Showing papers on "Matrix difference equation published in 1979"
TL;DR: A simple four-variable oscillator containing but one quadratic term produces a higher form of chaos with two (rather than one) directions of hyperbolic instability on the attractor as mentioned in this paper.
1,240 citations
TL;DR: A new method is proposed which differs from the Bartels-Stewart algorithm in that A is only reduced to Hessenberg form, and the resulting algorithm is between 30 and 70 percent faster depending upon the dimensions of the matrices A and B.
Abstract: One of the most effective methods for solving the matrix equation AX+XB=C is the Bartels-Stewart algorithm. Key to this technique is the orthogonal reduction of A and B to triangular form using the QR algorithm for eigenvalues. A new method is proposed which differs from the Bartels-Stewart algorithm in that A is only reduced to Hessenberg form. The resulting algorithm is between 30 and 70 percent faster depending upon the dimensions of the matrices A and B . The stability of the new method is demonstrated through a roundoff error analysis and supported by numerical tests. Finally, it is shown how the techniques described can be applied and generalized to other matrix equation problems.
795 citations
01 Jan 1979
736 citations
01 Jan 1979
TL;DR: In this article, the stochastic difference equation Y, =AY_-1+B, with i.i.d. random pairs (An,B) was considered and conditions under which Y converges in distribution.
Abstract: The present paper considers the stochastic difference equation Y,=AY_-1+B, with i.i.d. random pairs (An,B,) and obtains conditions under which Y. converges in distribution. This convergence is related to the d
409 citations
TL;DR: In this article, the extremal eigenvalues of the positive definite solution of the Riccati equation were derived for discrete algebraic matrix Riccaci equations. But these estimations appear to appear to be considerably tighter than previously available results in many cases.
Abstract: Given an algebraic matrix Riccati equation A'K+ KA - KBB'K + Q =0 , the fundamental inequalities which are satisfied by the extremal eigenvalues of the positive definite solution K , are established. It Is illustrated that these resultant estimations appear to be considerably tighter than previously available results in many cases. Similar results are obtained for the discrete algebraic matrix Riccati equation.
126 citations
TL;DR: In this paper, the necessary and sufficient condition for the existence of a unique, non-negative definite and periodic solution and the asymptotically stable closed-loop system is derived in terms of the stabilizability and detectability of matrix pairs, that is the natural generalization of those for constant matrix pairs.
Abstract: The matrix Riccati differential equation with periodic coefficients is discussed in this paper. As is well known, such equations are frequently encountered in the applications of the optimal filtering and control theory. The necessary and sufficient condition is derived for the existence of a unique, non-negative definite and periodic solution and the asymptotically stable closed-loop system. Such condition is stated in terms of the stabilizability and detectability of matrix pairs, that is the natural generalization of those for constant matrix pairs.
81 citations
TL;DR: In this article, a necessary and sufficient condition for solvability of the matrix equation AX − YB = C was established, which differs from that given by W.E. Roth.
72 citations
TL;DR: In this paper, a new derivation of the classification of the minimal Markovian representations of the given process z is presented which is based on a certain backward filter of the innovations.
Abstract: Invariant directions of the Riccati difference equation of Kalman filtering are shown to occur in a large class of prediction problems and to be related to a certain invariant subspace of the transpose of the feedback matrix. The discrete time stochastic realization problem is studied in its deterministic as well as probabilistic aspects. In particular a new derivation of the classification of the minimal Markovian representations of the given process z is presented which is based on a certain backward filter of the innovations. For each Markovian representation which can be determined from z the space of invariant directions is decomposed into two subspaces, one on which it is possible to predict the state process without error forward in time and one on which this can be done backward in time.
52 citations
TL;DR: In this paper, the homogeneous and nonhomogeneous second order linear difference equations are studied and the existence of so-called equivalence theorems based on the coefficients is proven.
Abstract: This paper studies homogeneous and nonhomogeneous second order linear difference equations. Comparison theorems based on the coefficients are proven for the homogeneous equation. Existence of so ca...
46 citations
TL;DR: It is proven analytically that this method uniquely determines both the correct topology and root in theory for unequal rates of sequence evolution.
Abstract: Evolutionary trees are usually calculated from comparisons of protein or nucleic acid sequences from present-day organisms by use of algorithms that use only the difference matrix, where the difference matrix is constructed from the sequence differences between pairs of sequences from the organisms. The difference matrix alone cannot define uniquely the correct position of the ancestor of the present-day organisms (root of the tree). Furthermore, methods using the difference matrix alone often fail to give the correct pattern of tree branching (topology) when the different sequences evolve at different rates. Only for equal rates of evolution can the difference matrix (when used with the so-called matrix method) yield exactly the correct topology and root. In this paper we present a method for calculating evolutionary trees from sequence data that uses, along with the difference matrix, the rate of evolution of the various sequences from their common ancestor. It is proven analytically that this method uniquely determines both the correct topology and root in theory for unequal rates of sequence evolution. How one would estimate an ancestral sequence to be used in the method is discussed in particular for the 5S RNA sequences from prokaryotes and eukaryotes and for ferredoxin sequences.
44 citations
TL;DR: In this paper, a simple constructive procedure of Berkhout, based on the backwards Levinson algorithm is discussed and an application of the result in stochastic control is mentioned, where the solution to the discrete-time Lyapunov matrix equation in controllable canonical form is shown to be the inverse of the Schur-Cohn matrix.
Abstract: The solution to the discrete-time Lyapunov matrix equation in controllable canonical form is shown to be the inverse of the Schur-Cohn matrix. A simple constructive procedure of Berkhout, based on the backwards Levinson algorithm is discussed and an application of the result in stochastic control is mentioned.
TL;DR: For the matrix neutral difference-differential equation, this paper constructed a quadratic Liapunov functional which gives necessary and sufficient conditions for the asymptotic stability of the solutions of that equation.
01 Jan 1979
TL;DR: In this paper, the authors determine which solutions K to the quadratic matrix equation XBX + XA DX C = 0 are "stable" in the sense that all small changes in the coefficients of the equation produce equations some of whose solutions are close to K (in the metric determined by the operator norm).
Abstract: ABSTRACr. The authors determine which solutions K to the quadratic matrix equation XBX + XA DX C = 0 are "stable" in the sense that all small changes in the coefficients of the equation produce equations some of whose solutions are close to K (in the metric determined by the operator norm). Our main result is that a solution is stable if and only if it is an isolated solution. (The isolated solutions already have a simple characterization in terms of the coefficient matrices.) It follows that each equation has only finitely many stable solutions. Equivalently, we identify the stable invariant subspaces for an operator T on a finite-dimensional space as the isolated invariant subspaces.
TL;DR: The resulting algorithm is suitable for solving (n+1)-th order recursions (n≥1) for which there existn independent solutions that are dominated by each solution that does not belong to the space spanned by thesen solutions.
Abstract: In one of his papers [5] Gautschi presents an algorithm for determining the minimal solution of a second-order homogeneous difference equation. The method is based on the connection between the existence of a minimal solution of such a difference equation and the convergence of a certain continued fraction. In the present paper, these results are generalized. For this purpose we use the concept of generalized continued fraction. The resulting algorithm is suitable for solving (n+1)-th order recursions (n≥1) for which there existn independent solutions that are dominated by each solution that does not belong to the space spanned by thesen solutions.
TL;DR: In this article, the solution to the Riccati equation is given in terms of the partition of the transition matrix and matrix differential equations for the partition are derived and are solved using computational methods.
Abstract: This paper is concerned with the solution of the finite time Riccati equation. The solution to the Riccati equation is given in terms of the partition of the transition matrix. Matrix differential equations for the partition of the transition matrix are derived and are solved using computational methods. Examples illustrating the method are presented and the computational algorithms are given.
TL;DR: It is shown that it is always possible to achieve a lower bound, while the upper bound can be obtained for a specified class of Q matrices which is given.
Abstract: In this paper, new bounds are given for the matrix solution of the Lyapunov equation A'P+ PA+Q=0 . It is shown that it is always possible to achieve a lower bound, while the upper bound can be obtained for a specified class of Q matrices which is given. The results are compared to those given in [1] through some examples.
TL;DR: In this paper, the application of the two variable expansion method for the study of a class of non-linear difference equations arising in discrete time systems is presented, where the two-variable expansion method, a multi-time perturbational technique which has been used to a considerable extent for analysis of nonlinear differential equations, has been adopted for the investigation of the response characteristics of non linear difference equations.
01 Dec 1979
TL;DR: In this article, a representation of the solution matrix P to the Lyapunov matrix equation PF + F'P = -LL' is derived, which is a proper subset of the generalized positive real matrices defined by Anderson and Moore.
Abstract: In this paper, a representation of the solution matrix P to the Lyapunov matrix equation PF + F'P = -LL' is derived. We consider the class of m×m matrices Z(?) of real rational functions of a complex variable s, bounded at s = ?, with Z (jw) + Z' (-j?) equal to a nonnegative definite Hermitian matrix for all real ?, and with ? + µ ? 0 for all poles, not necessarily distinct, of Z(s). This last condition is imposed because (1) has a unique solution if and only if ? + µ ? 0 for all eigenvalues of the matrix F. This means that the class {z(s)} is a proper subset of the generalized positive real matrices defined by Anderson and Moore.
TL;DR: In this paper, a continous-time age-structured population growth model is given in terms of a matrix differential equation and is compared with Leslie's earlier discrete time age-structure growth model which was given by using a matrix difference equation.
TL;DR: In this paper, it was shown that certain numerical problems encountered in the solution of the stationary discrete matrix Riccati equation by the eigenvalue-eigenvector method of Vanghan [1] can be avoided by a simple reformulation.
Abstract: The purpose of this correspondence is to point out that certain numerical problems encountered in the solution of the stationary discrete matrix Riccati equation by the eigenvalue-eigenvector method of Vanghan [1] can be avoided by a simple reformulation.
TL;DR: A,Q∈Rn×n(Cn×N) as mentioned in this paper, A,Q ∈Rnn×n, Qは 正 定 対称 行 列, Aは ǫ(A)<1)
Abstract: た だ し, A,Q∈Rn×n, Qは 正 定 対称 行 列, Aは 正 則 か つ|λi(A)|<1と す る. の 解 行 列Pの 適 当 な 尺度 に よ る “木 き さ”の 限 界 を, Q,Aな ど の “大 き さ” に よ り評 価 した もの で あ る. 連続 形 の リア プ ノ フ方程 式 に関 し て は, 同 様 の 趣 旨 の 報 告 がす で に い くつ か公 表 さ れて い る1)~5). 本 報 告 の 目 的 は, 離 散形 方 程 式 に対 して も連 続 形 方 程 式 に 対 す る結 果 と並 行 した 評 価 式 を得 て お くこ とに あ る. 以 下 で は, P,Qの 固 有 根 を順 に そ れ ぞ れ, 0<α1≦ ...≦ αn, 0<β1≦... ≦βnと し, 行 列X∈Rn×n(Cn×n)に 対 し て ||X||.で,・=1,2,∞な どの 適 当 な 行 列 ノ ル ム を 示 し, μ(X)を||X||. よ り 導 か れ た 行 列 メ ジ ャ ー7)と して u.[X]を 次 式 の よ うに定 め る. μ.[X]=max{-μ.(-X),-μ(X)} ま た, 行 列Aの ス ペ ク トル 半 径 を ρ(A)で 表 わ せ ば, ρ(A)<1で あ る.
TL;DR: In this article, the nonlinear process of the two-photon absorption from two beams is treated theoretically from the quantum statistical point of view, and a master equation based on a microscopically correct hamiltonian is derived and then solved analytically to give general expressions for the time dependence of off-diagonal matrix elements of the density operator for the two light beams involved in the process.
TL;DR: In this paper, the authors proposed an algorithm for solving the TB + CT =−A matrix equation, and extended the ideas behind this algorithm to solve (∗) for R. Some suggestions for the efficient implementation of the algorithms are also given.
TL;DR: In this paper, a functional of a trial matrix and a suitable correlation matrix is constructed which is an absolute maximum when the trial matrix satisfies the Wiener-Hopf equation for the filter matrix.
Abstract: A functional of a trial matrix and suitable correlation matrix is constructed which is an absolute maximum when the trial matrix satisfies the Wiener-Hopf equation for the filter matrix. The maximum value of the functional is essentially the sum of the squares of the minimum errors of the observations and is thus of interest in its own right.
TL;DR: In this paper, a design technique for determining a suboptimal feedback control law for linear systems actuated by quantized control signals is presented, which is easily implemented in the form of the switching hyperplanes characterized by a nonlinear matrix difference equation for discrete time systems or by a differential equation for continuous time systems.
Abstract: This short paper presents a design technique for determining a suboptimal feedback control law for linear systems actuated by the quantized control signals. This control law is easily implemented in the form of the switching hyperplanes characterized by a nonlinear matrix difference equation for discrete-time systems or by a differential equation for continuous-time systems. The properties of this control law are investigated in detail.
TL;DR: In this paper, the purpose of this correspondence is to point out that relation (19) in the above paper is invalid, and therefore, the relation is invalid in the present paper.
Abstract: The purpose of this correspondence is to point out that relation (19) in the above paper is invalid.
TL;DR: In this paper, the Riccati equation is solved using a linear equation solver and the convergence speed is independent of the dimension of the matrix and unaffected by the stability or unstability of the open loop system.
TL;DR: By performing the one-sided Laplace transform on the matrix integro-differential equation for a semi-infinite plane parallel imperfect Rayleigh scattering atmosphere, the authors derived an integral equation for the emergent intensity matrix.
Abstract: By performing the one-sided Laplace transform on the matrix integro-differential equation for a semi-infinite plane parallel imperfect Rayleigh scattering atmosphere we derive an integral equation for the emergent intensity matrix. Application of the Wiener-Hopf technique to this integral equation will give the emergent intensity matrix in terms of singularH-matrix and an unknown matrix. The unknown matrix has been determined considering the boundary condition at infinity to be identical with the asymptotic solution for the intensity matrix.