a unifying convergence analysis of second order methods for secular equations existing numerical methods of second order are considered for a so called secular equation we give a brief description of the most important of these methods and show that all of them can be interpreted as improvements of newton s method for an equivalent problem for which newton s method exhibits convergence from any point in a given interval this interpretation unifies the convergence analysis of these methods provides convergence proofs where they were lacking and furnishes ways to construct improved methods in addition we show that some of these methods are in fact equivalent a second secular equation is also briefly considered

characterization of the critical submanifolds in quantum ensemble control landscapes the quantum control landscape is defined as the functional that maps the control variables to the expectation values of an observable over the ensemble of quantum systems analyzing the topology of such landscapes is important for understanding the origins of the increasing number of laboratory successes in the optimal control of quantum processes this paper proposes a simple scheme to compute the characteristics of the critical topology of the quantum ensemble control landscapes showing that the set of disjoint critical submanifolds one to one corresponds to a finite number of contingency tables that solely depend on the degeneracy structure of the eigenvalues of the initial system density matrix and the observable whose expectation value is to be maximized the landscape characteristics can be calculated as functions of the table entries including the dimensions and the numbers of positive and negative eigenvalues of the hessian quadratic form of each of the connected components of the critical submanifolds typical examples are given to illustrate the effectiveness of this method

quadratically constrained least squares and quadratic problems we consider the following problem compute a vectorx such that ax b 2 min subject to the constraint x 2 a new approach to this problem based on gauss quadrature is given the method is especially well suited when the dimensions ofa are large and the matrix is sparse it is also possible to extend this technique to a constrained quadratic form for a symmetric matrixa we consider the minimization ofx t a x 2b t x subject to the constraint x 2 some numerical examples are given

fast and stable qr eigenvalue algorithms for generalized companion matrices and secular equations we introduce a class of n n structured matrices which includes three well known classes of generalized companion matrices tridiagonal plus rank one matrices comrade matrices diagonal plus rank one matrices and arrowhead matrices relying on the structure properties of we show that if a then a rq where a qr is the qr decomposition of a this allows one to implement the qr iteration for computing the eigenvalues and the eigenvectors of any a with o n arithmetic operations per iteration and with o n memory storage this iteration applied to generalized companion matrices provides new o n2 flops algorithms for computing polynomial zeros and for solving the associated rational secular equations numerical experiments confirm the effectiveness and the robustness of our approach

principal lines on surfaces immersed with constant mean curvature configurations of lines of principal curvature on constant mean curva ture immersed surfaces are studied analytical models for these configurations near general isolated umbilical points and particular types of ends are found from the existence of transversal invariant measures for the foliations by principal lines established here follows that the union of recurrent lines of principal curvature is an open set examples illustrating all possible cases are given 1 introduction this paper is devoted to the study of the possible configurations of lines of principal curvature and umbilical points of surfaces immersed in r3 with constant mean curvature except in the case where the image of the immersion is contained in a plane or a sphere the umbilical points are isolated 10 11 a very precise analytical model is found for the family of principal lines near the isolated umbilical points this model is related to a result of h hopf 10 11 which permits the computation of the index of the umbilical point the behaviour of the principal lines at infinity near particular types of ends called here elementary ends of the immersion is also described through precise analytical models in connection with the global behaviour of the principal lines it is shown that their foliations have a natural smooth transversal invariant measure this implies that the compact principal lines cycles appear in cylindrical open sets and that the closures of recurrent principal lines have nonempty interior examples illustrating all possible situations are given however the recurrent lines have been found only for immer sions of tori into spaces which are not more than locally isometric to r3 the existence of such immersions into r3 seems to be a very difficult question related to a conjecture of h hopf 24 p 684 problem 63 2 from the point of view of principal lines an immersion with constant mean curvature presents a global configuration which is quite opposite to that of the principally structurally stable immersions studied in 4 5 for a large class of this kind of immersions it is shown that the principal cycles are isolated and do not present recurrent lines 4 5 examples of immersions of the sphere and torus into

a divide and conquer algorithm for the eigendecomposition of symmetric block diagonal plus semiseparable matrices we present a fast and numerically stable algorithm for computing the eigendecomposition of a symmetric block diagonal plus semiseparable matrix we report numerical experiments that indicate that our algorithm is significantly faster than the standard method which treats the given matrix as a general symmetric dense matrix

convergence of the shifted qr algorithm for unitary hessenberg matrices this paper shows that for unitary hessenberg matrices the qr algorithm with an exceptional initial value modification of the wilkinson shift gives global convergence moreover the asymptotic rate of convergence is at least cubic higher than that which can be shown to be quadratic only for hermitian tridiagonal matrices under no further assumption a general mixed shift strategy with global convergence and cubic rates is also presented

on tridiagonal matrices unitarily equivalent to normal matrices abstract in this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied it is well known that any matrix is unitarily equivalent to a tridiagonal matrix in case of a normal matrix the resulting tridiagonal inherits a strong relation between its super and subdiagonal elements the corresponding elements of the super and subdiagonal will have the same absolute value in this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied both an iterative reduction based on krylov sequences as a direct tridiagonalization procedure via householder transformations are reconsidered this equivalence transformation is then applied to the normal case and equality of the absolute value between the super and subdiagonals is proved self adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved properties when applying the reduction on symmetric skew symmetric hermitian skew hermitian and unitary matrices and their relations with e g complex symmetric and pseudo symmetric matrices are presented it is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the takagi factorization finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given

fast inversion algorithms for a class of structured operator matrices we study linear complexity inversion algorithms for diagonal plus semiseparable operator matrices applications to integral and differential equations and block matrices are obtained comparison of different algorithms is performed complexity choice of parameters and results of numerical experiments are analyzed

control landscapes for two level open quantum systems a quantum control landscape is defined as the physical objective as a function of the control variables in this paper the control landscapes for two level open quantum systems whose evolution is described by general completely positive trace preserving maps i e kraus maps are investigated in detail the objective function which is the expectation value of a target system operator is defined on the stiefel manifold representing the space of kraus maps three practically important properties of the objective function are found a the absence of local maxima or minima i e false traps b the existence of multi dimensional sub manifolds of optimal solutions corresponding to the global maximum and minimum and c the connectivity of each level set all of the critical values and their associated critical sub manifolds are explicitly found for any initial system state away from the absolute extrema there are no local maxima or minima and only saddles may exist whose number and the explicit structure of the corresponding critical sub manifolds are determined by the initial system state there are no saddles for pure initial states one saddle for a completely mixed initial state and two saddles for partially mixed initial states in general the landscape analysis of critical points and optimal manifolds is relevant to explain the relative ease of obtaining good optimal control outcomes in the laboratory even in the presence of the environment

iteration schemes for the divide and conquer eigenvalue solver we propose globally convergent iteration schemes for updating the eigenvalues of a symmetric matrix after a rank 1 modification such calculations are the core of the divide and conquer technique for the symmetric tridiagonal eigenvalue problem we prove the superlinear convergence right from the start of our schemes which allows us to improve the complexity bounds of 3 the effectiveness of our algorithms is confirmed by numerical results which are reported and discussed

a modification of the dewilde van der veen method for inversion of finite structured matrices abstract we study a class of block structured matrices r r ij i j 1 n with a property that the solution of the corresponding system rx y of linear algebraic equations may be performed for o n arithmetic operations in this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms these algorithms are related to those suggested by p m dewilde and a j van der veen time varying systems and computations kluwer academic publishers new york 1998 for a class of finite and infinite matrices with a small hankel rank the algorithms presented here are more transparent and are a modification of the algorithms from the above reference the approach and the proofs are essentially different from those in the above mentioned reference the paper contains also analysis of complexity and results of numerical experiments

inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices abstract matrices represented as a sum of diagonal and semiseparable ones are considered here these matrices belong to the class of structured matrices which arises in numerous applications fast o n algorithms for their inversion were developed before under additional restrictions which were a source of instability our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms in this paper the case of semiseparable matrices of order one is considered

volume fractions of the kinematic near critical sets of the quantum ensemble control landscape an estimate is derived for the volume fraction of a subset c p u grad j u u n in the neighborhood of the critical set c pu n pu m of the kinematic quantum ensemble control landscape j u tr u u o where u represents the unitary time evolution operator is the initial density matrix of the ensemble and o is an observable operator this estimate is based on the hilbert schmidt geometry for the unitary group and a first order approximation ofgrad j u 2 an upper bound on these near critical volumes is conjectured and supported by numerical simulation leading to an asymptotic analysis as the dimension n of the quantum system rises in which the volume fractions of these near critical sets decrease to zero as n increases this result helps explain the apparent lack of influence exerted by the many saddles of j over the gradient flow

analysis of effectiveness of lyapunov control for non generic quantum states a lyapunov based control design for natural trajectory tracking problems is analyzed for quantum states where the analysis in the generic case is not applicable using dynamical systems tools we show almost global asymptotic stability for stationary target states subject to certain conditions on the hamiltonians and discuss effectiveness of the design when these conditions are not satisfied for pseudo pure target states the effectiveness of the design is studied further for both stationary and non stationary states using alternative tools

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

numerical solution of a secular equation a method is proposed for the solution of a secular equation arising in modified symmetric eigenvalue problems and in several other areas this equation has singularities which make the application of standard root finding methods difficult in order to solve the equation a class of transformations of variables is considered which transform the equation into one for which newton s method converges from any point in a certain given interval in addition the form of the transformed equation suggests a convergence accelerating modification of newton s method the same ideas are applied to the secant method and numerical results are presented

on computing the eigenvectors of a class of structured matrices a real symmetric matrix of order n has a full set of orthogonal eigenvectors the most used approach to compute the spectrum of such matrices reduces first the dense symmetric matrix into a symmetric structured one i e tridiagonal matrices or semiseparable matrices this step is accomplished in o n 3 operations once the latter symmetric structured matrix is available its spectrum is computed in an iterative fashion by means of the qr method in o n 2 operations in principle the whole set of eigenvectors of the latter structured matrix can be computed by means of inverse iteration in o n 2 operations the blemish in this approach is that the computed eigenvectors may not be numerically orthogonal if clusters are present in the spectrum to enforce orthogonality the gram schmidt procedure is used requiring o n 3 operations in the worst case in this paper a fast and stable method to compute the eigenvectors of tridiagonal and semiseparable matrices which does not suffer from the loss of orthogonality due to the presence of clusters in the spectrum is presented the algorithm requires o n 2 floating point operations if clusters of small size are present in the spectrum

structural and computational properties of possibly singular semiseparable matrices a classical result of structured numerical linear algebra states that the inverse of a nonsingular semiseparable matrix is a tridiagonal matrix such a property of a semiseparable matrix has been proved to be useful for devising linear complexity solvers for establishing recurrence relations among its columns or rows and moreover for efficiently evaluating its characteristic polynomial in this paper we provide sparse structured representations of a semiseparable matrix a which hold independently of the fact that a is singular or not these relations are found by pointing out the band structure of the inverse of the sum of a plus a certain sparse perturbation of minimal rank further they can be used to determine in a computationally efficient way both a reflexive generalized inverse of a and its characteristic polynomial

fast and stable eigendecomposition of symmetric banded plus semi separable matrices abstract a new fast and stable algorithm to reduce a symmetric banded plus semi separable matrix to tridiagonal form via orthogonal similarity transformations is presented

transformations to rank structures by unitary similarity we study transformations by unitary similarity of a nonderogatory matrix to certain rank structured matrices this kind of transformations can be described in a unified framework that involves krylov matrices the rank structures here addressed are preserved by qr iterations and every iterate can be associated with a suitable krylov matrix

rank structures preserved by the qr algorithm the singular case in an earlier paper we introduced the classes of polynomial and rank structures both of them preserved by applying a shifted qr step on a matrix a in the present paper we further investigate the case of rank structures we show that even if a is a singular matrix a new qr iterate can be constructed having the same rank structure as the matrix a itself to this end we introduce the concepts of effectively eliminating qr decompositions and sparse givens patterns both of them being of independent interest

control landscapes for open system quantum operations the reliable realization of control operations is a key component of quantum information applications in practice meeting this goal is very demanding for open quantum systems this paper investigates the landscape defined as the fidelity j between the desired and achieved quantum operations with an open system the goal is to maximize j as a functional of the control variables we specify the complete set of critical points of the landscape function in the so called kinematic picture an associated hessian analysis of the landscape reveals that upon the satisfaction of a particular controllability criterion the critical topology is dependent on the particular environment but no false traps i e suboptimal solutions exist thus a gradient type search algorithm should not be hindered in searching for the ultimate optimal solution with such controllable systems moreover the maximal fidelity is proven to coincide with uhlmann s fidelity between the environmental initial states associated with the achieved and desired quantum operations which provides a generalization of uhlmann s theorem in terms of kraus maps

convergence of the unitary qr algorithm with a unimodular wilkinson shift in applying the qr algorithm to compute the eigenvalues of a unitary hessenberg matrix a projected wilkinson shift of unit modulus is proposed and proved to give global convergence with at least a quadratic asymptotic rate for the qr iteration experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence

the cal d q r algorithm basic theory convergence and conditional stability we describe a fast matrix eigenvalue algorithm that uses a matrix factorization and reverse order multiply technique involving three factors and that is based on the symmetric matrix factorization as well as on d orthogonal reduction techniques where d mbox diag pm1 is computed from the given matrix a it operates on a similarity reduction of a real matrix a nn to general tridiagonal form t and computes all of a s eigenvalues in frac 4 3 n 3 o n 2 operations where the o n 2 part of the operations is possibly performed over bbb c instead of the 7 8 n 3 real flops required by the qr eigenvalue algorithm potential breakdo wn of the dqr algorithm can occur in the reduction to tridiagonal form and in the d orthogonal dqr reductions both however can be monitored during the computations the former occurs rather rarely for dimensions n leq 300 and can essentially be bypassed while the latter is extremely rare and can be bypassed as well in our conditionally stable implementation of the dqr steps we prove an implicit dq theorem which allows implicit shifts give a convergence proof for the dqr algorithm and show that dqr is conditionally stable for general balanced tridiagonal matrices t rm b

a qr method for computing the singular values via semiseparable matrices the standard procedure to compute the singular value decomposition of a dense matrix first reduces it into a bidiagonal one by means of orthogonal transformations once the bidiagonal matrix has been computed the qr method is applied to reduce the latter matrix into a diagonal one in this paper we propose a new method for computing the singular value decomposition of a real matrix in a first phase an algorithm for reducing the matrix a into an upper triangular semiseparable matrix by means of orthogonal transformations is described a remarkable feature of this phase is that depending on the distribution of the singular values after few steps of the reduction the largest singular values are already computed with a precision depending on the gaps between the singular values an implicit qr method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values the numerical tests show that the proposed method can compete with the standard method using an intermediate bidiagonal matrix for computing the singular values of a matrix

structures preserved by the qr algorithm in this paper we investigate some classes of structures that are preserved by applying a shifted qr step on a matrix a we will handle two classes of such structures the first we call polynomial structures for example a matrix being hermitian or hermitian up to a rank one correction and the second we call rank structures which are encountered for example in all kinds of what we could call hessenberg like and lower semiseparable like matrices an advantage of our approach is that we define a structure by decomposing it as a collection of building stones which we call structure blocks this allows us to state the results in their natural most general context

control landscapes for observable preparation with open quantum systems a quantum control landscape is defined as the observable as a function al of the system control variables such landscapes were introduced to provide a basis to understand the increasing number of successful experiments controlling quantum dynamics phenomena this paper extends the concept to encompass the broader context of the environment having an influence for the case that the open system dynamics are fully controllable it is shown that the control landscape for open systems can be lifted to the analysis of an equivalent auxiliary landscape of a closed composite system that contains the environmental interactions this inherent connection can be analyzed to provide relevant information about the topology of the original open system landscape application to the optimization of an observable expectation value reveals the same landscape simplicity observed in former studies on closed systems in particular no false suboptimal traps exist in the system control landscape when seeking to optimize an observable even in the presence of complex environments moreover a quantitative study of the control landscape of a system interacting with a thermal environment shows that the enhanced controllability attainable with open dynamics significantly broadens the range of the achievable observable values over the control landscape

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

measuring the distance from saddle points and driving to locate them over quantum control landscapes optimal control of quantum phenomena involves the introduction of a cost functional j to characterize the degree of achieving a physical objective by a chosen shaped electromagnetic field the cost functional dependence upon the control forms a control landscape two theoretically important canonical cases are the landscapes associated with seeking to achieve either a physical observable or a unitary transformation upon satisfaction of particular assumptions both landscapes are analytically known to be trap free yet possess saddle points at precise suboptimal j values the presence of saddles on the landscapes can influence the effort needed to find an optimal field as a foundation to future algorithm development and analyzes we define metrics that identify the distance from a given saddle based on the sufficient and necessary conditions for the existence of the saddles algorithms are introduced utilizing the metrics to find a control such that the dynamics arrive at a targeted saddle the saddle distance metric and saddle seeking methodology is tested numerically in several model systems

analysis of lyapunov method for control of quantum states the natural trajectory tracking problem is studied for generic quantum states represented by density operators a control design based on the hilbert schmidt distance as a lyapunov function is considered the control dynamics is redefined on an extended space where the lasalle invariance principle can be correctly applied even for non stationary target states lasalle s invariance principle is used to derive a general characterization of the invariant set which is shown to always contain the critical points of the lyapunov function critical point analysis of the latter is used to show that for generic states it is a morse function with n isolated critical points including one global minimum one global maximum and n 2 saddles it is also shown however that the actual dynamics of the system is not a gradient flow and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points this analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular and in fact convergence to the target state trajectory in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space on the other hand if either of these sufficient conditions is not satisfied the target state ceases to be asymptotically stable a center manifold emerges around the target state and the control design ceases to be effective

random orthogonal matrix simulation this paper introduces a method for simulating multivariate samples that have exact means covariances skewness and kurtosis we introduce a new class of rectangular orthogonal matrix which is fundamental to the methodology and we call these matrices l matrices they may be deterministic parametric or data specific in nature the target moments determine the l matrix then infinitely many random samples with the same exact moments may be generated by multiplying the l matrix by arbitrary random orthogonal matrices this methodology is thus termed rom simulation considering certain elementary types of random orthogonal matrices we demonstrate that they generate samples with different characteristics rom simulation has applications to many problems that are resolved using standard monte carlo methods but no parametric assumptions are required unless parametric l matrices are used so there is no sampling error caused by the discrete approximation of a continuous distribution which is a major source of error in standard monte carlo simulations for illustration we apply rom simulation to determine the value at risk of a stock portfolio

a constrained eigenvalue problem in this paper we consider the following mathematical and computational problem given the quantities a n m by n m matrix symmetric n 0 n n m by m matrix with full rank t vector of dimension m with nt t 1 determine an x such that operatorname x t ax min subject to the constraints n t x t i x t x 1 ii variants of this problem occur in many applications 1 5 7 8 11 the problem has been studied previously when t 0 the null vector cf 4 6 when t 0 then the problem becomes more complicated

characterization of the critical sets of quantum unitary control landscapes this work considers various families of quantum control landscapes i e objective functions for optimal control for obtaining target unitary transformations as the general solution of the controlled schrodinger equation we examine the critical point structure of the kinematic landscapes j f u u w a 2 and j p u a 4 tr aa w u 2 defined on the unitary group u h of a finite dimensional hilbert space h the parameter operator a e h is allowed to be completely arbitrary yielding an objective function that measures the difference in the actions of u and the target w on a subspace of state space namely the column space of a the analysis of this function includes a description of the structure of the critical sets of these kinematic landscapes and characterization of the critical points as maxima minima and saddles in addition we consider the question of whether these landscapes are morse bott functions on u h landscapes based on the intrinsic geodesic distance on u h and the projective unitary group pu h are also considered these results are then used to deduce properties of the critical set of the corresponding dynamical landscapes

a composite polynomial zerofinding matrix algorithm abstract a globally convergent matrix algorithm is presented for finding the real and complex zeros of a complex polynomial p x a combination of schmeisser s and fiedler s matrices is used in the algorithm first the greatest common divisor g x of p x and its derivative p x is obtained and used to reduce p x to a polynomial q x p x q x with simple zeros second the zeros of q x are computed either by finding the eigenvalues of the first block of schmeisser s matrix or by applying fiedler s algorithm recursively in either case the eigenvalues are obtained by the qr algorithm third the multiplicity of each zero of p x is calculated by means of lagouanelle s modified limiting formula sample numerical results are given to demonstrate the effectiveness of the algorithm

a unitary hessenberg qr based algorithm via semiseparable matrices in this paper we present a novel method for solving the unitary hessenberg eigenvalue problem in the first phase an algorithm is designed to transform the unitary matrix into a diagonal plus semiseparable form then we rely on our earlier adaptation of the qr algorithm to solve the dpss eigenvalue problem in a fast and robust way exploiting the structure of the problem enables us to yield a quadratic time using a linear memory space nonetheless the algorithm remains robust and converges as fast as the customary qr algorithm numerical experiments confirm the effectiveness and the robustness of our approach

a levinson like algorithm for symmetric strongly nonsingular higher order semiseparable plus band matrices in this paper we will derive a solver for a symmetric strongly nonsingular higher order generator representable semiseparable plus band matrix the solver we will derive is based on the levinson algorithm which is used for solving strongly nonsingular toeplitz systems in the first part an o p2n solver for a semiseparable matrix of semiseparability rank p is derived and in a second part we derive an o l2n solver for a band matrix with bandwidth 2l 1 both solvers are constructed in a similar way firstly a yule walker like equation needs to be solved and secondly this solution is used for solving a linear equation with an arbitrary right hand side finally a combination of the above methods is presented to solve linear systems with semiseparable plus band coefficient matrices the overall complexity of this solver is 6 l p 2n plus lower order terms in the final section numerical experiments are performed attention is paid to the timing and the accuracy of the described methods

on estimating the condition of eigenvalues and eigenvectors abstract a method is developed for estimating the accuracy of computed eigenvalues and eigenvectors that are obtained via certain eispack subroutines it does this at a cost of o n 2 flops per eigenpair assuming that the eigenpair is known and assuming that the original matrix has been reduced to hessenberg form the heart of the technique involves estimating the smallest singular value of a certain nearly triangular submatrix this is accomplished by some standard zero chasing with givens transformations and with a 2 norm version of the linpack condition estimator an eispack compatible code has been developed and its performance is discussed

expressing a polynomial as the characteristic polynomial of a symmetric matrix abstract we show that given a polynomial one can without knowing the roots construct a symmetric matrix whose characteristics polynomial is the given polynomial appropriately normed for a real polynomial the matrix can have purely imaginary off diagonal entries for a polynomial with only real distinct roots the matrix can be chosen real

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

dynamic homotopy and landscape dynamical set topology in quantum control we examine the topology of the subset of controls taking a given initial state to a given final state in quantum control where state may mean a pure state an ensemble density matrix or a unitary propagator u 0 t the analysis consists in showing that the endpoint map acting on control space is a hurewicz fibration for a large class of affine control systems with vector controls exploiting the resulting fibration sequence and the long exact sequence of basepoint preserving homotopy classes of maps we show that the indicated subset of controls is homotopy equivalent to the loopspace of the state manifold this not only allows us to understand the connectedness of dynamical sets realized as preimages of subsets of the state space through this endpoint map but also provides a wealth of additional topological information about such subsets of control space

the restarted qr algorithm for eigenvalue computation of structured matrices the qr algorithm is a popular numerical method for the computation of eigenvalues of matrices all eigenvalues of a general n n upper hessenberg matrix typically can be computed in o n3 arithmetic floating point operations using o n2 storage locations when the upper hessenberg matrix is hermitian or unitary then it can be represented by o n parameters and there are variants of the qr algorithm that reduce the operation count for computing all eigenvalues to o n2 arithmetic floating point operations and the storage requirement to o n locations however for many structured matrices that can be represented with o n storage locations available implementations of the qr algorithm require o n3 arithmetic floating point operations and o n2 storage locations to determine all eigenvalues this note shows that for some of the latter matrices the operation count can be reduced to o n2 arithmetic floating point operations and the memory requirement to o n storage locations by periodically restarting the qr algorithm