual polynomial for block gmres and the matrix e pseudospectrum are derived and illustrated with numerical experiments the role of invariant subspaces in the effectiveness of block methods is also discussed

a thick restarted block arnoldi algorithm with modified ritz vectors for large eigenproblems the block arnoldi method is one of the most commonly used techniques for large eigenproblems in this paper we exploit certain modified ritz vectors to take the place of ritz vectors in the thick restarted block arnoldi algorithm and propose a modified thick restarted block arnoldi algorithm for large eigenproblems we then consider how to periodically combine the refined subspace iterative method with the modified thick restarting block arnoldi algorithm for computing a few dominant eigenpairs of a large matrix the resulting algorithm is called a subspace block arnoldi algorithm numerical experiments show the efficiency of our new algorithms

justification of the stationary phase approximation in time domain asymptotics a rigorous proof is supplied for the validity of an asymptotic approximation to the integral i ab g p f x dx where f x and g x are sufficiently smooth functions on a b and p x is a piece wise smooth periodic function with mean zero in addition a two dimensional generalization is given problems concerning coalescence of two stationary points and a stationary point near an end point are also considered

a modified harmonic block arnoldi algorithm with adaptive shifts for large interior eigenproblems the harmonic block arnoldi method can be used to find interior eigenpairs of large matrices given a target point or shift t to which the needed interior eigenvalues are close the desired interior eigenpairs are the eigenvalues nearest t and the associated eigenvectors however it has been shown that the harmonic ritz vectors may converge erratically and even may fail to do so to do a better job a modified harmonic block arnoldi method is coined that replaces the harmonic ritz vectors by some modified harmonic ritz vectors the relationships between the modified harmonic block arnoldi method and the original one are analyzed moreover how to adaptively adjust shifts during iterations so as to improve convergence is also discussed numerical results on the efficiency of the new algorithm are reported

matrix recursive interpolation algorithm for block linear systems direct methods abstract this paper presents an unified algorithm for solving systems of matrix and block linear equations solving a system of block linear equations will be interpreted as a matrix interpolation problem this new approach gives us a general algorithm called the matrix recursive interpolation algorithm mria which includes block direct methods the orthogonal version of the mria will be given and denoted by omria we will also show how to choose two free parameters in the mria for obtaining known and new block direct methods

the complete far field asymptotic description of a point source acting on a transversely isotropic half space the elastic wavefield generated by a point source of tractions acting on the surface of a transversely isotropic half space is studied the symmetry axis of the solid is oriented arbitrarily with respect to the surface of the half space first the integral representation of the time harmonic green9s tensor is given then the complete far field asymptotic approximation of a quasi longitudinal qp and two quasi shear qsh and qsv waves is derived the qp wave is described by the leading term of the ray series since there is only one arrival of this wave the qsh wave is treated similarly everywhere apart from the so called kissing point boundary layer where the qsh and qsv wavefronts are tangentially close to each other a special asymptotic formula is obtained for this case the qsv sheet of the wave surface is allowed to have conical points and cuspidal edges thus the far field approximation of the qsv wave involves ray asymptotic expressions while inside the geometrical regions where either one or three qsv arrivals exist or else boundary layer asymptotics inside conical point cuspidal edge and kissing point boundary layers at the end of the paper we present numerical results of the simulation of pulse propagation a good agreement between the asymptotic and direct numerical codes is achieved but the former is orders of magnitude faster

a modified block newton iteration for approximating an invariant subspace of a symmetric matrix in this paper we propose a modified block newton method mbnm for approximating an invariant subspace j and the corresponding eigenvalues of a symmetric matrix a the method generates a sequence of matrices z k which span subspaces jk approximating j the matrices z k are calculated via a newton step applied to a special formulation of the block eigenvalue problem for the matrix a followed by a rayleigh ritz step which also yields the corresponding eigenvalue approximations we show that for sufficiently good initial approximations the subspaces jk converge to j in the sense that sin k with k jk j q quadratically converges to zero under appropriate conditions