backward errors for the inverse eigenvalue problem backward errors for the symmetric matrix inverse eigenvalue problem with respect to an approximate solution are defined and explicit expressions of the backward errors are derived the expressions may be useful for testing the stability of practical algorithms

five six and seven term karatsuba like formulae the karatsuba ofman algorithm starts with a way to multiply two 2 term i e linear polynomials using three scalar multiplications there is also a way to multiply two 3 term i e quadratic polynomials using six scalar multiplications these are used within recursive constructions to multiply two higher degree polynomials in subquadratic time we present division free formulae which multiply two 5 term polynomials with 13 scalar multiplications two 6 term polynomials with 17 scalar multiplications and two 7 term polynomials with 22 scalar multiplications these formulae may be mixed with the 2 term and 3 term formulae within recursive constructions leading to improved bounds for many other degrees using only the 6 term formula leads to better asymptotic performance than standard karatsuba the new formulae work in any characteristic but simplify in characteristic 2 we describe their application to elliptic curve arithmetic over binary fields we include some timing data

a note on the backward errors for hermite eigenvalue problems we discuss the backward error analysis of the hermite eigenproblems three kinds of problems are included generalized eigenproblem matrix polynomial eigenproblem and multiparameter eigenproblem in the case of the approximate eigenvalue being real these problems have already been solved by higham and higham siam j matrix anal 20 1998 493 tisseur lin alg appl 309 2002 339 and hachstenbach and plestenjak lin alg appl 375 2003 63 this note as a complement of their results considers the complex case

brief constrained linear state estimation a moving horizon approach this article considers moving horizon strategies for constrained linear state estimation additional information for estimating state variables from output measurements is often available in the form of inequality constraints on states noise and other variables formulating a linear state estimation problem with inequality constraints however prevents recursive solutions such as kalman filtering and consequently the estimation problem grows with time as more measurements become available to bound the problem size we explore moving horizon strategies for constrained linear state estimation in this work we discuss some practical and theoretical properties of moving horizon estimation we derive sufficient conditions for the stability of moving horizon state estimation with linear models subject to constraints on the estimate we also discuss smoothing strategies for moving horizon estimation our framework is solely deterministic

on eigenvalue variations of rayleigh quotient matrix pencils of a definite pencil abstract let a b be a definite matrix pencil of order n i e both a and b are n n hermitian and c a b def min xe c n x 2 1 x h a ib x 0 suppose y is an n l matrix with full column rank whose column vectors span an approximate invariant subspace for a b this note investigates the relation between eigenvalues of a b and those of y h ay y h by our result for the spectral norm improves sun s linear algebra appl 139 253 267 1990 we also present bounds in frobenius norms and in general unitarily invariant norms

on interval estimates of perturbations of generalized eigenvalues for diagonalizable pairs abstract on generalized eigenvalue perturbation bounds for diagonalizable matrix pairs upper bounds are discussed all the time in this paper we mainly consider not only upper bounds but also lower bounds of perturbation of generalized eigenvalues for diagonalizable matrix pairs sharper upper bounds and sharp lower bounds on perturbation of generalized eigenvalues for diagonalizable matrix pairs are obtained in terms of the distances of two points in grassmann manifold the main results improve the existing results of chen 2007 1 and numerical examples illustrate the efficiency of the theoretical results

new perturbation bounds of partitioned generalized hermitian eigenvalue problem abstract in this paper some new absolute perturbation bounds of partitioned generalized hermitian positive definite eigenvalue problem are established by two different ways numerical results show that our bounds are sharper than the ones in the literature

receding horizon estimation for discrete time linear systems the problem of estimating the state of a discrete time linear system can be addressed by minimizing an estimation cost function dependent on a batch of recent measure and input vectors this problem has been solved by introducing a receding horizon objective function that includes also a weighted penalty term related to the prediction of the state for such an estimator convergence results and unbiasedness properties have been proved the issues concerning the design of this filter are discussed in terms of the choice of the free parameters in the cost function the performance of the proposed receding horizon filter is evaluated and compared with other techniques by means of a numerical example

a numerically robust state space approach to stable predictive control strategies recent work with predictive control strategies has shown that the stable gpc sgpc approach has significant computational and numerical advantages however sgpc is cast in the transfer function framework which limits its application here we develop a means of extending the results to state space models and show that improved numerical conditioning can be obtained for many stable predictive control strategies

observer design for nonlinear systems with discrete time measurements this paper focuses on the development of asymptotic observers for nonlinear discrete time systems it is argued that instead of trying to imitate the linear observer theory the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations in particular it is shown that the discrete newton method properly interpreted yields an asymptotic observer for a large class of discrete time systems while the continuous newton method may be employed to obtain a global observer furthermore it is analyzed how the use of broyden s method in the observer structure affects the observer s performance and its computational complexity an example illustrates some aspects of the proposed methods moreover it serves to show that these methods apply equally well to discrete time systems and to continuous time systems with sampled outputs

on estimation error bounds for receding horizon filters using quadratic boundedness quadratic boundedness is used to deal with stability and design of receding horizon estimators upper bounds on the norm of the estimation error have been found by means of invariant sets that can be constructed by using quadratic boundedness moreover these bounds are expressed in terms of linear matrix inequalities and are well suited to being minimized for the purpose of design

on some perturbation inequalities for operators abstract an estimate for the norm of the solution to the equation ax xb s obtained by r bhatia c davis and a mcintosh for normal operators a and b is shown to be valid for a larger class some other inequalities in the same spirit are obtained including a sin theorem for singular vectors some inequalities concerning the continuity of the map a ita obtained recently by kittaneh and kosaki are extended using these ideas

on the local convergence of an iterative approach for inverse singular value problems the purpose of this paper is to provide the convergence theory for the iterative approach given by m t chu numerical methods for inverse singular value problems siam j numer anal 29 1992 pp 885 903 in the context of solving inverse singular value problems we provide a detailed convergence analysis and show that the ultimate rate of convergence is quadratic in the root sense numerical results which confirm our theory are presented it is still an open issue to prove that the method is q quadratic convergent as claimed by m t chu

a note on stewart s theorem for definite matrix pairs abstract let a and b be n n hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b inf x 1 h a ib x 0 in this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by g w stewart 2 moreover we prove that this upper bound is a projective metric in the set of n n definite pairs

linear constrained moving horizon estimator with pre estimating observer abstract in this paper a constrained moving horizon estimation mhe strategy for linear systems is proposed recently the use of a pre estimating linear observer in the forward prediction equations in the mhe cost function has been proposed in order to reduce the effects of uncertainty here we introduce state constraints within this formulation and investigate stability properties in the presence of bounded disturbances and noise the robustness and performance of the proposed observer is demonstrated with a simulation example

efficient hardware implementation of fp arithmetic for pairing friendly curves this paper describes a new method to speed up ifp arithmetic in hardware for pairing friendly curves such as the well known barreto naehrig bn curves we explore the characteristics of the modulus defined by these curves and choose curve parameters such that ifp multiplication becomes more efficient the proposed algorithm uses montgomery reduction in a polynomial ring combined with a coefficient reduction phase using a pseudo mersenne number as an application we show that the performance of pairings on bn curves in hardware can be significantly improved resulting in a factor 2 5 speedup compared with state of the art hardware implementations

optimal pairings in this paper we introduce the concept of an optimal pairing which by definition can be computed using only log 2 r k basic miller iterations with r the order of the groups involved and k the embedding degree we describe an algorithm to construct optimal ate pairings on all parametrized families of pairing friendly elliptic curves finally we conjecture that any nondegenerate pairing on an elliptic curve without efficiently computable endomorphisms different from powers of frobenius requires at least log 2 r k basic miller iterations

numerical methods for inverse singular value problems3 two numerical methods one continuous and the other discrete are proposed for solving inverse singular value problems the first method consists of solving an ordinary differential equation obtained from an explicit calculation of the projected gradient of a certain objective function the second method generalizes an iterative process proposed originally by friedland nocedal and overton siam j numer anal 24 1987 pp 634 667 for solving inverse eigenvalue problems with the geometry understood from the first method it is shown that the second method also the method proposed by friedland nocedal and overton for inverse eigenvalue problems is a variation of the newton method while the continuous method is expected to converge globally at a slower rate in finding a stationary point of the objective function the discrete method is proved to converge locally at a quadratic rate if there is a solution some numerical examples are presented

on perturbations of matrix pencils with real spectra a revisit this paper continues earlier studies by bhatia and li on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectra a unifying framework for creating crucial perturbation equations is developed with the help of a recent result on generalized commutators involving unitary matrices new and much sharper bounds are obtained

a perturbation bound for definite pencils we attempt to generalize a well known result on spectral variations of a hermitian matrix due to mirsky to the definite generalized eigenvalue problem we also point out that some results on perturbations of definite pencils due to stewart can be slightly improved

a backward error for the inverse singular value problem a backward error for inverse singular value problems with respect to an approximate solution is defined and an explicit expression for the backward error is derived by extending the approach described in j g sun backward errors for the inverse eigenvalue problem numer math 82 1999 339 349 the expression may be useful for testing the stability of practical algorithms

generalized eigenvalues of a definite hermitian matrix pair abstract in this note we study some basic properties of generalized eigenvalues of a definite hermitian matrix pair in particular we prove an interlacing theorem and a minimax theorem we also obtain upper bounds for the variation of the generalized eigenvalues under perturbation these results extend and improve those of r c li j g sun and g w stewart on the topic

an algorithm for symmetric generalized inverse eigenvalue problems abstract using qr like decomposition with column pivoting and least squares techniques we propose a new and efficient algorithm for solving symmetric generalized inverse eigenvalue problems and give its locally quadratic convergence analysis we also present some numerical experiments which illustrate the behaviour of our algorithm

on perturbations of matrix pencils with real spectra perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil a ab with real spectrum are developed it is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices the applications of these bounds to the spectral variations of definite pencils are conducted in such a way that extra attention is paid to their peculiarities so as to derive more sophisticated perturbation bounds our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of hermitian matrices such as the weyl lidskii theorem and the hoffman wielandt theorem and those for generalized eigenspaces are counterparts of the celebrated davis kahan sin 0 sin 20 theorems for the eigenspace variations of hermitian matrices the paper consists of two parts part i is for generalized eigenvalue perturbations while part ii deals with generalized eigenspace perturbations

a concept of local observability abstract a notion of local observability which is natural in the context of nonlinear input output regulation is introduced a simple characterization is provided a comparison is made with other local nonlinear observability definitions and its behavior under constant rate sampling is analyzed

optimization based non linear observers revisited optimization based non linear observers are those where the integrated output prediction error is used to define the dynamic of the observer s state their main advantage is genericity and the natural assumptions that are needed for the related convergence results to hold here we present some new theoretical results concerning these observers despite the title we use our observer is not based on the search for a global minimum of a cost function it is based on a descent like approach without the use of the hessian matrix a nice feature is that the resulting observer takes a classical form of in the sense that it is defined by a set of ordinary differential equations besides the theoretical results some implementation issues are discussed in particular we propose to use the post stabilization approach well known when dealing with differential systems with invariant submanifolds this enables to increase the sampling period while keeping very good precision two illustrative examples are presente

some inequalities for norm ideals several inequalities for norms of operators are extended to more operators and or to more norms these include results of halmos and bouldin on approximating a normal operator by another with restricted spectrum the powers stormer and the van hemmen ando inequalities for the distance between the square roots of two positive operators and also some recent generalisations of these latter results by kittaneh

finding a positive definite linear combination of two hermitian matrices abstract let a and b be n by n hermitian matrices over the complex field a result of au yeung 1 and stewart 8 states that if x a ib x 0 for all nonzero n vectors x then there is a linear combination of a and b which is positive definite in this article we present an algorithm which finds such a linear combination in a finite number of steps we also discuss the implementation of the algorithm in case a and b are real symmetric sparse matrices

absolute and relative weyl theorems for generalized eigenvalue problems abstract weyl type eigenvalue perturbation theories are derived for hermitian definite pencils a b in which b is positive definite the results provide a one to one correspondence between the original and perturbed eigenvalues and give a uniform perturbation bound we give both absolute and relative perturbation results defined in the standard euclidean metric instead of the chordal metric that is often used

the perturbation bounds for eigenspaces of a definite matrix pair leta andb be hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b equiv mathop min limits left x right 1 x h a ib x 0 the perturbation bounds for eigenspaces of a definite pair on every unitary invariant matrix norm were obtained by imposing additional restrictions on the location of the generalized eigenvalues thus it gives a positive answer for an open question proposed by stewart 7 the famous davis kahan sin theorems and sin 2 theorem 2 can also be deduced from the present results

the nearest definite pair for the hermitian generalized eigenvalue problem the generalized eigenvalue problem ax lambda bx has special properties when a b is a hermitian and definite pair given a general hermitian pair a b it is of interest to find the nearest definite pair having a specified crawford number delta 0 we solve the problem in terms of the inner numerical radius associated with the field of values of a ib we show that once the problem has been solved it is trivial to rotate the perturbed pair a da b db to a pair widetilde a widetilde b for which lambda min widetilde b achieves its maximum value delta which is a numerically desirable property when solving the eigenvalue problem by methods that convert to a standard eigenvalue problem by inverting b numerical examples are given to illustrate the analysis

constrained state estimation for nonlinear discrete time systems stability and moving horizon approximations state estimator design for a nonlinear discrete time system is a challenging problem further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances one strategy for constrained state estimation is to employ online optimization using a moving horizon approximation we propose a general theory for constrained moving horizon estimation sufficient conditions for asymptotic and bounded stability are established we apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation examples are used to illustrate the benefits of constrained state estimation our framework is deterministic

a backward error for the symmetric generalized inverse eigenvalue problem abstract in this paper we discuss the backward error for the symmetric generalized inverse eigenvalue problem which extends the result of sun 1999 10 the optimal backward error is defined for symmetric generalized inverse eigenvalue problem with respect to an approximate solution and the upper and lower bounds are derived for the optimal backward error the expressions may be useful for testing the stability of practical algorithms

stochastic h infty we consider stochastic linear plants which are controlled by dynamic output feedback and subjected to both deterministic and stochastic perturbations our objective is to develop an h infty type theory for such systems we prove a bounded real lemma for stochastic systems with deterministic and stochastic perturbations this enables us to obtain necessary and sufficient conditions for the existence of a stabilizing compensator which keeps the effect of the perturbations on the to be controlled output below a given threshhold gamma 0 in the deterministic case the analogous conditions involve two uncoupled linear matrix inequalities but in the stochastic setting we obtain coupled nonlinear matrix inequalities instead the connection between h infty theory and stability radii is discussed and leads to a lower bound for the radii which is shown to be tight in some special cases

low weight polynomial form integers for efficient modular multiplication in 1999 solinas introduced families of moduli called the generalized mersenne numbers gmns which are expressed in low weight polynomial form p f t where t is limited to a power of 2 gmns are very useful in elliptic curve cryptosystems over prime fields since modular reduction by a gmn requires only integer additions and subtractions however since there are not many gmns and each gmn requires a dedicated implementation gmns are hardly useful for other cryptosystems here we modify gmn by removing restriction on the choice of t and restricting the coefficients of f t to 0 and plusmn1 we call such families of moduli low weight polynomial form integers lwpfis we show an efficient modular multiplication method using lwpfi moduli lwpfis allow general implementation and there exist many lwpfi moduli one may consider lwpfis as a trade off between general integers and gmns

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

moving horizon estimation for large scale interconnected systems we present computationally efficient centralized and distributed moving horizon estimation mhe methods for large scale interconnected systems that are described by sparse banded or sparse multibanded system matrices both of these mhe methods are developed by approximating a solution of the mhe problem using the chebyshev approximation method by exploiting the sparsity of this approximate solution we derive a centralized mhe method which computational complexity and storage requirements scale linearly with the number of local subsystems of an interconnected system furthermore on the basis of the approximate solution of the mhe problem we develop a novel distributed mhe method this distributed mhe method estimates the state of a local subsystem using only local input output data in contrast to the existing distributed algorithms for the state estimation of large scale systems the proposed distributed mhe method is not relying on the consensus algorithms and has a simple analytic form we have studied the stability of the proposed mhe methods and we have performed numerical simulations that confirm our theoretical results

moving horizon state estimation for nonlinear discrete time systems new stability results and approximation schemes a moving horizon state estimation problem is addressed for a class of nonlinear discrete time systems with bounded noises acting on the system and measurement equations as the statistics of such disturbances and of the initial state are assumed to be unknown we use a generalized least squares approach that consists in minimizing a quadratic estimation cost function defined on a recent batch of inputs and outputs according to a sliding window strategy for the resulting estimator the existence of bounding sequences on the estimation error is proved in the absence of noises exponential convergence to zero is obtained moreover suboptimal solutions are sought for which a certain error is admitted with respect to the optimal cost value the approximate solution can be determined either on line by directly minimizing the cost function or off line by using a nonlinear parameterized function simulation results are presented to show the effectiveness of the proposed approach in comparison with the extended kalman filter

a neural state estimator with bounded errors for nonlinear systems a neural state estimator is described acting on discrete time nonlinear systems with noisy measurement channels a sliding window quadratic estimation cost function is considered and the measurement noise is assumed to be additive no probabilistic assumptions are made on the measurement noise nor on the initial state novel theoretical convergence results are developed for the error bounds of both the optimal and the neural approximate estimators to ensure the convergence properties of the neural estimator a minimax tuning technique is used the approximate estimator can be designed offline in such a way as to enable it to process on line any possible measure pattern almost instantly

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

the eta pairing revisited in this paper we simplify and extend the eta pairing originally discovered in the setting of supersingular curves by barreto to ordinary curves furthermore we show that by swapping the arguments of the eta pairing one obtains a very efficient algorithm resulting in a speed up of a factor of around six over the usual tate pairing in the case of curves that have large security parameters complex multiplication by an order of qopf radic 3 and when the trace of frobenius is chosen to be suitably small other more minor savings are obtained for more general curves

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

a computer algorithm for calculating the product ab modulo m it is possible to find the smallest nonnegative integer r congruent modulo m to the product ab of two nonnegative integers without dividing by m in multiple precision arithmetic doing away with the division cuts the calculation time by varying amounts depending on machine architecture it also cuts storage space

on perturbations of matrix pencils with real spectra ii a well known result on spectral variation of a hermitian matrix due to mirsky is the following let a and a be two n x n hermitian matrices and let 1 n and 1 n be their eigenvalues arranged in ascending order then diag 1 1 n n a a for any unitarily invariant norm in this paper we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra the much studied case of definite pencils is included in this

a quadratically convergent algorithm based on matrix equations for inverse eigenvalue problems abstract we propose a quadratically convergent algorithm for inverse symmetric eigenvalue problems based on matrix equations the basic idea is seen in a recent study by ogita and aishima while they derive an efficient iterative refinement algorithm for symmetric eigenvalue problems using special matrix equations in other words this study is interpreted as a unified view on quadratically convergent algorithms for eigenvalue problems and inverse eigenvalue problems based on matrix equations to the best of our knowledge such a unified development of algorithms is provided for the first time since the proposed algorithm for the inverse eigenvalue problems can be regarded as the newton s method for the matrix equations the quadratic convergence is naturally proved our algorithm is interpreted as an improved version of the cayley transform method for the inverse eigenvalue problems although the cayley transform method is one of the effective iterative methods the cayley transform takes o n 3 arithmetic operations to produce an orthogonal matrix using a skew symmetric matrix in each iteration our algorithm can refine orthogonality without the cayley transform which reduces the operations in each iteration it is worth noting that our approach overcomes the limitation of the cayley transform method to the inverse standard eigenvalue problems resulting in an extension to inverse generalized eigenvalue problems

inequalities for the schatten p norm iv we present some inequalities for the schattenp norm of operators on a hilbert space it is shown among other things that ifa is an operator such that rea a 0 then for any operatorx ax xa p 2a x p also for any two operatorsa andb a b 2 2 a b 2 2 2 a b 2 2

linear moving horizon estimation with pre estimating observer in this note a moving horizon estimation mhe strategy for detectable linear systems is proposed like the idea of pre stabilizing model predictive control the states are estimated by a forward simulation with a pre estimating observer in the mhe formulation compared with standard linear mhe approaches it has more degrees of freedom to optimize the noise filtering tuning parameters are chosen to minimize the effects of measurement noise and model errors which is implemented by finding tightest estimation error bounds the performance of the proposed observer is demonstrated on one linear discrete time example