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Matrices with dominant diagonal blocks and economic theory
Abstract
In this paper we shall define matrices with quasi-strictly-dominant diagonal blocks or quasi-dominant diagonal blocks on the basis of general matrix norms including the spectral norm. Several theorems on such matrices are derived and a simple numerical example and economic models are considered to apply our theorems.
... The linear weakly coupled systems were introduced to the control audience in [1] and since then have been studied by many control researchers (see [2] and [3] and references therein). In addition, the weakly coupled systems have been studied in mathematics [5]- [7], economics [8], [9], and power system engineering [10]- [12] under the name of block diagonally dominant matrices and block diagonally dominant systems. In addition, weak coupling linear structures also appear in nearly completely decomposable continuous-and discrete-time Markov chains [13]- [15]. ...
... The main problem that we are faced with is the solution of the system of algebraic equations (8). This system has the form ...
In this paper a novel transformation is introduced for block-diagonalization of weakly coupled linear systems composed of N subsystems. The block-diagonalization transformation matrix is obtained by successive solution of reduced-order nonsquare, nonsymmetric, algebraic Riccati equations. The nonsquare, nonsymmetric, algebraic Riccati equations can be efficiently solved by iterative methods.
... The linear weakly coupled systems were introduced to the control audience in [1] and since then have been studied by many control researchers (see [2] and [3] and references therein). In addition, the weakly coupled systems have been studied in mathematics [5]- [7], economics [8], [9], and power system engineering [10]- [12] under the name of block diagonally dominant matrices and block diagonally dominant systems. In addition, weak coupling linear structures also appear in nearly completely decomposable continuous-and discrete-time Markov chains [13]- [15]. ...
... The main problem that we are faced with is the solution of the system of algebraic equations (8). This system has the form ...
A transformation is introduced for exact decomposition
(block-diagonalization) of linear weakly coupled systems composed of N
subsystems. This transformation can also be used for block
diagonalization of block-diagonally dominant matrices and, under certain
assumptions, it can be applied for block diagonalization of nearly
completely decomposable Markov chains. A twelfth-order real-world power
system example is included to demonstrate the efficiency of the proposed
method
... He showed that H is non-singular. Okuguchi (1976) modi…ed the condition in (1) and showed that if there exist d j > 0 such that d j jh jj j > P i6 =j d i jh ij j, where j = 1; :::; n, then H has a quasi-dominant diagonal and is nonsingular (See also Pearce, 1974;Okuguchi, 1978). ...
We show that in an tridiagonal matrix that has a positive dominant diagonal and negative super- and sub-diagonals, there exists a strictly positive that satisfies the system , and . Furthermore, if is symmetric, the components of can be ranked under certain conditions. We apply these results to characterize the comparative-statics properties in an optimization problem.
... Weakly coupled linear systems have been studied extensively because their introduction to the control systems community by Kokotovic et al. [2] (see also, for example, [3,4] and the references therein). Those systems have also been studied in mathematics [5,6], economics [7], power system engineering [8][9][10], and in nearly complete decomposable continuous-time and discrete-time Markov chains [11][12][13]. ...
This paper proposes a new approximate dynamic programming algorithm to solve the infinite-horizon optimal control problem for weakly coupled nonlinear systems. The algorithm is implemented as a three-critic/four-actor approximators structure, where the critic approximators are used to learn the optimal costs, while the actor approximators are used to learn the optimal control policies. Simultaneous continuous-time adaptation of both critic and actor approximators is implemented, a method commonly known as synchronous policy iteration. The adaptive control nature of the algorithm requires a persistence of excitation condition to be a priori guaranteed, but this can be relaxed by using previously stored data concurrently with current data in the update of the critic approximators. Appropriate robustifying terms are added to the controllers to eliminate the effects of the residual errors, leading to asymptotic stability of the equilibrium point of the closed-loop system. Simulation results show the effectiveness of the proposed approach for a sixth-order dynamical example. Copyright
... Relationship (6.11) and other versions of the "revealed preference" condition [35,38,24] (see also [8, p. 138]) lead to convexity of the set of equilibrium prices or to uniqueness of equilibrium, and also to stability of the processes (6.1) (for ~=~) ). Another condition, more general than strict g.s. in the smooth case, stipulates that the Jacobian matrix of the excess demand has a dominant diagonal at each point [63,72] (see also [i0, 36]). This condition also ensures uniqueness and stability of equilibrium. ...
A survey of the results relating to the application of gross substitutability in economic equilibrium theory. The topics considered include existence and uniqueness of equilibrium, comparative statics, coalition stability, and stability of price-adjustment tatonnement processes. The main theorems cover the case of multivalued demand satisfying the gross substitutability condition and, in particular, are applicable to linear exchange models.
This chapter briefly explains life and works of Professor Koji Okuguchi. This chapter also contains a list of his selected publications.
The main objective of this paper is to formulate a generalization of block diagonal dominance, which can be used to establish nonsingularity of matrices via overlapping diagonal blocks. A number of stability results are derived in the new setting by exploiting the well-known M-matrix properties, as well as extensions of the normalization, scaling, and alternative norm utilization. A link between generalized block diagonal dominance and vector Liapunov functions is established, which can be applied in the stability analysis of interconnected dynamic systems.
Some simple tests for the stability properties of dynamic systems represented by differential or difference equations are given. The aim is to make such tests accessible to biologists, economists, etc. for whom the stability properties of social and physical systems are frequently of great interest. The paper is in the nature of a survey and the exposition has deliberately been kept simple.
This paper shows the following properties for the waveform relaxation-Newton method: (1) the iterative solution converges uniformly and globally under the same condition as the convergence condition for the waveform relaxation method; (2) the waveform Newton method used in the inner loop of the iteration converges locally with the second-order; (3) the waveform relaxation-Newton method is slower in the convergence both globally and locally than the waveform relaxation method; (4) the time interval where the error monotone decreases is shorter in the waveform relaxation-Newton method than in the waveform relaxation method; and (5) the convergence speed asymptotically approaches that of the waveform relaxation method.
We provide a short review of the literature on design of decentralized control based on weak coupling measures for transfer function models. The notion of diagonal dominance for transfer functions has been suggested as a measure of weak coupling for system decomposition. Various generalizations of this notion for partitioned transfer function matrices are discussed as they pertain to design of decentralized control. Such weak coupling techniques permit a decentralized design procedure based on an approximate decoupled model. The accuracy of these decoupled approximations for control design is highlighted.
This paper considers the convergence problem of parallel asynchronous block-iterative computation schemes. A new mathematical state-space model for a class of nonlinear time-varying parallel iterative schemes is proposed. Using this model, which generalizes several models of the Chazan-Miranker type, together with large-scale systems and Liapunov techniques, it is shown that the well-known quasidominance condition on a certain aggregated matrix guarantees exponential convergence of this class of methods.
In this note we shall present some sufficient conditions for D-stability on the basis of fundamental property of matrices with quasi-dominant diagonal blocks and, as an economic application, the tâtonnement process with adaptive expectations [Arrow and Nerlove (1958)] is shown to be totally stable in the small.
Introduces an electrical power network decomposition methodology
for the network equation solution by block-iterative methods implemented
on parallel computers. The proposed methodology is based on the building
up of a fixed number of subnetworks by aggregating network nodes to a
given number of so-called seed nodes. The aggregation method is
controlled by a node ranking criterion based on the weights assigned to
the network nodes reflecting the magnitude of their electrical coupling.
The resulting decompositions are usually conducive to well-conditioned
iterative processes, with acceptable computational load balancing, and
relatively low communication requirements for the parallel solution of
the network equations. Results of computational tests performed on
medium-size real networks are presented
The purpose of this paper is to present a generalization of the Nyquist array method to blockwise decompositions, which is based upon a new version of block diagonal dominance. An additional flexibility of the proposed method is in partitioning of the system matrices into disjoint as well as overlapping submatrices, which increases considerably the class of control systems which can be designed via block diagonal dominance. Within this framework, the controllers for individual subsystems can be designed independently of each other, so that their union represents a decentralized controller for the overall system.
In mathematical economics we appeal frequently to the theory of matrices with dominating diagonal elements. This paper generalizes that theory to the case of matrices with dominating diagonal submatrices which appear to possess the same fundamental properties. Matrix dominance is defined, and its connection with “bigness” of a matrix is established. Quick checks for dominance are developed. The standard theorems associated with dominating diagonal matrices are shown to be special cases of more general results. Examples are presented to illustrate the wide range of possible application.