the mathematics of eigenvalue optimization optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge such problems arise often in theory and practice particularly in engineering design and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory this essay presents a personal choice of some central mathematical ideas outlined for the broad optimization community i discuss the convex analysis of spectral functions and invariant matrix norms touching briefly on semidefinite representability and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and lie algebra analogous nonconvex notions lead into eigenvalue perturbation theory the last third of the article concerns stability for polynomials matrices and associated dynamical systems ending with a section on robustness the powerful and elegant language of nonsmooth analysis appears throughout as a unifying narrative thread

a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm the i th singular value of a transfer matrix need not be a differentiable function of frequency where its multiplicity is greater than one we show that near a local maximum however the largest singular value has a lipschitz second derivative but need not have a third derivative using this regularity result we give a quadratically convergent algorithm for computing the l norm of a transfer matrix

lowner based blind signal separation of rational functions with applications a new blind signal separation bss technique is proposed enabling a deterministic separation of signals into rational functions rational functions can take on a wide range of forms such as the well known pole like shape the approach is a possible alternative for the well known independent component analysis when the theoretical sources are not independent such as for frequency spectra or when only a small number of samples is available the technique uses a low rank decomposition on the tensorized version of the observed data matrix the deterministic tensorization with lowner matrices is comprehensively analyzed in this paper uniqueness properties are investigated and a connection with the separation into exponential polynomials is made finally the technique is illustrated for fetal electrocardiogram extraction and with an application in the domain of fluorescence spectroscopy enabling the identification of chemical analytes using only a single excitation emission matrix

cal l _ infty norm computation for continuous time descriptor systems using structured matrix pencils in this technical note we discuss an algorithm for the computation of the l norm of transfer functions related to descriptor systems we show how one can achieve this goal by computing the eigenvalues of certain skew hamiltonian hamiltonian matrix pencils and analyze arising problems we also formulate and prove a theoretical result which serves as a basis for testing a transfer function matrix for properness finally we illustrate our results using a descriptor system related to mechanical engineering

frame theory based analysis and design of oversampled filter banks direct computational method this paper studies the frames corresponding to oversampled filter banks fbs for this class of fb frames we present a state space parameterization of perfect reconstruction fb frames and explicit and numerically efficient formulas to compute the tightest frame bounds to obtain the dual fb frame and to construct a tight paraunitary fb frame from a given untight nonparaunitary fb frame the derivation uses well developed techniques from modern control theory which results in the unified formulas for generic infinite impulse response iir and finite impulse response fir fbs these formulas involve only algebraic manipulations of real matrices and can be computed efficiently reliably and exactly without the approximation required in the existing methods for generic fbs the results provide a unified framework for frame theory based analysis and systematic design of oversampled filter banks

a holonomic systems approach to special functions identities abstract we observe that many special functions are solutions of so called holonomic systems bernstein s deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps this is partially substantiated by an algorithm that proves terminating hypergeometric series identities and that is given both in english and in maple

generalized confluent cauchy vandermonde matrices displacement structures inversion formulas and tangential interpolations generalized confluent cauchy and cauchy vandermonde matrices are introduced these two kinds of matrices generalize the ordinary cauchy and cauchy vandermonde matrices with multiple nodes studied earlier by various authors by using displacement structure theory fast inversion formulas for these matrices are derived the tangential interpolation interpretations for associated linear systems with such matrices are given the fast algorithm for solving this kind of linear systems are also considered

on quantum separation of variables we present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models the basis is generated by the repeated action of the transfer matrix itself on a generically chosen state of the hilbert space the fusion relations for the transfer matrix stemming from the yang baxter algebra properties provide the necessary closure relations to define the action of the transfer matrix on such a basis in terms of elementary local shifts leading to a separate transfer matrix spectral problem hence our scheme extends to the quantum case a key feature of the liouville arnold classical integrability framework where the complete set of conserved charges defines both the level manifold and the flows on it leading to the construction of action angle variables we work in the framework of the quantum inverse scattering method as a first example of our approach we give the construction of such a basis for models associated with y gln and argue how it extends to their trigonometric and elliptic versions then we show how our general scheme applies concretely to fundamental models associated with the y gl2 and y gl3 r matrices leading to the full characterization of their spectrum for y gl2 and its trigonometric deformation a particular case of our method reproduces sklyanin s construction of separate variables for y gl3 it gives new results in particular through the proper identification of the shifts acting on the separate basis we stress that our method also leads to the full characterization of the spectrum of other known quantum integrable lattice models including in particular trigonometric and elliptic spin chains open chains with general integrable boundaries and further higher rank cases that we will describe in forthcoming publications

displacement structure approach to cauchy and cauchy vandermonde matrices inversion formulas and fast algorithms the confluent cauchy and cauchy vandermonde matrices are considered which were studied earlier by various authors in different ways in this paper we use another way called displacement structure approach to deal with matrices of this kind we show that the cauchy and cauchy vandermonde matrices satisfy some special type of matrix equations this leads quite naturally to the inversion formulas and fast algorithms for matrices of this kind

computation of the structured stability radius via matrix sign function robustness of perturbed state space models of the form x a bdc x is considered where b c are given matrices a is an asymptotically stable matrix and d is the unknown perturbation matrix an efficient algorithm to compute the complex structured stability radius which is based on the properties of the matrix sign function is presented a comparison with previous algorithms shows the efficiency of the new algorithm

confluent polynomial vandermonde like matrices displacement structures inversion formulas and fast algorithm in the present paper we use the displacement structure approach to introduce a new class of what are called confluent polynomial vandermonde like matrices which generalize the ordinary simple nodes polynomial vandermonde matrices studied earlier by different authors the displacement structure approach leads to the explicit inversion formulas for confluent polynomial vandermonde like matrices and fast algorithms for inversion and for solving the associated linear systems

confluent cauchy and cauchy vandermonde matrices abstract the s c confluent cauchy and confluent cauchy vandermonde matrices are introduced corresponding to an interpolation problem with multiple interpolation nodes inversion formulas are derived and a factorization of confluent loewner matrices by means of confluent cauchy vandermonde matrices is described

hankel and loewner matrices abstract mutual relations between the hankel toeplitz bezout and loewner matrices as well as further connections to rational interpolation and projective geometry are investigated

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

on the real stability radius of sparse systems abstract in this paper we study robust stability of sparse lti systems using the stability radius sr as a robustness measure we consider real perturbations with an arbitrary and pre specified sparsity pattern of the system matrix and measure their size using the frobenius norm we formulate the sr problem as an equality constrained minimization problem using the lagrangian method for optimization we characterize the optimality conditions of the sr problem thereby revealing the relation between an optimal perturbation and the eigenvectors of an optimally perturbed system further we use the sylvester equation based parametrization to develop a penalty based gradient newton descent algorithm which converges to the local minima of the optimization problem finally we illustrate how our framework provides structural insights into the robust stability of sparse networks

computing the distance to continuous time instability of quadratic matrix polynomials a bisection method is used to compute lower and upper bounds on the distance from a quadratic matrix polynomial to the set of quadratic matrix polynomials having an eigenvalue on the imaginary axis each bisection step requires to check whether an even quadratic matrix polynomial has a purely imaginary eigenvalue first an upper bound is obtained using frobenius type linearizations it takes into account rounding errors but does not use the even structure then lower and upper bounds are obtained by reducing the quadratic matrix polynomial to a linear palindromic pencil the bounds obtained this way also take into account rounding errors numerical illustrations are presented

a newton based method for the calculation of the distance to instability in this paper a new fast algorithm for the computation of the distance of a stable matrix to the unstable matrices is provided the method uses newton s method to find a two dimensional jordan block corresponding to a pure imaginary eigenvalue in a certain two parameter hamiltonian eigenvalue problem introduced by byers r byers a bisection method for measuring the distance of a stable matrix to the unstable matrices siam j sci statist comput 9 1988 875 881 this local method is augmented by a test step previously used by other authors to produce a global method numerical results are presented for several examples and comparison is made with the methods of boyd and balakrishnan s boyd v balakrishnan a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm systems control lett 15 1990 1 7 and he and watson c he g a watson an algorithm for computing the distance to instability siam j matrix anal appl 20 1999 101 116

an efficient algorithm for computing the cal h _ infty norm this technical note addresses the computation of the h norm by directly computing the isolated common zeros of two bivariate polynomials unlike the iteration algorithm that is currently used to find the h norm the proposed method to h norm calculation is compared with the existing method by bruinsma and steinbuch 1990 using numerical experiments on random transfer functions of orders upto 240 the time taken is better by 15 to 30 times in addition to improved accuracy the proposed method uses techniques involving structured linearization of the bezoutian matrix constructed from two bivariate polynomials

a unified approach to loewner and hankel matrices abstract we introduce so called generalized loewner matrices the definition is almost the same as antoulas and anderson s our class of generalized loewner matrices contains both classical loewner and hankel matrices as special cases the properties of such matrices are studied together with the associated interpolation problem the detailed characterization of its nonsolvable cases is given by means of the s c minimal generalized solution by means of rank properties of the loewner matrix and by means of the s c l polynomial introduced by fiedler

on computing the infinity norm an iterative algorithm is given for computing the infinity norm of the transfer matrix of a bounded linear time invariant finite dimensional system the procedure which is based on the numerically stable qz algorithm enables the infinite norm to be computed with a guaranteed accuracy no ad hoc procedures such as those for selecting a suitable grid of frequencies for example are used the degree of accuracy is under the direct control of the user with each additional decimal place of accuracy requiring approximately 3 3 calls to the qz algorithm experience has shown the algorithm to function reliably and in reasonable time for models of order as large as 56

h norm computation for ltv systems using nonlinear optimal control algorithms we discuss the computation for 2 induced norms of finite horizon linear time varying systems it is well known that 2 induced norm calculating problems can be converted into optimal control problems so with the help of optimal control algorithms we propose a numerical technique for 2 induced norm computation our approach is as follows first we present a new computational method for nonlinear optimal control problems that is numerically reliable stable and robust analysing the convergence property under some assumptions next by modifying the presented method we propose an algorithm for the 2 induced norm calculation of finite horizon linear time varying systems which is essentially different from all other existing methods finally the effectiveness of our algorithm is demonstrated by simulation

computation of extremum singular values and the strong h infinity norm of siso time delay systems we consider the computation of h infinity norms for single input single output siso time delay systems which are described by delay differential algebraic equations unlike the iterative level set methods in the literature we present a novel numerical method to compute the h infinity norm this method requires solving one eigenvalue problem of at most twice the size of the eigenvalue problem in every iteration of a level set method but in practice often considerably lower we first show that the computation of extrema of the transfer function can be turned into the computation of the imaginary axis zeros of a transcendental function we compute these zeros by a predictor corrector type algorithm it is known that the h infinity norm of delay differential algebraic systems which can model both retarded and neutral type systems might be sensitive with respect to arbitrarily small delay perturbations this recently led to the concept of strong h infinity norms which explicitly take into account such small delay perturbations we present a direct numerical method to compute the strong h infinity norm of siso time delay systems our algorithm is applicable to the closed loop system of interconnections series parallel feedback junctions of time delay systems and or controllers

a lagrange matrices approach to confluent cauchy matrices we present a new approach to study inversion and factorization properties of confluent cauchy matrices we consider a class of generalized vandermonde matrices called lagrange matrices that are connected in a simple way with the cauchy matrices we apply the methods introduced in adv appl math 10 1989 348 to obtain inversion and factorization theorems for lagrange matrices and their confluent forms and then derive corresponding results for cauchy matrices a lagrange matrix v is associated with a pair of vectors x0 x1 xn and y0 y1 yn v is the matrix representation of the substitution operator that sends the vector p x0 p x1 p xn t to the vector p y0 p y1 p yn t for any polynomial p of degree at most n

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

quasidirect decompositions of hankel and toeplitz matrices decompositions over an algebraically closed field of a hankel matrix into a sum of hankel matrices the sum of the ranks of which is equal to the rank of the original matrix are completely described similar results hold for toeplitz matrices

an inversion formula and fast algorithms for cauchy vandermonde matrices abstract matrices of composed type consisting of a vandermonde and a cauchy part and their connection to rational interpolation are investigated an inversion formula is presented and fast solution algorithms with o n log2 n complexity for cauchy vandermonde systems are developed

h control by state feedback an iterative algorithm and characterization of high gain occurrence we present an iterative algorithm to compute the achievable h norm by state feedbak for a standard regular problem and give a characterization when high gain feedback is necessary to approach the optimal value if there is no need for high gain components to approximate the optimal value we can reduce the problem to the computation of the h norm of a certain stable transfer matrix

brief a fast algorithm for the computation of an upper bound on the µ norm a fast algorithm for the computation of the optimally frequency dependent scaled h norm of a finite dimensional lti system is presented it is well known that this quantity is an upper bound to the m norm furthermore it was recently shown to play a special role in the context of slowly time varying uncertainty numerical experimentation suggests that the algorithm generally converges quadratically

second order nonsmooth optimization for h synthesis the standard way to compute h feedback controllers uses algebraic riccati equations and is therefore of limited applicability here we present a new approach to the h output feedback control design problem which is based on nonlinear and nonsmooth mathematical programming techniques our approach avoids the use of lyapunov variables and is therefore flexible in many practical situations

robustness measures for linear systems with application to stability radii of hurwitz and schur polynomials in this paper we consider robustness measures stability radii for system matrices which are subjected to structured real and complex perturbations of the form a a bdc where b c are given matrices our object is twofold a to present a number of new results mainly concerning the real stability radius and its differences from the complex one b to give an overview of our approach to the robustness analysis of linear state space systems including basic properties and characterizations of the complex stability radius applying the results to the special case where a is in companion form and b 0 0 0 1 t we are able to determine stability radii for hurwitz and schur polynomials under arbitrary complex and real affine perturbations of the coefficient vector computable formulae are obtained and illustrated by several examples

polynomials and hankel matrices abstract compatibility of a hankel n n matrix h and a polynomial f of degree m m n is defined if m n compatibility means that hc t f c f h where c f is t companion matrix of f with a suitable generalization of c f this theorem is generalized to the case that m n

simple covariance approach to mathcal h _ infty analysis we revisit the classical mathcal h infty analysis problem of computing the ell 2 induced norm and the mathcal l 2 induced norm of a linear time invariant system for the discrete time case and the continuous time case respectively we follow an approach based on converting the problem of maximization over signals to that of maximization over a sort of deterministic covariance matrices the reformulation in terms of these covariance matrices greatly simplifies the dynamic analysis problem and converts the computation to a convex constrained matrix maximization problem furthermore the equivalence is for the actual mathcal h infty norm of the system rather than a bound and thus does not require the typical gamma iterations we argue that this approach is both attractive elementary and constructive in that the worst case disturbance is also easily obtained as a state feedback constructed from the solution of the matrix problem

polynomials compatible with a symmetric loewner matrix abstract several mutually equivalent conditions are proved for a polynomial of degree n and a symmetric n n matrix which form a basis for so called compatibility of these two objects one of the conditions yields a new parametric expression for a symmetric loewner matrix

evaluating and approximating fir filters an approach based on functions of matrices most of the controllers designed for systems with timedelays involve finite impulse response fir filters in their structure in particular prediction based controllers for systems with input output delays we present theory algorithms and software to evaluate the frequency response to approximate the filters by stable linear timeinvariant systems and to compute their induced cal l 2 gain the approach is based on the theory of functions of matrices in the approach pole zero cancelations in frequency domain representations are explicitly taken into account

robust stability of linear systems described by higher order dynamic equations the stability radius of higher order differential and difference systems with respect to various classes of complex affine perturbations of the coefficient matrices is studied different perturbation norms are considered the aim is to derive robustness criteria that are expressed directly in terms of the original data previous results on robust stability of hurwitz and schur polynomials are extended to monic matrix polynomials for disturbances acting via a uniform input matrix computable formulas are obtained whereas for perturbations with multiple input matrices structured singular values are involved

displacement structures and fast inversion formulas for confluent polynomial vandermonde like matrices in the present paper confluent polynomial vandermonde like matrices with general recurrence structure are introduced three kinds of displacement structure equations and two kinds of fast inversion formulas for this class of matrices are derived by using displacement structure matrix method a relationship between confluent polynomial vandermonde like matrices and confluent cauchy like matrices is pointed out

intertwining and testing matrices corresponding to a polynomial abstract given a natural number n and a polynomial of degree m n a description is given of a class r n of n by n matrices the intertwining matrices for with the following properties let h n be the family of all n by n hankel matrices compatible with and consider the relation hr t rh then an n by n matrix h belongs to h n if and only if this relation holds for all r r n a characterization is also given of testing matrices a matrix r is said to be a testing matrix if the fulfillment of the above relation already implies that h h n

interconnected systems with uncertain couplings explicit formulae for mu values spectral value sets and stability radii in this paper we study the variation of the spectrum of block diagonal systems under perturbations of compatible block structure with fixed zero blocks at arbitrarily prescribed locations gershgorin type perturbations we derive explicit and computable formulae for the associated mu values the results are then applied to characterize spectral value sets and stability radii for such perturbed systems by specializing our results to the scalar diagonal case the classical eigenvalue inclusion theorems of gershgorin brauer and brualdi are obtained as corollaries moreover it follows that the inclusion regions of brauer and brualdi are optimal for the corresponding perturbation structures

generalized confluent cauchy and cauchy vandermonde matrices abstract the so called generalized confluent cauchy vandermonde gccv matrices of the form c v consisting of a generalized confluent cauchy part c and a generalized confluent vandermonde part v are considered a simple relationship between gccv and classical confluent cauchy vandermonde ccv matrices is given this leads to the reduction of the displacement structure inversion formulas and factorizations of gccv matrices and the interpolation interpretations of linear systems with such gccv coefficient matrices as tangential interpolation problems to the corresponding results of ccv matrices the criteria of invertibility and left right inversion formulas for such matrices are given all results are stated for the general case of multiple interpolation nodes extending the work of heinig and vavrin among others

on the computation of the optimal h norms for two feedback control problems abstract numerical computation of the optimal h norms for the state feedback and the output feedback control problems is considered in 1989 doyle et al derived for a given parameter r 0 necessary and sufficient conditions under which an admissible output feedback controller exists as well as conditions for the existence of a state feedback controller we explore numerical methods to find the infimum of parameter r that satisfies those conditions

error estimation in reduced basis method for systems with time varying and nonlinear boundary conditions many physical phenomena such as mass transport and heat transfer are modeled by systems of partial differential equations with time varying and nonlinear boundary conditions control inputs and disturbances typically affect the system dynamics at the boundaries and a correct numerical implementation of boundary conditions is therefore crucial however numerical simulations of high order discretized partial differential equations are often too computationally expensive for real time and many query analysis for this reason model complexity reduction is essential in this paper it is shown that the classical reduced basis method is unable to incorporate time varying and nonlinear boundary conditions to address this issue it is shown that by using a modified surrogate formulation of the reduced basis ansatz combined with a feedback interconnection and a input related term the effects of the boundary conditions are accurately described in the reduced order model the results are compared with the classical reduced basis method unlike the classical method the modified ansatz incorporates boundary conditions without generating unphysical results at the boundaries moreover a new approximation of the bound and a new estimate for the error induced by model reduction are introduced the effectiveness of the error measures is studied through simulation case studies and a comparison with existing error bounds and estimates is provided the proposed approximate error bound gives a finite bound of the actual error unlike existing error bounds that grow exponentially over time finally the proposed error estimate is more accurate than existing error estimates

on some norms for descriptor systems we present generalizations of the impulse and frequency responses as well as convolution and hankel operators for continuous time and discrete time descriptor systems some norms for descriptor systems are introduced and their representations via the different linear system concepts are considered

on isotropic rank 1 convex functions let f be a rotationally invariant function defined on the set lin of all tensors with positive determinant on a vector space of arbitrary dimension necessary and sufficient conditions are given for the rank 1 convexity of f in terms of its representation through the singular values for the global rank 1 convexity on lin the result is a generalization of a two dimensional result of aubert generally the inequality on contains products of singular values of the type encountered in the definition of polyconvexity but is weaker it is also shown that the rank 1 convexity is equivalent to a restricted ordinary convexity when f is expressed in terms of signed invariants of the deformation

factorizations of confluent cauchy vandermonde matrices abstract the confluent cauchy vandermonde matrices are considered which were studied earlier by various authors in different way in this paper we analyze the factorization of the inverse of a special type of confluent cauchy vandermonde matrix as a product of block bidiagonal matrices by using block neville elimination factorizations of confluent cauchy vandermonde matrices are also considered which generalize the results of j j martinez j m pena factorizations of cauchy vandermonde matrices linear algebra appl 284 1998 229 237

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

on the computation of the real hurwitz stability radius recently qiu et al 1995 obtained a computationally attractive formula for the evaluation of the real stability radius this formula involves a global maximization over frequency here for the hurwitz stability case we show that the frequency range can be limited to a certain finite interval numerical experimentation suggests that this interval is often reasonably small

a fast algorithm to computer the h norm of a transfer function matrix abstract a fast algorithm is presented to compute the h norm of a transfer function matrix based on the relation between the singular values of the transfer function matrix and the eigenvalues of a related hamiltonian matrix the norm is computed with guaranteed accuracy a very simple method to compute a rather close lower bound on the h norm is given

doubly cuspidal cohomology for principal congruence subgroups of gl 3 z the cohomology of arithmetic groups is made up of two pieces the cuspidal and noncuspidal parts within the cuspidal cohomology is a subspacethe f cuspidal cohomology spanned by the classes that generate representations of the associated finite lie group which are cuspidal in the sense of finite lie group theory few concrete examples of f cuspidal cohomology have been computed geometrically outside the cases of rational rank 1 or where the symmetric space has a hermitian structure this paper presents new computations of the f cuspidal cohomology of principal congruence subgroups f p of gl 3 2 of prime level p we show that the f cuspidal cohomology of f p vanishes for all p 19 with p 11 but that it is nonzero for p 11 we give a precise description of the f cuspidal cohomology for jt 11 in terms of the f cuspidal representations of the finite lie group gl 3 e 11 we obtained the result ultimately by proving that a certain large complex matrix m is rank deficient computation with the svd algorithm gave strong evidence that m was rank deficient but to prove it we mixed ideas from numerical analysis with exact computation in algebraic number fields and finite fields

brief paper non parametric methods for l2 gain estimation using iterative experiments in this paper we develop non parametric methods to estimate the l 2 gain h norm of a linear dynamical system from iterative experiments this work is mainly motivated by model error modeling where the error dynamics are more complex than can be captured by a low order parametric model the standard system identification approach to the gain estimation problem is to estimate a parametric model of the system which is then used to calculate the gain if it is possible to update the input signal during the experiment an alternative way is to iteratively optimize the input signal in order to maximize the estimated input to output gain a key observation is that the gradient of the gain with respect to the input signal can without knowing a model be found from two experiments iterative numerical methods for calculation of eigenvalues of matrices e g the power method or the lanczos method can then be applied to update the input signal sequence between experiments in order to find the maximum gain the main difficulty compared to the corresponding eigenvalue problem in numerical analysis is the effects of additive measurement noise which require modified schemes that avoid bias errors three such related methods are derived and evaluated by a numerical example partial results on convergence and statistical properties of the gain estimator are given a constrained stochastic gradient method with local optimization of step length gives the best numerical results in the case of noisy data

rank 1 perturbations of deformation gradients rank 1 connections i e pairs f g of tensors matrices differing by a rank 1 tensor g f f n play an important role in the nonlinear elasticity and in particular in the theory of solid to solid phase transformations if f g are rank 1 connected g is said to be a rank 1 perturbation of f this paper describes the set of all rank 1 perturbations of f with prescribed singular values in an n dimensional space this set is shown to consist of 2n families of dimension n 1 and within each family g and f may be expressed as functions of the unit vector n given by explicit formulas these are then specialized to dimensions n 2 and 3 some illustrative examples are given