block arnoldi based methods for large scale discrete time algebraic riccati equations in the present paper we present block arnoldi based methods for the computation of low rank approximate solutions of large discrete time algebraic riccati equations dare the proposed methods are projection methods onto block or extended block krylov subspaces we give new upper bounds for the norm of the error obtained by applying these block arnoldi based processes we also introduce the newton method combined with the block arnoldi algorithm and present some numerical experiments with comparisons between these methods

passivity preserving model reduction via interpolation of spectral zeros abstract an algorithm is developed for passivity preserving model reduction of linear time invariant systems implementation schemes are described for both medium scale dense and large scale sparse applications the algorithm is based upon interpolation at selected spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros these interpolation conditions are satisfied through the computation of a basis for a selected invariant subspace of a certain block matrix which has the spectral zeros as its spectrum explicit interpolation is avoided and passivity of the reduced model is established instead through satisfaction of the necessary conditions of the positive real lemma it is also shown that this procedure indirectly solves the associated controllability and observability riccati equations and how to select the interpolation points to give maximal or minimal solutions of these equations from these a balancing transformation may be constructed to give a reduced model that is balanced as well as passive and stable

halley s method for the matrix sector function the matrix n sector function is a generalization of the matrix sign function it can be used to determine the number of eigenvalues of a matrix in a specific sector of the complex plane and to extract the eigenpairs belonging to this sector without explicitly computing the eigenvalues it is known that newton s method which can be used for computing the matrix sign function is not globally convergent for the matrix sector function the only existing algorithm for computing the matrix sector function is based on the continued fraction expansion approximation to the principal nth root of an arbitrary complex matrix in this paper we introduce a new algorithm based on halley s generalized iteration formula for solving nonlinear equations it is shown that the iteration has good error propagation properties and high accuracy finally we give two application examples and summarize the results of our numerical experiments comparing newton s the continued fraction and halley s method

on discrete algebraic riccati equations a rank characterization of solutions abstract we give a rank characterization of the solution set of discrete algebraic riccati equations dares involving real matrices our characterization involves the use of invariant subspaces of the system matrix we observe that a dare can be considered as a specific type of a non symmetric continuous time algebraic riccati equation we show how to extract the low rank solutions from the full rank solution of a homogeneous dare we also study the eigen structure of various feedback closed loop matrices associated with solutions of a dare we show as a consequence of our observations that a solution x of a dare for controllable systems and certain specific types of uncontrollable systems satisfies the inequality x m i n x x m a x we characterize conditions under which the solution set of dare is bounded unbounded we use the method of invariant subspaces to identify some invariant sets of a difference riccati equation

the matrix sign function a survey of the matrix sign function is presented including some historical background definitions and properties approximation theory and computational methods and condition theory and estimation procedures applications to areas such as control theory eigendecompositions and roots of matrices are outlined and some new theoretical results are also given

numerical solution of projected algebraic riccati equations we consider the numerical solution of projected algebraic riccati equations using newton s method such equations arise for instance in model reduction of descriptor systems based on positive real and bounded real balanced truncation we also discuss the computation of low rank cholesky factors of the solutions of projected riccati equations numerical examples are given that demonstrate the properties of the proposed algorithms

solving the matrix differential riccati equation a lyapunov equation approach in this technical note we investigate a solution of the matrix differential riccati equation that plays an important role in the linear quadratic optimal control problem unlike many methods in the literature the approach that we propose employs the negative definite anti stabilizing solution of the matrix algebraic riccati equation and the solution of the matrix differential lyapunov equation an illustrative numerical example is provided to show the efficiency of our approach

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

separating eigenvalues using the matrix sign function abstract howland has used the matrix sign function to separate the eigenvalues of a given matrix the derived method required complex arithmetic even in the case of real matrices a modification of the method is presented that does not suffer from this drawback moreover it is shown that the problem may be decomposed into smaller subproblems which leads to considerable savings in computational work

an inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems we discuss an inverse free highly parallel spectral divide and conquer algorithm it can compute either an invariant subspace of a nonsymmetric matrix a or a pair of left and right deflating subspaces of a regular matrix pencil a lambda b this algorithm is based on earlier ones of bulgakov godunov and malyshev but improves on them in several ways this algorithm only uses easily parallelizable linear algebra building blocks matrix multiplication and qr decomposition but not matrix inversion similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function which requires matrix inversion and is faster but can be less stable than the new algorithm

lur e equations and even matrix pencils abstract in this work we consider the so called lur e matrix equations that arise e g in model reduction and linear quadratic infinite time horizon optimal control we characterize the set of solutions in terms of deflating subspaces of even matrix pencils in particular it is shown that there exist solutions which are extremal in terms of definiteness it is shown how these special solutions can be constructed via deflating subspaces of even matrix pencils

successive refinement of large multicell models this paper presents and analyzes a new relaxation method for minimizing a function of many variables the minimized function is associated with a model that is composed of a large number of cells in some applications the terms elements mesh points or facilities are more appropriate such problems often arise in models that are based on finite difference or finite element methods the new method which we call the successive refinement method utilizes the special structure of the model to save on both storage and computations although the s r method has many features in common with the nonlinear s o r method there is a basic difference between the two methods the difference is that the steps of the s r method are not using relaxation parameters instead each of its steps is aimed to reduce the objective function value as much as possible this gives us a robust algorithm that is able to solve problems that the s o r method cannot handle such problems are for example minimization of a

the generalized newton iteration for the matrix sign function in this paper we present modified algorithms for computing deflating subspaces of matrix pairs using the matrix sign function our new algorithms achieve a considerable reduction of the computational cost of the generalized newton iteration for the matrix sign function and improve the accuracy of the computed deflating subspaces the matrix sign function is thus revealed as an effective technique for applications in which bases for the deflating subspaces are required when partial or complete information about the eigenspectrum is desired the matrix sign function can be used as an initial divide and conquer technique the basic kernels involved in this iteration are especially appropriate for current high performance architectures

fourth order approximations at first time level linear stability analysis and the numerical solution of multidimensional second order nonlinear hyperbolic equations in polar coordinates in this article three level implicit difference schemes of o k4 k2h2 h4 where k 0 h 0 are grid sizes in time and space coordinates respectively are proposed for the numerical solution of one two and three space dimensional nonlinear wave equations in polar coordinates subject to appropriate initial and dirichlet boundary conditions we also discuss fourth order approximation at first time level for more general case we also obtain the stability range of the difference scheme when applied to a test equation utt urr aruu ar2u g r t a 1 and 2 numerical examples are provided to demonstrate the required order of convergence of the methods

spectral decomposition of a matrix using the generalized sign matrix an algorithm for spectral decomposition is presented which does not require knowledge of eigenvalues and eigenvectors a set of eigenprojectors are defined which covers the entire spectrum of a matrix and special attention is given to the projection on the zero eigenvalue some useful applications are discussed in the paper

the matrix sign decomposition and its relation to the polar decomposition the sign function of a square matrix was introduced by roberts in 1971 we show that it is useful to regard s sign a as being part of a matrix sign decomposition a sn where n a2 1 2 this decomposition leads to the new representation sign a a a2 1 2 most results for the matrix sign decomposition have a counterpart for the polar decomposition a uh and vice versa to illustrate this we derive best approximation properties of the factors u h and s determine bounds for a s and a u and describe integral formulas for s and u we also derive explicit expressions for the condition numbers of the factors s and n an important equation expresses the sign of a block 2 2 matrix involving a in terms of the polar factor u of a we apply this equation to a family of iterations for computing s by pandey kenney and laub to obtain a new family of iterations for computing u the iterations have some attractive properties including suitability for parallel computation

structured doubling algorithms for weakly stabilizing hermitian solutions of algebraic riccati equations in this paper we propose structured doubling algorithms for the computation of the weakly stabilizing hermitian solutions of the continuous and discrete time algebraic riccati equations respectively assume that the partial multiplicities of purely imaginary and unimodular eigenvalues if any of the associated hamiltonian and symplectic pencil respectively are all even and the c dare and the dual c dare have weakly stabilizing hermitian solutions with property p under these assumptions we prove that if these structured doubling algorithms do not break down then they converge to the desired hermitian solutions globally and linearly numerical experiments show that the structured doubling algorithms perform efficiently and reliably

the moser veselov equation abstract we study the orthogonal solutions of the matrix equation xj jx t m where j is symmetric positive definite and m is skew symmetric this equation arises in the discrete version of the dynamics of a rigid body investigated by moser and veselov commun math phys 139 1991 217 we show connections between orthogonal solutions of this equation and solutions of a certain algebraic riccati equation this will bring out the symplectic geometry of the moser veselov equation and also reduces most computational issues about solutions to finding invariant subspaces of a certain hamiltonian matrix necessary and sufficient conditions for the existence of orthogonal solutions and methods to compute them are presented our method is contrasted with the moser veselov approach commun math phys 139 1991 217 we also exhibit explicit solutions of a particular case of the moser veselov equation which appears associated with the continuous version of the dynamics of a rigid body

a jacobi like method for solving algebraic riccati equations on parallel computers an algorithm to solve continuous time algebraic riccati equations through the hamiltonian schur form is developed it is an adaption for hamiltonian matrices of an asymmetric jacobi method of eberlein 1987 it uses unitary symplectic similarity transformations and preserves the hamiltonian structure of the matrix each iteration step needs only local information about the current matrix thus admitting efficient parallel implementations on certain parallel architectures convergence performance of the algorithm is compared with the hamiltonian jacobi algorithm of byers 1990 the numerical experiments suggest that the method presented here converges considerably faster for non hermitian hamiltonian matrices than byers hamiltonian jacobi algorithm besides that numerical experiments suggest that for the method presented here the number of iterations needed for convergence can be predicted by a simple function of the matrix size

computing enclosures for the inverse square root and the sign function of a matrix abstract we study computational methods for obtaining rigorous a posteriori error bounds for the inverse square root and the sign function of an n n matrix a given a computed approximation for the inverse square root of a our methods work by using interval arithmetic to obtain a narrow interval matrix which with mathematical certainty is known to contain the exact inverse square root particular emphasis is put on the computational efficiency of the method which has complexity o n 3 and which uses almost exclusively matrix matrix operation a key to the efficient use of available software for interval computations the standard formulation of the method assumes that a can be diagonalized and that the eigenvector matrix of a is well conditioned a modification relying on a stable similarlity transformation to block diagonal form is also developed

matrix sign function methods for solving projected generalized continuous time sylvester equations in this technical note we investigate the numerical solution of the projected generalized sylvester equations via a matrix sign function method such equations arise in stability analysis and control problems for descriptor systems including model reduction based on balanced truncation unlike the classical matrix sign function iteration we propose a modification of the matrix sign function method that converges quadratically for pencils of arbitrary index numerical experiments report the effectiveness of the modified method

an extended block golub kahan algorithm for large algebraic and differential matrix riccati equations abstract in this paper we propose a new projection method to solve both large scale continuous time matrix riccati equations and differential matrix riccati equations the new approach projects the problem onto an extended block krylov subspace and gets a low dimensional equation we use the block golub kahan procedure to construct the orthonormal bases for the extended krylov subspaces for matrix riccati equations the reduced problem is then solved by means of a direct riccati scheme such as the schur method when we solve differential matrix riccati equations the reduced problem is solved by the backward differentiation formula bdf method and the obtained solution is used to build the low rank approximate solution of the original problem finally we give some theoretical results and present numerical experiments

closed form h infinity optimal control for a class of infinite dimensional systems h infinity optimal control and estimation are addressed for a class of systems governed by partial differential equations with bounded input and output operators diffusion equations are an important example in this class explicit formulas for the optimal state feedback controller as well as the optimal state estimator are given unlike traditional methods for h infinity synthesis no iteration is needed to obtain the optimal solution moreover the optimal performance for both the state feedback and state estimation problems are explicitly calculated this is shown to be useful for problems of h infinity optimal actuator and sensor location furthermore the results can be used in testing and bench marking of general purpose algorithms for h infinity synthesis the results also apply to finite dimensional systems

the sign matrix and the separation of matrix eigenvalues abstract the sign matrices uniquely associated with the matrices m j i 2 where j are the corners of a rectangle oriented at 4 to the axes of a cartesian coordinate system may be used to compute the number of eigenvalues of the arbitrarily chosen matrix m which lie within the rectangle and to determine the left and right invariant subspaces of m associated with these eigenvalues this paper is concerned with the proof of this statement and with the details of the computation of the required sign matrices

a newton squaring algorithm for computing the negative invariant subspace of a matrix by combining newton s method for the matrix sign function with a squaring procedure a basis for the negative invariant subspace of a matrix can be computed efficiently the algorithm presented is a variant of multiplication rich schemes for computing the matrix sign function such as the well known inversion free schulz method which requires two matrix multiplications per step however by avoiding a complete computation of the matrix sign and instead concentrating only on the negative invariant subspace the final newton steps can be replaced by steps which require only one matrix squaring each this efficiency is attained without sacrificing the quadratic convergence of newton s method

componentwise analysis of direct factorization of real symmetric and hermitian matrices abstract we derive componentwise error bound for the factorization h gjg t where h is a real symmetric matrix g has full column rank and j is diagonal with 1 s on the diagonal we also derive a componentwise forward error bound that is we bound the difference between the exact and the computed factor g in the cases where such a bound is possible we extend these results to the hermitian case and to the well known bunch parlett factorization finally we prove bounds for the scaled condition of the matrix g and show that the factorization can have the rank revealing property

solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

an iterative algorithm for the solution of the discrete time algebraic riccati equation abstract the discrete time algebraic riccati equation is solved in this study by an iterative algorithm for the square root of a squared hamiltonian matrix which is obtained from the s 1 transformation of the symplectic pencil associated with the riccati equation the symplectic givens and n n block diagonal orthogonal transformations are used before the iterative process so that the iteration is structure preserving and can achieve on average 60 reduction of computation time compared with the qz algorithm a formal analysis for roundoff errors and some numerical examples are also given