bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices abstract let a monic polynomial pn x xn a1xn 1 an aj plane1d 49e j 1 2 n be given bounds for the eigenvalues and the computation of the singular values of some special companion matrices give bounds for the zeros of the polynomial pn which improve classical and recently found ones also some new bounds for the zeros are given

error analysis of signal zeros a projected companion matrix approach an error analysis of so called signal zeros of polynomials associated with exponentially damped undamped signals is performed and zero error bounds are derived the bounds are in terms of the angle between the exact and approximate signal subspace the signal parameter themselves the polynomial degree and the error on the polynomial coeficients the key idea behind the analysis is to regard signal zeros as eigenvalues of projected companion matrices and then to generate error bounds by exploiting perturbation theorem for eigenvalues the conclusion drawn from the bounds is that the signal zeros become relatively insensitive to small perturbations on the polynomial coefficients as long as the polynomial degree is large enough and the zeros are extracted as eigenvalues of projected companion matrices also the bounds suggests that signal zero estimates derived from projected companion matrices are more accurate than those obtained from the companion matrices themselves illustrative numerical results are provided

seislet transform and seislet frame we introduce a digital waveletlike transform which is tailored specifically for representing seismic data the transform provides a multiscale orthogonal basis with basis functions aligned along seismic events in the input data it is defined with the help of the wavelet lifting scheme combined with local plane wave destruction in the 1d case the seislet transform is designed to follow locally sinusoidal components in the 2d case it is designed to follow local plane wave components with smoothly variable slopes if more than one component is present the transform turns into an overcomplete representation or a tight frame in these terms the classic digital wavelet transform is simply a seislet transform for a zero frequency in one dimension or zero slope in two dimensions the main objective of the new transform is an effective seismic data compression for designing efficient data analysis algorithms traditional signal processing tasks such as noise attenuation and trace interpolation become simply defined in the seislet domain when applied in the offset direction on common midpoint or common image point gathers the seislet transform finds an additional application in optimal stacking of seismic records

accurate solution of polynomial equations using macaulay resultant matrices we propose an algorithm for solving two polynomial equations in two variables our algorithm is based on the macaulay resultant approach combined with new techniques including randomization to make the algorithm accurate in the presence of roundoff error the ultimate computation is the solution of a generalized eigenvalue problem via the qz method we analyze the error due to roundoff of the method showing that with high probability the roots are computed accurately assuming that the input data that is the two polynomials are well conditioned our analysis requires a novel combination of algebraic and numerical techniques

continuous solutions for divergence type equations associated to elliptic systems of complex vector fields abstract in this paper we characterize all the distributions f d u such that there exists a continuous weak solution v c u c n with u to the divergence type equation l 1 v 1 l n v n f where l 1 l n is an elliptic system of linearly independent vector fields with smooth complex coefficients defined on r n in case where l 1 l n is the usual gradient field on r n we recover the classical result for the divergence equation proved by t de pauw and w pfeffer its proof is based on the closed range theorem and inspired by 3 and 6 in the classical case our method slightly differs from theirs by relying on the banach grothendieck theorem and introducing tools from pseudodifferential operators useful in our local setting of a system of complex vector fields with variable coefficients

pseudozeros of multivariate polynomials the pseudozero set of a system f of polynomials in n complex variables is the subset of cn which is the union of the zero sets of all polynomial systems g that are near to f in a suitable sense this concept is made precise and general properties of pseudozero sets are established in particular it is shown that in many cases of natural interest the pseudozero set is a semialgebraic set also estimates are given for the size of the projections of pseudozero sets in coordinate directions several examples are presented illustrating some of the general theory developed here finally algorithmic ideas are proposed for solving multivariate polynomials

the reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform this paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its fourier transform the first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude although phase or magnitude information alone is not sufficient in general to uniquely specify a sequence a large class of sequences are shown to be recoverable from their phase or magnitude the second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude for those sequences which are uniquely specified by their phase several practical algorithms are described which may be used to reconstruct a sequence from its phase several examples of phase only reconstruction are also presented unfortunately however even for those sequences which are uniquely defined by their magnitude it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude nevertheless an iterative procedure which has been proposed is briefly discussed and evaluated

representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples it is shown that the electromagnetic em field radiated or scattered by bounded sources can be accurately represented over a substantially arbitrary surface by a finite number of samples even when the observation domain is unbounded the number of required samples is nonredundant and essentially coincident with the number of degrees of freedom of the field this result relies on the extraction of a proper phase factor from the field expression and on the use of appropriate coordinates to parameterize the domain it is demonstrated that the number of degrees of freedom is independent of the observation domain and depends only on the source geometry the case of spheroidal sources and observation domains with rotational symmetry is analyzed in detail and the particular cases of spherical and planar sources are explicitly considered for these geometries precise and fast sampling algorithms of central type are presented which allow an efficient recovery of em fields from a nonredundant finite number of samples such algorithms are stable with respect to random errors affecting the data

phaseless bi polar planar near field measurements and diagnostics of array antennas antenna near field measurements typically require very accurate measurement of the near field phase there are applications where an accurate phase measurement may not be practically achievable phaseless measurements are beginning to emerge as an alternative microwave antenna measurements technique when phase cannot be directly measured there are many important aspects for successful implementation of a phaseless measurement algorithm this paper presents appropriate phaseless measurement requirements and a phase retrieval algorithm tailored for the bi polar planar near field antenna measurement technique two amplitude measurements and a squared amplitude optimal sampling interpolation method are integrated with an iterative fourier procedure to first retrieve the phase information and then construct both the far field pattern and diagnostic characteristics of the antenna under test in order to critically examine the methodologies developed in this paper phaseless measurement results for two different array antennas are presented and compared to results obtained when the near field amplitude and phase are directly measured

a klein bottle based dictionary for texture representation a natural object of study in texture representation and material classification is the probability density function in pixel value space underlying the set of small patches from the given image inspired by the fact that small n times n n n high contrast patches from natural images in gray scale accumulate with high density around a surface fancyscript k subset mathbb r n 2 k r n 2 with the topology of a klein bottle carlsson et al international journal of computer vision 76 1 1 12 2008 we present in this paper a novel framework for the estimation and representation of distributions around fancyscript k k of patches from texture images more specifically we show that most n times n n n patches from a given image can be projected onto fancyscript k k yielding a finite sample s subset fancyscript k s k whose underlying probability density function can be represented in terms of fourier like coefficients which in turn can be estimated from s s we show that image rotation acts as a linear transformation at the level of the estimated coefficients and use this to define a multi scale rotation invariant descriptor we test it by classifying the materials in three popular data sets the curet uiuctex and kth tips texture databases

polynomial zerofinders based on szego polynomials abstract the computation of zeros of polynomials is a classical computational problem this paper presents two new zerofinders that are based on the observation that after a suitable change of variable any polynomial can be considered a member of a family of szeg polynomials numerical experiments indicate that these methods generally give higher accuracy than computing the eigenvalues of the companion matrix associated with the polynomial

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

projection matrix method for shaped beam synthesis in phased arrays and reflectors we present the projection matrix method for shaped beam synthesis in a general array of different size elements the method relies on orthogonal projection of the desired far field intensity vector onto the space spanned by the far field intensity vectors of the array elements it is found that for a uniform convergence of the solution the far field sample space must be extended beyond the coverage region otherwise the projection matrix becomes ill conditioned a general guideline for the far field sample space is provided the method with necessary amendments is then employed successfully for a reflector surface synthesis the method is found to be several times faster than the gradient search method commonly used for beam synthesis numerical results for array and shaped reflector syntheses are shown and the advantages discussed

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

an improved toeplitz algorithm for polynomial matrix null space computation in this paper we present an improved algorithm to compute the minimal null space basis of polynomial matrices a problem which has many applications in control and systems theory this algorithm takes advantage of the block toeplitz structure of the sylvester matrix associated with the polynomial matrix the analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well known pencil state space algorithms found in the literature

diagnosis of array faults from far field amplitude only data the diagnosis of the faulty elements of a planar array from noisy far field power pattern data is considered in the case of on off faults the possible ambiguities of the solutions are considered both in the theoretical and practical sense and are shown to be intrinsically less relevant than in the widely studied continuous case the probability of the occurrence of the practical ambiguities is inferred from a number of numerical examples and is shown to be negligible in all cases of interest an effective algorithm is presented here based on an intersection set finding approach and involving the minimization of a suitable objective functional the global minimization of the functional has been successfully performed by applying a properly modified genetic algorithm a number of numerical examples shows the effectiveness of the approach whose computational complexity essentially increases linearly with the array size

an effective deterministic procedure for the synthesis of shaped beams by means of uniform amplitude linear sparse arrays a rigorous deterministic approach is proposed for the mask constrained power pattern synthesis of shaped beams through one dimensional sparse arrays having uniform amplitude excitations and variable phases the approach includes the well known density taper procedure as a special case the corresponding technique which exploits an analytical formulation of the problem and convex programming routines results in fast computation and is able to achieve optimal solutions without recurring to any global optimization numerical results assessing the capabilities of the proposed design method are provided opening the way to the development of an effective approach for the synthesis of circularly symmetric shaped beams via uniformly excited circular ring arrays

a question of rentschler and the dixmier problem the following theorem is the main result of the paper theorem let r and t be somewhat commutative algebras with the same holonomic number and let r t be an algebra homomorphism then every holonomic t module m is via a holonomic r module and has finite length as an r module when applied to the weyl algebra this result gives a positive answer to a question of rentschler in the important case where r d x and t d y are rings of differential operators on smooth irreducible algebraic affine varieties x and y of the same dimension the result means that holonomicity is preserved by twisting by an arbitrary algebra homomorphism a short proof is given of the well known fact that an affirmative solution to the dixmier problem whether every algebra endomorphism of the weyl algebra an is an automorphism implies the jacobian conjecture the dixmier problem has a positive answer if and only if the twisted from both sides a n bimodule an is simple for each to begin with it is not even clear whether it is finitely generated the theorem implies that it has finite length moreover the bimodule is holonomic thus simple subfactors of it have the least possible gelfand kirillov dimension

fast qr iterations for unitary plus low rank matrices some fast algorithms for computing the eigenvalues of a block companion matrix have recently appeared in the literature in this paper we generalize the approach to encompass unitary plus low rank matrices of the form a u xy h where u is a general unitary matrix three important cases for applications are u unitary diagonal u unitary block hessenberg and u unitary in block cmv form our extension exploits the properties of a larger matrix hat a obtained by a certain embedding of the hessenberg reduction of a suitable to maintain its structural properties we show that hat a can be factored as product of lower and upper unitary hessenberg matrices possibly perturbed in the first k rows and moreover such a data sparse representation is well suited for the design of fast eigensolvers based on the qr iteration the resulting eigenvalue algorithm is fast and backward stable

stratification of full rank polynomial matrices we show that perturbations of polynomial matrices of full normal rank can be analyzed viathe study of perturbations of companion form linearizations of such polynomial matrices it is proved that a

a companion matrix resultant for bernstein polynomials abstract a closed form expression for a companion matrix m of a bernstein polynomial is obtained and this is used to derive an expression for a resultant matrix of two bernstein polynomials it is shown that m differs from its equivalent form for a power basis polynomial because an upper triangular hankel matrix does not define a similarity transformation between m and m t a measure of the numerical condition of a resultant matrix for polynomials in an arbitrary basis is reviewed and this is used to compare the stability of two resultant matrices in particular computational tests are performed and it is shown that the resultant matrix of two bernstein polynomials is numerically better conditioned than the resultant matrix that is obtained when a simple parameter substitution is used to transform the polynomials to the power basis

dixmier s problem 5 for the weyl algebra abstract the purpose of this paper is to give a negative answer to problem 5 and a positive answer in the case of homogeneous elements to problem 4 of the article of j dixmier bull soc math france 96 1968 209 242 on the first weyl algebra

a global local synthesis approach for large non regular arrays a method is proposed for the synthesis of large planar non regular arrays assuming constant magnitude for the excitation of the elements the approach which combines global and local optimization is based on distinguishing aperture type and non coherent parts of the array factor the following constraints are considered minimal separation of the antennas maximum size of the array and fixed mainbeam width the level of the sidelobes is reduced via the minimization of a new type of flexible averaging cost function based on an lp norm possible applications of this model lie in the field of radio astronomy satellite communications and radar systems the proposed optimization strategy consists of three successive steps designed to be independent of each other starting with an equivalent continuous aperture the first two steps act as global transformations of the radial and azimuthal positions of the elements in the initial array while the third step performs a local optimization of individual elements of the array this last step heavily relies on computations of the gradient of the cost function which can be done quickly using an fft based procedure the method is illustrated for several large scale examples by considering as inputs four different types of non regular arrays

vectorial phase retrieval of 1 d signals reconstruction of signals from measurements of their spectral intensities also known as the phase retrieval problem is of fundamental importance in many scientific fields in this paper we present a novel framework denoted as vectorial phase retrieval for reconstruction of pairs of signals from spectral intensity measurements of the two signals and of their interference we show that this new framework can alleviate some of the theoretical and computational challenges associated with classical phase retrieval from a single signal first we prove that for compactly supported signals in the absence of measurement noise this new setup admits a unique solution next we present a statistical analysis of vectorial phase retrieval and derive a computationally efficient algorithm to solve it finally we illustrate via simulations that our algorithm can accurately reconstruct signals even at considerable noise levels

modified gershgorin disks for companion matrices all the zeros of a polynomial are contained in the union of gershgorin disks derived from its companion matrix a consequence of gershgorin s theorem however this theorem does not exploit the structure of the companion matrix we will use this structure to obtain smaller zero inclusion regions thereby providing some nonstandard results to accompany and illustrate this frequently covered topic in numerical and matrix analysis

radiation pattern evaluation from near field intensities on planes this paper describes analyzes and implements an approach to far field determination from phaseless measurements over two planar surfaces a proper formulation of the problem is considered as a quadratic inverse one whose data is the square amplitude of the near field a solution is introduced as the global minimum of an appropriate functional next to perform such a minimization procedure a finite dimensional representation of the field radiated by sources whose plane wave spectrum becomes negligible outside a fixed angular domain is used a detailed investigation of the properties of the mapping connecting the unknowns to the data makes it possible to analyze how to escape from the local minima possibly met in the course of the minimization procedure to this end the crucial role of the availability of phaseless data over two different sarfa ces and of appropriate weights in the functional definition is emphasized and a reliable iterative procedure converging on the solution regardless of the starting point is thus obtained this property is confirmed by experimental results concerning near zone data from a shaped reflector at 9 ghz it can be readily appreciated that when only the field intensity is detected the complexity and the cost of the equipment required for near field techniques in antenna testing and diagnostics can be reduced to a very large extent

perturbation bounds for polynomials using two different elementary approaches we derive a global and a local perturbation theorem on polynomial zeros that significantly improve the results of ostrowski acta math 72 99 257 1940 elsner et al linear algebra appl 142 195 209 1990 a comparison of different perturbation bounds shows that our results are better in many cases than the similar local result of beauzamy can math bull 42 1 3 12 1999 using the matrix theoretical approach we also improve the backward stability result of edelman and murakami proceedings of the fifth siam conference on applied linear algebra siam philapdelphia 1994 math comput 64 210 763 1995

simple holonomic modules over the second weyl algebra a2 abstract for simple generalized weyl algebras of gelfand kirillov dimension 4 a class including the second weyl algebra a 2 some simple factor algebras of the universal enveloping algebra of the lie algebra sl 2 sl 2 and of u q sl 2 u q sl 2 etc the simple holonomic modules are classified up to pairs of irreducible elements of certain noncommutative euclidean ring

robust and efficient solution of the drum problem via nystrom approximation of the fredholm determinant the drum problem finding the eigenvalues and eigenfunctions of the laplacian with dirichlet boundary condition has many applications yet remains challenging for general domains when high accuracy or high frequency is needed boundary integral equations are appealing for large scale problems yet certain difficulties have limited their use we introduce the following two ideas to remedy this 1 we solve the resulting nonlinear eigenvalue problem using boyd s method for analytic root finding applied to the fredholm determinant and we show that this is many times faster than the usual iterative minimization of a singular value 2 we fix the problem of spurious exterior resonances via a combined field representation this also provides the first robust boundary integral eigenvalue method for non simply connected domains we implement the new method in two dimensions using spectrally accurate nystrom product quadrature we prove exponential convergence of the determinant at roots for domains with

on data increasing in phase retrieval via quadratic inversion flattening manifold and local minima in this article we address the question of traps in phase retrieval via quadratic inversion first we clarify the genesis of local minima from a geometrical perspective by showing how the shape of the manifold of data and the position of the data point are related to the presence of local minima in the objective functional later we recall a mathematical condition for the lack of traps afterward we illustrate how the ratio between the dimension of data space m and the number of real unknowns n r impacts on the manifold of data and consequently on the presence of local minima in particular we show that if the dimension of data space is high enough the manifold of data is sufficiently large and no traps appear in the functional in the last part of this article with reference to a particular unknown sequence we establish the minimum value of the ratio m n r such that no traps should appear in the functional in this condition phase retrieval via quadratic inversion exhibits a global convergence behavior consequently the actual solution of the problem can be reached regardless of the initial guess which can be chosen completely at random

polynomial roots from companion matrix eigenvalues in classical linear algebra the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial an algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition we derive a first order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm our error analysis expands on work by van dooren and dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully regarding the stronger concept of a small componentwise backward error our analysis predicts a small such error on a test suite of eight random polynomials suggested by toh and trefethen however we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice we extend our results to polynomial matrices where the result is essentially the same but the geometry becomes more complicated

probability of local bifurcation type from a fixed point a random matrix perspective results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems real random matrices time delay dynamical systems companion matrices and a set of mappings known for their properties as universal approximators neural networks the eigenvalue spectrum is considered both numerically and analytically using previous work of edelman et al based upon the numerical evidence various conjectures are presented the conclusion is that in many circumstances most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues nevertheless surprising situations are presented for which the aforementioned conclusion does not hold e g real random matrices with gaussian elements with a large positive mean and finite variance

efficient ehrlich aberth iteration for finding intersections of interpolating polynomials and rational functions abstract we analyze the problem of carrying out an efficient iteration to approximate the eigenvalues of some rank structured pencils obtained as linearization of sums of polynomials and rational functions expressed in possibly different interpolation bases the class of linearizations that we consider has been introduced by robol vandebril and van dooren in 17 we show that a traditional qz iteration on the pencil is both asymptotically slow since it is a cubic algorithm in the size of the matrices and sometimes not accurate since in some cases the deflation of artificially introduced infinite eigenvalues is numerically difficult to solve these issues we propose to use a specifically designed ehrlich aberth iteration that can approximate the eigenvalues in o k n 2 flops where k is the average number of iterations per eigenvalue and n the degree of the linearized polynomial we suggest possible strategies for the choice of the initial starting points that make k asymptotically smaller than o n thus making this method less expensive than the qz iteration moreover we show in the numerical experiments that this approach does not suffer of numerical issues and accurate results are obtained

an effective approach for the optimal focusing of array fields subject to arbitrary upper bounds a new approach to the optimal focusing of array fields subject to arbitrary upper bounds is presented the approach formulates the problem as the minimization of a linear function in a convex set unlike other approaches this one guarantees the achievement of the global optimum by using local optimization techniques and can moreover deal with any convex constraint on the unknowns such as near field constraints optimization is performed by two ad hoc developed solution algorithms which exploit the geometrical characteristics of the problem at hand thus leading to extremely effective and computationally efficient numerical codes an extensive numerical analysis has been performed in all cases of linear planar and circular arc arrays the enhanced performance of the proposed technique with respect to the solution algorithms available in the literature fully confirms the effectiveness of the approach

constrained decentralized estimation over noisy channels for sensor networks decentralized estimation of a noise corrupted source parameter by a bandwidth constrained sensor network feeding through a noisy channel a fusion center is considered the sensors due to bandwidth constraints provide binary representatives of a noise corrupted source parameter recently proposed decentralized distributed estimation and power scheduling methods do no consider errors occurring during the transmission of binary observations from the sensors to fusion center in this paper we extend the decentralized estimation model to the case where imperfect transmission channels are considered the proposed estimator which operates on additive channel noise corrupted versions of quantized noisy sensor observations is approached from maximum likelihood ml perspective the resulting ml estimate is a root in the region of interest roi of a derivative polynomial function we analyze the natural logarithm of the polynomial within the roi showing that the function is log concave thereby indicating that numerical methods such as newton s algorithm can be utilized to obtain the optimal solution due to complexity and implementation issues associated with the numerical methods we derive and analyze simpler suboptimal solutions i e the two stage and mean estimators the two stage estimator first estimates the binary observations from noisy fusion center observations utilizing a threshold operation followed by an estimate of the source parameter the optimal threshold is the maximum a posteriori map detector for binary detection and minimizes the probability of binary observation estimation error optimal threshold expressions for commonly utilized light gaussian and heavy tailed cauchy channel noise models are derived the mean estimator simply averages the noisy fusion center observations the output variances of means of the proposed suboptimal estimators are derived in addition a computational complexity analysis is presented comparing the proposed ml optimal and suboptimal two stage and mean estimators numerical examples evaluating and comparing the performance of proposed ml two stage and mean estimators are also presented

pseudozeros of polynomials and pseudospectra of companion matrices it is well known that the zeros of a polynomial p are equal to the eigenvalues of the associated companion matrix a in this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding the epsilon pseudozero set z epsilon p is the set of zeros of all polynomials hat p obtained by coefficientwise perturbations of p of size leq epsilon this is a subset of the complex plane considered earlier by mosier and is bounded by a certain generalized lemniscate the epsilon pseudospectrum lambda epsilon a is another subset of bbb c defined as the set of eigenvalues of matrices hat a a e with vert e vert leq epsilon it is bounded by a level curve of the resolvent of a we find that if a is first balanced in the usual eispack sense then z epsilon vert p vert p and lambda epsilon vert a vert a are usually quite close to one another it follows that the matlab roots algorithm of balancing the companion matrix then computing its eigenvalues is a stable algorithm for polynomial zerofinding experimental comparisons with the jenkins traub imsl and madsen reid harwell fortran codes confirm that these three algorithms have roughly similar stability properties

gelfand kirillov dimension of commutative subalgebras of simple inﬁnite dimensional algebras and their quotient division algebras

block kronecker linearizations of matrix polynomials and their backward errors we introduce a new family of strong linearizations of matrix polynomials which we call block kronecker pencils and perform a backward stability analysis of complete polynomial eigenproblems these problems are solved by applying any backward stable algorithm to a block kronecker pencil such as the staircase algorithm for singular pencils or the qz algorithm for regular pencils this analysis allows us to identify those block kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial these favorable pencils include the famous fiedler linearizations which are just a very particular case of block kronecker pencils thus our analysis offers the first proof available in the literature of global backward stability for fiedler pencils in addition the theory developed for block kronecker pencils is much simpler than the theory available for fiedler pencils especially in the case of rectangular matrix polynomials the global backward error analysis in this work presents for the first time the following key properties it is a rigorous analysis valid for finite perturbations i e it is not a first order analysis it provides precise bounds it is valid simultaneously for a large class of linearizations and it establishes a framework that may be generalized to other classes of linearizations these features are related to the fact that block kronecker pencils are a particular case of the new family of strong block minimal bases pencils which include certain perturbations of block kronecker pencils this allows us to extend the results in this paper to other contexts

an analogue of the conjecture of dixmier is true for the algebra of polynomial integro differential operators abstract let a 1 k x d d x be the weyl algebra and i 1 k x d d x be the algebra of polynomial integro differential operators over a field k of characteristic zero the conjecture problem of dixmier 1968 still open is an algebra endomorphism of the weyl algebra a 1 an automorphism the aim of the paper is to prove that each algebra endomorphism of the algebra i 1 is an automorphism notice that in contrast to the weyl algebra a 1 the algebra i 1 is a non simple non noetherian algebra which is not a domain moreover it contains infinite direct sums of nonzero left and right ideals

material optimization in transverse electromagnetic scattering applications a class of algorithms for the solution of discrete material optimization problems in electromagnetic applications is discussed the idea behind the algorithm is similar to that of the sequential programming however in each major iteration a model is established on the basis of an appropriately parametrized material tensor the resulting nonlinear parametrization is treated on the level of the subproblem for which globally optimal solutions can be computed due to the block separability of the model although global optimization of nonconvex design problems is generally prohibitive a well chosen combination of analytic solutions along with standard global optimization techniques leads to a very efficient algorithm for most relevant material parametrizations a global convergence result for the overall algorithm is established the effectiveness of the approach in terms of both computation time and solution quality is demonstrated by numerical examples based on a two dimensional helmholtz equation inclu

perturbations of invariant subspaces of unreduced hessenberg matrices non structured perturbation of invariant subspaces of unreduced i e nonderogatory hessenberg matrices is considered some perturbation results for the generalized eigenvectors and the characteristic polynomial of unreduced upper hessenberg matrices are given two theorems are on the perturbation of invariant subspaces which are somewhat similar to the sin q theorems of davis and kahan apart from the residual which we do not have here dense perturbations of unreduced hessenberg matrices are also considered finally we prove an invariant subspace perturbation theorem for nonderogatory matrices

phase recovery maxcut and complex semidefinite programming phase retrieval seeks to recover a signal x in mathbb c p x c p from the amplitude a x a x of linear measurements ax in mathbb c n a x c n we cast the phase retrieval problem as a non convex quadratic program over a complex phase vector and formulate a tractable relaxation called phasecut similar to the classical maxcut semidefinite program we solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in gerchberg and saxton optik 35 237 246 1972 where each iteration is a matrix vector product numerical results show the performance of this approach over three different phase retrieval problems in comparison with greedy phase retrieval algorithms and matrix completion formulations

a deterministic two dimensional density taper approach for fast design of uniform amplitude pencil beams arrays the synthesis of electrically large aperiodic planar arrays with equi amplitude excitations plays an increasingly relevant role in satellite applications unfortunately for such a kind of synthesis problems local optimization procedures may be ineffective whereas global optimization procedures involve a severe computational effort to circumvent these problems a new deterministic approach for fast design of aperiodic concentric ring arrays is proposed the method exploits easily obtained optimal continuous planar solutions accounts for the geometric properties of the array element and allows once the central geometry of the array that is at least the most internal ring is fixed to compute in a deterministic iterative very fast way the whole geometry of the density tapered concentric ring array numerical examples show the effectiveness of the method which provides in a few seconds layouts of hundreds of elements able to produce directivity patterns that satisfy realistic satellite project requirements

designing unimodular sequence sets with good correlations including an application to mimo radar a multiple input multiple output mimo radar system that transmits orthogonal waveforms via its antennas can achieve a greatly increased virtual aperture compared with its phased array counterpart this increased virtual aperture enables many of the mimo radar advantages including enhanced parameter identifiability and improved resolution practical radar requirements such as unit peak to average power ratio and range compression dictate that we use mimo radar waveforms that have constant modulus and good auto and cross correlation properties we present in this paper new computationally efficient cyclic algorithms for mimo radar waveform synthesis these algorithms can be used for the design of unimodular mimo sequences that have very low auto and cross correlation sidelobes in a specified lag interval and of very long sequences that could hardly be handled by other algorithms previously suggested in the literature a number of examples are provided to demonstrate the performances of the new waveform synthesis algorithms

mask constrained power synthesis of linear arrays with even excitations an existing effective approach for the synthesis of linear arrays providing power patterns lying in arbitrary masks is extended to the case where an even distribution is required on element excitations the approach consists of a linear programming optimization plus a polynomial factorization and it allows one to deal with a considerably reduced computational burden with respect to the previous approaches by so doing it is now possible to solve problems that cannot be successfully tackled by the previous techniques an extension of the overall strategy to the synthesis of fields guaranteeing optimal performances in terms of directivity or ripple in the main beam region is also given

the eigenstructure of an arbitrary polynomial matrix computational aspects we give a new numerical method to compute the eigenstructure i e the zero structure the polar structure and the left and right null space structure of a polynomial matrix p these structural elements are of fundamental importance in several systems and control problems involving polynomial matrices the approach is more general than previous numerical methods because it can be applied to an arbitrary m n polynomial matrix p with normal rank r smaller than m and or n the algorithm is then shown to compute the structure of the left and right null spaces of p as well the speed and accuracy of this new approach are also discussed

krull gelfand kirillov and filter dimensions of simple affine algebras the aim of the paper is to prove the following two results 1 letabe a finitely partitive simple affine algebra with gk a then the krull dimension where gk and fil dim are the gelfand kirillov and the filter dimension respectively 2 letbbe an integral regular domain affine over a field of characteristic zero and let d b be its ring of differential operators the filter dimension fil dim d b 1

optimal synthesis of circularly symmetric shaped beams the problem of the optimal power pattern synthesis of circularly symmetric shaped beams is addressed and solved in particular an existence criterion is provided to test a priori i e without solving the overall synthesis problem if a non superdirective source of given size can radiate or not a circularly symmetric power pattern lying in a given arbitrary mask in the affirmative case the proposed approach guarantees the achievement of the optimal solution i e that of maximum directivity or minimum size by using convex optimization techniques in general a number of equivalent source distributions i e radiating the same power pattern do exist which can be explicitly determined in a deterministic way numerical examples concerning applications of actual interest support the given theory and confirm the effectiveness of the developed synthesis procedure

excitation retrieval of microwave linear arrays from phaseless far field data a methodology to recover the excitations of microwave linear arrays from the measurements of far field magnitude only is proposed the approach combines tools from convex optimization to solve the phase retrieval problem and a simple measurement procedure to mitigate the non uniqueness of the solution numerical simulations in various representative and realistic configurations of noise and measurement sampling are presented and discussed they show that it is possible to perfectly retrieve the array excitations from only the knowledge of far field magnitude by simply solving a convex optimization problem in addition the proposed approach that only calls for readily available routines is stable with respect to noise since the reconstruction performances degrades gracefully as the signal to noise ratio decreases

implicit double shift qr algorithm for companion matrices in this paper an implicit double shifted qr method for computing the eigenvalues of companion and fellow matrices will be presented companion and fellow matrices are hessenberg matrices that can be decomposed into the sum of a unitary and a rank 1 matrix the hessenberg the unitary as well as the rank 1 structures are preserved under a step of the qr method this makes these matrices suitable for the design of a fast qr method several techniques already exist for performing a qr step the implementation of these methods is highly dependent on the representation used unfortunately for most of the methods compression is needed since one is not able to maintain all three unitary hessenberg and rank 1 structures in this manuscript an implicit algorithm will be designed for performing a step of the qr method on the companion or fellow matrix based on a new representation consisting of givens transformations moreover no compression is needed as the specific representation of the involved matrices is maintained finally also a double shift version of the implicit method is presented

inverse power and durand kerner iterations for univariate polynomial root finding abstract univariate polynomial root finding is the oldest classical problem of mathematics and computational mathematics and is still an important research topic due to its impact on computational algebra and geometry the weierstrass durand kerner approach and its variations as well as matrix methods based on the qr algorithm are among the most popular practical choices for simultaneous approximation of all roots of a polynomial we propose an alternative application of the inverse power iteration to generalized companion matrices for polynomial root finding demonstrate its effectiveness and relate its study to unifying the derivation of the weierstrass durand kerner algorithm having quadratic convergence and its extensions having convergence rates 4 6 8 our experiments show substantial improvement versus the latter algorithm even though the inverse power iteration is most effective for the more limited tasks of approximating a single root or a few selected roots

min max elementwise backward error for roots of polynomials and a corresponding backward stable root finder abstract a new measure called min max elementwise backward error is introduced for approximate roots of scalar polynomials p z compared with the elementwise relative backward error this new measure allows for larger relative perturbations on the coefficients of p z that do not participate much in the overall backward error by how much these coefficients can be perturbed is determined via an associated max times polynomial and its tropical roots an algorithm is designed for computing the roots of p z it uses a companion linearization c z a z b of p z to which we added an extra zero leading coefficient and an appropriate two sided diagonal scaling that balances a and makes b graded in particular when there is variation in the magnitude of the coefficients of p z an implementation of the qz algorithm with a strict deflation criterion for eigenvalues at infinity is then used to obtain approximations to the roots of p z under the assumption that this implementation of the qz algorithm exhibits a graded backward error when b is graded we prove that our new algorithm is min max elementwise backward stable several numerical experiments show the superior performance of the new algorithm compared with the matlab roots function extending the algorithm to polynomial eigenvalue problems leads to a new polynomial eigensolver that exhibits excellent numerical behaviour compared with other existing polynomial eigensolvers as illustrated by many numerical tests

phaselift exact and stable signal recovery from magnitude measurements via convex programming suppose we wish to recover a signal input amssym font abc cmmib10 def bi 1 hbox abc 1 bi x in bbb c n from m intensity measurements of the form that is from data in which phase information is missing we prove that if the vectors are sampled independently and uniformly at random on the unit sphere then the signal x can be recovered exactly up to a global phase factor by solving a convenient semidefinite program a trace norm minimization problem this holds with large probability provided that m is on the order of and without any assumption about the signal whatsoever this novel result demonstrates that in some instances the combinatorial phase retrieval problem can be solved by convex programming techniques finally we also prove that our methodology is robust vis a vis additive noise 2012 wiley periodicals inc