separation theorem for linearly constrained lqg optimal control we solve the linearly constrained linear quadratic lq and linear quadratic gaussian lqg optimal control problems closed form expressions of the optimal controls are derived and the separation theorem is generalized

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

minimal quasi separable realizations for the inverse of a quasi separable operator we derive minimal quasi separable i e state space representations for the upper and lower parts of the inverse of an invertible but otherwise general operator t which itself is given by its upper and lower minimal quasi separable representations we show that if the original representation is given in an adequate normal form then the computation of the representation of the inverse can be done in a single downward or upward pass involving only small local computations so called intrinsic factors play an essential role in the derivation we define them and show how they can be extracted the results are given in closed form provided one accepts the computation of a basis for a space and its orthogonal complement as numerically closed qr type factorizations common in array computations the central workhorse is the classical square root algorithm utilized here in a generalized form

a note on generalized diagonally dominant matrices abstract we obtain a sufficient condition for generalized diagonally dominant matrices under the assumption that a cn n is weakly diagonally dominant an equivalent condition for a to be a generalized diagonally dominant matrix is proved

bounds for the perron root of nonnegative matrices and spectral radius of iteration matrices abstract new bounds for the perron root a of a nonnegative matrix a are proposed we prove that min 1 i n r i a b r i b a max 1 i n r i a b r i b where b is an arbitrary matrix with row sums r i b k b i k 0 i 1 n the bounds of h minc 1 and shulin liu 6 are both special cases of this result and based on this result we also get some bounds for the spectral radius of iteration matrices

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

the numerically stable reconstruction of jacobi matrices from spectral data we present an expose of the elementary theory of jacobi matrices and in particular their reconstruction from the gaussian weights and abscissas many recent works propose use of the diagonal hermitian lanczos process for this purpose we show that this process is numerically unstable we recall rutishauser s elegant and stable algorithm of 1963 based on plane rotations implement it efficiently and discuss our numerical experience we also apply rutishauser s algorithm to reconstruct a persymmetric jacobi matrix from its spectrum in an efficient and stable manner

bounds for the greatest characteristic root of a nonnegative matrix abstract new bounds for the greatest characteristic root of a nonnegative matrix are obtained they generalize and improve the bounds of g frobenius and h minc

eigenstructure of order one quasiseparable matrices three term and two term recurrence relations abstract this paper presents explicit formulas and algorithms to compute the eigenvalues and eigenvectors of order one quasiseparable matrices various recursive relations for characteristic polynomials of their principal submatrices are derived the cost of evaluating the characteristic polynomial of an n n matrix and its derivative is only o n this leads immediately to several versions of a fast quasiseparable newton iteration algorithm in the hermitian case we extend the sturm property to the characteristic polynomials of order one quasiseparable matrices which yields to several versions of a fast quasiseparable bisection algorithm conditions guaranteeing that an eigenvalue of a order one quasiseparable matrix is simple are obtained and an explicit formula for the corresponding eigenvector is derived the method is further extended to the case when these conditions are not fulfilled several particular examples with tridiagonal almost unitary hessenberg and toeplitz matrices are considered the algorithms are based on new three term and two term recurrence relations for the characteristic polynomials of principal submatrices of order one quasiseparable matrices r it turns out that the latter new class of polynomials generalizes and includes two classical families i polynomials orthogonal on the real line that play a crucial role in a number of classical algorithms in numerical linear algebra and ii the szego polynomials that play a significant role in signal processing moreover new formulas can be seen as generalizations of the classical three term recurrence relations for the real orthogonal polynomials and of the two term recurrence relations for the szego polynomials

uncoupling the perron eigenvector problem abstract for a nonnegative irreducible matrix a with spectral radius this paper is concerned with the determination of the unique normalized perron vector which satisfies a 62 0 j j 1 it is explained how to uncouple a large matrix a into two or more smaller matrices say p 11 p 22 p kk such that this sequence of smaller matrices has the following properties 1 each p ii is also nonnegative and irreducible so that each pii has a unique perron vector i 2 each pii has the same spectral radius as a 3 it is possible to determine the i s completely independently of each other so that one can execute the computation of the i s parallel 4 it is easy to couple the smaller perron vectors i back together in order to produce the perron vector for the original matrix a

brief multiple objective risk sensitive control and its small noise limit this paper is concerned with a minimizing multiple objective risk sensitive control problem asymptotic analysis leads to the introduction of a new class of two player zero sum deterministic differential games the distinguishing feature of this class of games is that the cost functional is multiple objective in nature being composed of the risk neutral integral costs associated with the original risk sensitive problem more precisely the opposing player in such a game seeks to maximize the most vulnerable member of a given set of cost functionals while the original controller seeks to minimize the worst damage that the opponent can do over this set it is then shown that the problem of finding an efficient risk sensitive controller is equivalent asymptotically to solving this differential game surprisingly this differential game is proved to be independent of the weights on the different objectives in the original multiple objective risk sensitive problem as a by product our results generalize the existing results for the single objective risk sensitive control problem to a substantially larger class of nonlinear systems including those with control dependent diffusion terms

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

schur complements and its applications to symmetric nonnegative and z matrices in linear algebra appl 177 1992 137 smith proved that if h is a hermitian semidefinite matrix and a is a nonsingular principal submatrix then the eigenvalues of the schur complement h a interlace those of h in this paper we refine the latter result and use it to derive eigenvalues interlacing results on an irreducible symmetric nonnegative matrix that involve perron complements for an irreducible symmetric nonnegative matrix we give lower and upper bounds for its spectral radius and also a lower bound for the maximal spectral radius of its principal submatrices of a fixed order we apply our results to an irreducible symmetric z matrix and to the adjacency matrix or the general laplacian matrix of a connected weighted graph the equality cases for the bounds for spectral radii or least eigenvalues are also examined

localization of perron roots abstract this paper is concerned with the localization of the perron root of a nonnegative irreducible matrix a a new localization method that utilizes the relationship between the perron root of a nonnegative matrix and the estimates of the row sums of its generalized perron complement is presented the method is efficient because it gives the bounds on a only by computing the estimates of the row sums of the generalized perron complement rather than the generalized perron complement itself several numerical examples are given to illustrate the effectiveness of our method

an implicit multishift qr algorithm for hermitian plus low rank matrices hermitian plus possibly non hermitian low rank matrices can be efficiently reduced into hessenberg form the resulting hessenberg matrix can still be written as the sum of a hermitian plus low rank matrix in this paper we develop a new implicit multishift qr algorithm for hessenberg matrices which are the sum of a hermitian plus a possibly non hermitian low rank correction the proposed algorithm exploits both the symmetry and low rank structure to obtain a qr step involving only mathcal o n floating point operations instead of the standard mathcal o n 2 operations needed for performing a qr step on a hessenberg matrix the algorithm is based on a suitable mathcal o n representation of the hessenberg matrix the low rank parts present in both the hermitian and low rank part of the sum are compactly stored by a sequence of givens transformations and a few vectors due to the new representation we cannot apply classical deflation techniques for hessenberg matrices a new efficient technique is developed to overcome this problem some numerical experiments based on matrices arising in applications are performed the experiments illustrate effectiveness and accuracy of both the qr algorithm and the newly developed deflation technique

perron complement and perron root for a nonnegative irreducible matrix a this paper is concerned with the estimation and determination of the unique perron root or spectral radius of a we present a new method that utilizes the relation between perron roots of the nonnegative matrix and its generalized perron complement several numerical examples are given to show that our method is effective at least for some classes of nonnegative matrices

a quasi separation theorem for lqg optimal control with iq constraints we consider the deterministic the full observation and the partial observation lqg optimal control problems with finitely many iq integral quadratic constraints and show that wohnam s famous separation theorem for stochastic control has a generalization to this case although the problems of filtering and control are not independent we show that the interdependence of these two problems is so superficial that in effect they are problems which can be treated separately it is in this context that the label quasi separation theorem is to be understood we conclude with a discussion of computation issues and show how gradient type optimization algorithms can be used to solve these problems in this way a systematic computation algorithm is derived

the qr iteration method for hermitian quasiseparable matrices of an arbitrary order abstract the qr iteration method for tridiagonal matrices is in the heart of one classical method to solve the general eigenvalue problem in this paper we consider the more general class of quasiseparable matrices that includes not only tridiagonal but also companion comrade unitary hessenberg and semiseparble matrices a fast qr iteration method exploiting the hermitian quasiseparable structure and thus generalizing the classical tridiagonal scheme is presented the algorithm is based on an earlier work y eidelman and i gohberg a modification of the dewilde van der veen method for inversion of finite structured matrices linear algebra appl 343 344 2002 419 450 and it applies to the general case of hermitian quasiseparable matrices of an arbitrary order the algorithm operates on generators i e a linear set of parameters defining the quasiseparable matrix and the storage and the cost of one iteration are only linear the results of some numerical experiments are presented an application of this method to solve the general eigenvalue problem via quasiseparable matrices will be analyzed separately elsewhere

on the fast reduction of a quasiseparable matrix to hessenberg and tridiagonal forms in this paper we design a fast new algorithm for reducing an n n quasiseparable matrix to upper hessenberg form via a sequence of n 2 unitary transformations the new reduction is especially useful when it is followed by the qr algorithm to obtain a complete set of eigenvalues of the original matrix in particular it is shown that in a number of cases some recently devised fast adaptations of the qr method for quasiseparable matrices can benefit from using the proposed reduction as a preprocessing step yielding lower cost and a simplification of implementation

stochastic optimal lqr control with integral quadratic constraints and indefinite control weights a standard assumption in traditional deterministic and stochastic optimal minimizing linear quadratic regulator lqr theory is that the control weighting matrix in the cost functional is strictly positive definite in the deterministic case this assumption is in fact necessary for the problem to be well posed because positive definiteness is required to make it a convex optimization problem however it has recently been shown that in the stochastic case when the diffusion term is dependent on the control the control weighting matrix may have negative eigenvalues but the problem remains well posed in this paper the completely observed stochastic lqr problem with integral quadratic constraints is studied sufficient conditions for the well posedness of this problem are given indeed we show that in certain cases these conditions may be satisfied even when the control weighting matrices in the cost and all of the constraint functionals have negative eigenvalues it is revealed that the seemingly nonconvex problem with indefinite control weights can actually be a convex one by virtue of the uncertainty in the system finally when these conditions are satisfied the optimal control is explicitly derived using results from duality theory