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This paper describes a method to solve a system of N linear equations in N steps. A quadratic form is developed involving the sum of the squares of the residuals of the equations. Equating the quadratic form to a constant yields a surface which is an ellipsoid. For different constants, a family of similar ellipsoids can be generated. Starting at an arbitrary point an orthogonal basis is constructed and the center of the family of similar ellipsoids is found in this basis by a sequence of projections. The coordinates of the center in this basis are the solution of linear system of equations. A quadratic form in N variables requires N projections. That is, the current method is an exact method. It is shown that the sequence of projections is equivalent to a special case of the Gram-Schmidt orthogonalization process. The current method enjoys an advantage not shared by the classic Method of Conjugate Gradients. The current method can be extended to nonlinear systems without modification. For nonlinear equations the Method of Conjugate Gradients has to be augmented with a line-search procedure. Results for linear and nonlinear problems are presented.

This paper describes a method to solve a system of N linear equations in N steps. A quadratic form is developed involving the sum of the squares of the residuals of the equations. Equating the quadratic form to a constant yields a surface which is an ellipsoid. For different constants, a family of similar ellipsoids can be generated. Starting at an arbitrary point an orthogonal basis is constructed and the center of the family of similar ellipsoids is found in this basis by a sequence of projections. The coordinates of the center in this basis are the solution of linear system of equations. A quadratic form in N variables requires N projections. That is, the current method is an exact method. It is shown that the sequence of projections is equivalent to a special case of the Gram-Schmidt orthogonalization process. The current method enjoys an advantage not shared by the classic Method of Conjugate Gradients. The current method can be extended to nonlinear systems without modification. For nonlinear equations the Method of Conjugate Gradients has to be augmented with a line-search procedure. Results for linear and nonlinear problems are presented.

This paper describes a method to solve a system of N linear equations in N steps. A quadratic form is developed involving the sum of the squares of the residuals of the equations. Equating the quadratic form to a constant yields a surface which is an ellipsoid. For different constants, a family of similar ellipsoids can be generated. Starting at an arbitrary point an orthogonal basis is constructed and the center of the family of similar ellipsoids is found in this basis by a sequence of projections. The coordinates of the center in this basis are the solution of linear system of equations. A quadratic form in N variables requires N projections. That is, the current method is an exact method. It is shown that the sequence of projections is equivalent to a special case of the Gram-Schmidt orthogonalization process. The current method enjoys an advantage not shared by the classic Method of Conjugate Gradients. The current method can be extended to nonlinear systems without modification. For nonlinear equations the Method of Conjugate Gradients has to be augmented with a line-search procedure. Results for linear and nonlinear problems are presented.

This paper describes a method to solve a system of N linear equations in N steps. A quadratic form is developed involving the sum of the squares of the residuals of the equations. Equating the quadratic form to a constant yields a surface which is an ellipsoid. For different constants, a family of similar ellipsoids can be generated. Starting at an arbitrary point an orthogonal basis is constructed and the center of the family of similar ellipsoids is found in this basis by a sequence of projections. The coordinates of the center in this basis are the solution of linear system of equations. A quadratic form in N variables requires N projections. That is, the current method is an exact method. It is shown that the sequence of projections is equivalent to a special case of the Gram-Schmidt orthogonalization process. The current method enjoys an advantage not shared by the classic Method of Conjugate Gradients. The current method can be extended to nonlinear systems without modification. For nonlinear equations the Method of Conjugate Gradients has to be augmented with a line-search procedure. Results for linear and nonlinear problems are presented.

This paper describes a method to solve a system of N linear equations in N steps. A quadratic form is developed involving the sum of the squares of the residuals of the equations. Equating the quadratic form to a constant yields a surface which is an ellipsoid. For different constants, a family of similar ellipsoids can be generated. Starting at an arbitrary point an orthogonal basis is constructed and the center of the family of similar ellipsoids is found in this basis by a sequence of projections. The coordinates of the center in this basis are the solution of linear system of equations. A quadratic form in N variables requires N projections. That is, the current method is an exact method. It is shown that the sequence of projections is equivalent to a special case of the Gram-Schmidt orthogonalization process. The current method enjoys an advantage not shared by the classic Method of Conjugate Gradients. The current method can be extended to nonlinear systems without modification. For nonlinear equations the Method of Conjugate Gradients has to be augmented with a line-search procedure. Results for linear and nonlinear problems are presented.

The problem of solving banded linear systems by direct (non-iterative) techniques on the Vector Processor System (VPS) 32 supercomputer is considered. Two efficient direct methods for solving banded linear systems on the VPS 32 are described. The vector cyclic reduction (VCR) algorithm is discussed in detail. The performance of the VCR on a three parameter model problem is also illustrated. The VCR is an adaptation of the conventional point cyclic reduction algorithm. The second direct method is the Customized Reduction of Augmented Triangles' (CRAT). CRAT has the dominant characteristics of an efficient VPS 32 algorithm. CRAT is tailored to the pipeline architecture of the VPS 32 and as a consequence the algorithm is implicitly vectorizable.

A hierarchically-structured, suboptimal controller for a linear stochastic system composed of fast and slow subsystems is considered. The controller is optimal in the limit as the separation of time scales of the subsystems becomes infinite. The methodology is illustrated by design of a controller to suppress the phugoid and short period modes of the longitudinal dynamics of the F-8 aircraft.

From the March 1976 edition of Electronics Australia. 

This paper introduces some cheaper and faster variants of the classical additive Schwarz preconditioner (AS) for general sparse linear systems. Using numerical examples, the paper shows that the new methods are superior to AS in terms of both iteration counts and CPU time, as well as the communication cost when implemented on distributed memory computers. This is especially true for harder problems such as indefinite complex linear systems and systems of convection-diffusion equations from three-dimensional compressible flows. Both sequential and parallel results are reported.

Flexible structures containing a large number of modes can benefit from adaptive control techniques which are well suited to applications that have unknown modeling parameters and poorly known operating conditions. In this paper, we focus on a direct adaptive control approach that has been extended to handle adaptive rejection of persistent disturbances. We extend our adaptive control theory to accommodate troublesome modal subsystems of a plant that might inhibit the adaptive controller. In some cases the plant does not satisfy the requirements of Almost Strict Positive Realness. Instead, there maybe be a modal subsystem that inhibits this property. This section will present new results for our adaptive control theory. We will modify the adaptive controller with a Residual Mode Filter (RMF) to compensate for the troublesome modal subsystem, or the Q modes. Here we present the theory for adaptive controllers modified by RMFs, with attention to the issue of disturbances propagating through the Q modes. We apply the theoretical results to a flexible structure example to illustrate the behavior with and without the residual mode filter.

Systems Of Homogeneous Linear Equations Linear Albegra

Systems Of Homogeneous Linear Equations Linear Albegra

Thesis (MS)--U.S. Naval Postgraduate School, 1960

Damage sustained by equipment used in a moving carrier suddenly changing its velocity can sometimes be attributed to inadequate protective cushioning. Proper design of the protective cushioning can be realized by reducing the equipment and components to idealized mechanical systems and studying the responses as the system parameters are varied. This study considers the response to velocity shock with elastic impact. A set of design curves showing maximum acceleration of a flexible and vulnerable element of an equipment and the maximum deflection of the equipment's cushioning are produced using an analog computer. Additionally the computer's results are compared with analytical results for a few particular cases. The writer wishes to express his appreciation for the advice and assistance of Professor R. E. Newton in the selection and performance of this study. The writer wishes to express his appreciation for the advice and assistance of Professor R. E. Newton in the selection and performance of this study.

Damage sustained by equipment used in a moving carrier suddenly changing its velocity can sometimes be attributed to inadequate protective cushioning. Proper design of the protective cushioning can be realized by reducing the equipment and components to idealized mechanical systems and studying the responses as the system parameters are varied. This study considers the response to velocity shock with elastic impact. A set of design curves showing maximum acceleration of a flexible and vulnerable element of an equipment and the maximum deflection of the equipment's cushioning are produced using an analog computer. Additionally the computer's results are compared with analytical results for a few particular cases. The writer wishes to express his appreciation for the advice and assistance of Professor R. E. Newton in the selection and performance of this study. The writer wishes to express his appreciation for the advice and assistance of Professor R. E. Newton in the selection and performance of this study.

Takens' Embedding Theorem remarkably established that concatenating M previous outputs of a dynamical system into a vector (called a delay coordinate map) can be a one-to-one mapping of a low-dimensional attractor from the system state space. However, Takens' theorem is fragile in the sense that even small imperfections can induce arbitrarily large errors in this attractor representation. We extend Takens' result to establish deterministic, explicit and non-asymptotic sufficient conditions for a delay coordinate map to form a stable embedding in the restricted case of linear dynamical systems and observation functions. Our work is inspired by the field of Compressive Sensing (CS), where results guarantee that low-dimensional signal families can be robustly reconstructed if they are stably embedded by a measurement operator. However, in contrast to typical CS results, i) our sufficient conditions are independent of the size of the ambient state space, and ii) some system and measurement pairs have fundamental limits on the conditioning of the embedding (i.e., how close it is to an isometry), meaning that further measurements beyond some point add no further significant value. We use several simple simulations to explore the conditions of the main results, including the tightness of the bounds and the convergence speed of the stable embedding. We also present an example task of estimating the attractor dimension from time-series data to highlight the value of stable embeddings over traditional Takens' embeddings.

We improve on Jarn\'{\i}k's inequality between uniform Diophantine exponent $\alpha $ and ordinary Diophantine exponent $\beta$ for a system of $ n\ge 2$ real linear forms in two integer variables. Jarn\'{\i}k (1949, 1954) proved that $\beta \ge \alpha (\alpha -1)$. In the present paper we give a better bound in the case $\alpha >1$. We prove that \beta \ge 1/2(\alpha^2-\alpha+1+\sqrt{(\alpha^2-\alpha+1)^2 +4\alpha^2(\alpha-1)}) if 1\le \alpha \le 2 1/2(\alpha^2-1+\sqrt{(\alpha^2-1)^2+4\alpha (\alpha-1)}) if \alpha \ge 2

Identification of fractional order systems is considered from an algebraic point of view. It allows for a simultaneous estimation of model parameters and fractional (or integer) orders from input and output data. It is exact in that no approximations are required. Using Mikusinski's operational calculus, algebraic manipulations are performed on the operational representation of the system. The unknown parameters and (fractional) orders are calculated solely from convolutions of known signals. A generalized Voigt model describing a viscoelastic material is used to illustrate the approach.

We consider the optimal control design problem for discrete-time LTI systems with state feedback, when the actuation signal is subject to unmeasurable switching propagation delays, due to e.g. the routing in a multi-hop communication network and/or jitter. In particular, we set up a constrained optimization problem where the cost function is the worst-case $\mathcal{L}_2$ norm for all admissible switching delays. We first show how to model these systems as pure switching linear systems, and as main contribution of the paper we provide an algorithm to compute a sub-optimal solution.

In this text I present a couple of new principles and thereon based iterative methods for numerical solution of sequences of systems of linear equations with fixed system matrix and changing right-hand-sides. The use of the new methods is to recycle all subspace information that is obtained anyway in the solution process of a former system, to solve subsequent systems. All these principles and methods are based on short recurrences and small storage requirements. The principles are based on the IDR-theorem and the Horner scheme for polynomials.

This paper proposes a new approach to analyze and synthesize robust consensus control laws for general linear leaderless multi-agent systems (MASs) subjected to input constraints or uncertainties. First, the MAS under input constraints or uncertainties is reformulated as a network of Lur'e systems. Next, two scenarios of communication topology are considered, namely undirected and directed cyclic structures. In each case, a sufficient condition for consensus and the design of consensus controller gain are derived from solutions of a distributed LMI convex problem. Finally, a numerical example is introduced to illustrate the effectiveness of the proposed theoretical approach.

We calculate the probability that random polynomial matrices over a finite field with certain structures are right prime or left prime, respectively. In particular, we give an asymptotic formula for the probability that finitely many nonsingular polynomial matrices are mutually left coprime. These results are used to estimate the number of reachable and observable linear systems as well as the number of non-catastrophic convolutional codes. Moreover, we are able to achieve an asymptotic formula for the probability that a parallel connected linear system is reachable.

A new functional-based approach is developed for the stability analysis of linear impulsive systems. The new method, which introduces looped-functionals, considers non-monotonic Lyapunov functions and leads to LMIs conditions devoid of exponential terms. This allows one to easily formulate dwell-times results, for both certain and uncertain systems. It is also shown that this approach may be applied to a wider class of impulsive systems than existing methods. Some examples, notably on sampled-data systems, illustrate the efficiency of the approach.

Performance measures and associated criteria for linear constant coefficient systems forced by deterministic inputs are investigated, with particular reference to flight control systems. It is shown that the application of performance measures is facilitated by substituting for the actual flight control system an equivalent loworder linearized system having similar dynamic characteristics. A critical survey of current performance measures is given, and new methods for the analytic evaluation of some indicial error measures are presented.

This report involves the application of ideas in adaptive stochastic control to economics. We investigate the control problem for a linear, multivariable, dynamic system with purely random (i.e. white) parameters. The quadratic cost criterion is formulated to make the problem a tracking problem. Since the parameters are modelled as white stochastic processes, there is no posterior learning and no dual effect. The certainty-equivalence principle does not hold. We find that the extension of the Uncertainty Threshold Principle from scalar systems to multidimensional ones turns out to be analytically intractable. Next, we derive sensitivity equations for the above optimal system to study the effects of small variations in parameter uncertainties on the optimal performance of the system. These equations enable us to rank parameters in order of the sensitivity of the performance to variations in their variances. This makes it possible to locate the 'pressure' points in a model, if any exist. We then convert an economic policy problem into a stochastic optimal control tracking problem and analyse it with the equations we have derived. We study the different elements that enter into a tracking problem and then discuss the empirical results obtained from the sensitivity equations. The model we choose for the analysis turns out to be insensitive to variations in parameter variances which makes it reasonably reliable. We also analyse in detail the structure of the model and the interdependences of the state and control variables.

A simple approach is given to determine conditions for stability of linear feedback systems subject to additive and multiplicative perturbations in the operators describing these systems. The approach is based on techniques used in functional analysis, and provides an alternative development and generalization of some conditions for the time-invariant case that have appeared in the literature very recently. As an example of the application of the conditions, we consider the determination of finite regions of stability for singularly perturbed systems. (Author)

Free viscously damped vibrations of linear discrete structural systems are studied. The amount of damping varies among the various structural elements of the system resulting in several critical damping possibilities. A general method is developed for determining the critical damping surfaces of a system. These surfaces represent the loci of combinations of damping values corresponding to critically damped motions, and thus separate regions of partial or complete underdamping for those of overdamping. The dimension of a critical damping surface is equal to the number of independent amounts of damping present in the system. The determination of the surface point corresponding to equal amounts of damping is considerably simplified for systems which, on the assumption that all amounts of damping are equal, possess a damping matrix of the Rayleigh type. Three examples presented in detail illustrate the proposed technique and some of the important characteristics of critical damping surfaces. (Author)

The objective of this report is to survey many of the popular methods for the solution of large matrix equations with the hope of finding an efficient method suitable for both electromagnetic scattering and radiation problems and system identification problems.

This paper reports on recent work on time scale decomposition and aggregation of large-scale linear systems containing weak couplings and finite-state Markov processes containing rare transitions. This work builds on that of Coderch, et. al.. The focus of the work is on the asymptotic approximatin of the linear system. (Author)

The purpose of this application was to create a supplement to an undergraduate course in linear algebra. This application was drawn from the field physics and shows how linear algebra is used to solve systems of second order linear differential equations, which could be used to model small vibrations in molecules. This applications was designed so that an instructor of linear algebra could use it either as an independent study project or as an integrated part of the course.

The purpose of this application was to create a supplement to an undergraduate course in linear algebra. This application was drawn from the field physics and shows how linear algebra is used to solve systems of second order linear differential equations, which could be used to model small vibrations in molecules. This applications was designed so that an instructor of linear algebra could use it either as an independent study project or as an integrated part of the course.

A general description of linear, switched-parameter systems is developed in terms of the mathematics of state variables. The restrictions imposed by the grouping of state variables and by' the \"core\" states of the system upon the coefficient matrices are considered. Two procedures for determining the element values of those matrices are each illustrated with an example. A general expression for a Cost function to be used to measure system quality is developed and illustrated with two examples. Extensive recommendations for future work are made. Several examples of the utility of a cost function minimization technique for the improvement of the step responses of some switched parameter systems are presented.

A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and $H$ an ample divisor of square 2 for Beauville's quadratic form. If $n=2$ then according to the conjecture $X$ is a double cover of a (singular) sextic 4-fold in $\PP^5$. It follows from the conjecture that a deformation of $(K3)^{[n]}$ carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture. In doing so we bump into some interesting geometry: examples of two anti-symplectic involutions generating an interesting dynamical system, a case of Strange duality and what is probably an involution on the moduli space of degree-2 quasi-polarized $(X,H)$ where $X$ is a deformation of $(K3)^{[2]}$.

We investigate the propagation of random fluctuations through biochemical networks in which the concentrations of species are large enough so that the unperturbed problem is well-described by ordinary differential equation. We characterize the behavior of variance as fluctuations propagate down chains, study the effect of side chains and feedback loops, and investigate the asymptotic behavior as one rate constant gets large. We also describe how the ideas can be applied to the study of methionine metabolism.

We study different extended formulations for the set $X = \{x\in\mathbb{Z}^n \mid Ax = Ax^0\}$ in order to tackle the feasibility problem for the set $X_+=X \cap \mathbb{Z}^n_+$. Here the goal is not to find an improved polyhedral relaxation of conv$(X_+)$, but rather to reformulate in such a way that the new variables introduced provide good branching directions, and in certain circumstances permit one to deduce rapidly that the instance is infeasible. For the case that $A$ has one row $a$ we analyze the reformulations in more detail. In particular, we determine the integer width of the extended formulations in the direction of the last coordinate, and derive a lower bound on the Frobenius number of $a$. We also suggest how a decomposition of the vector $a$ can be obtained that will provide a useful extended formulation. Our theoretical results are accompanied by a small computational study.

The spinor representation of the Lorentz group does not accept simple generalization with the group GL(4,R) of general linear coordinate transformations. The Dirac equation may be written for an arbitrary choice of a coordinate system and a metric, but the covariant linear transformations of the four-component Dirac spinor exist only for isometries. For usual diagonal Minkowski metric the isometry is the Lorentz transformation. On the other hand, it is possible to define the Dirac operator on the space of anti-symmetric (exterior) forms, and in such a case the equation is covariant for an arbitrary general linear transformation. The space of the exterior forms is 16-dimensional, but usual Dirac equation is defined for four-dimensional complex space of Dirac spinors. Using suggested analogy, in present paper is discussed possibility to consider the space of Dirac spinors as some "subsystem" of a bigger space, where the group GL(4,R) of general relativity acts in a covariant way. For such purposes in this article is considered both Grassmann algebra of complex anti-symmetric forms and Clifford algebra of Dirac matrices. Both algebras have same dimension as linear spaces, but different structure of multiplication. The underlying sixteen-dimensional linear space also may be considered either as space of complex 4 x 4 matrices, or as space of states of two particles: the initial Dirac spinor and some auxiliary system. It is shown also, that such approach is in good agreement with well known idea to consider Dirac spinor as some ideal of Clifford algebra. Some other possible implications of given model are also discussed.

Large sparse linear systems occur in many scientific and engineering applications encountered in military and civilian domains. Such systems are typically solved using either iterative or direct methods. We are developing parallel formulations of computationally intensive algorithms that underly these methods. Direct methods for solving sparse linear systems are important because of their generality and robustness. For linear systems arising in certain applications, such as linear programming and some structural engineering applications, they are the only feasible methods. Although highly parallel formulations of dense matrix factorization are well known, it has been a challenge to implement efficient sparse linear system solvers using direct methods, even on moderately parallel computers. We have recently achieved a breakthrough in developing a highly parallel sparse Cholesky factorization algorithm that substantially improves the state of the art in parallel direct solution of sparse linear systems-both in terms of scalability and overall performance. Experiments have shown that this algorithm can easily speedup Cholesky factorization by a factor of at least a few hundred up to 1024 processors, and achieve levels of performance that were unheard of and unimaginable for this problem until very recently.

A guidance law for the control of a system in the neighborhood of a nominal suboptimal trajectory is developed. The guidance law is demonstrated using a lunar launch problem with constraints at orbit entry. A set of precomputed gains is used by the guidance law to operate on an extremal path in the neighborhood of the suboptimal trajectory. The guidance law and gains are designed to minimize the change in the desired performance index while still satisfying the final path constraints. In the lunar launch problem, the nominal suboptimal trajectory minimizes the final time using piecewise linear control. This trajectory is obtained to provide a nominal control history. The guidance law is found by minimizing the second variation of the suboptimal trajectory performance index subject to the final constraints being satisfied. For the lunar launch problem, the guidance law leads to a set of gains that relates deviations from the suboptimal trajectory to required changes in the nominal control history. The deviations from the suboptimal trajectory, used together with the precomputed gains, determines the change in the nominal control history required to meet the final constraints while minimizing the change in the final time.

Main Result: Let $(M,L)$ be a smooth complex polarized threefold. Then the linear system $| K+tL|$ separates any two different points on $M$ for any $t\ge 6$, where $K$ is the canonical bundle of $M$. The argument in the proof is a variant of Ein-Lazarsfeld method. Although it is less powerful, it is cheaper, i.e., needs fewer pages. Unfortunately, at present, I cannot show the very ampleness because of technical difficulties. Some related topics are also discussed. A hard copy is available on request to the author.

We consider {\it small solutions} of a vibrating system with smooth non-linearities for which we provide an approximate solution by using a double scale analysis; a rigorous proof of convergence of a double scale expansion is included; for the forced response, a stability result is needed in order to prove convergence in a neighbourhood of a primary resonance.

In this paper we propose an asymptotic preserving scheme for a family of Friedrichs systems on unstructured meshes based on a decomposition between the hyperbolic heat equation and a linear hyperbolic which not involved in the di usive regime. For the hyperbolic heat equation we use asymptotic preserving schemes recently designed previously. To discretize the second part we use classical Rusanov or upwind schemes. To nish we apply this method for the discretization of the PN and SN models which are widely used in transport codes.

This research investigated the application of a linear systems model to two parameters of vision: the contrast threshold and the spatial frequency difference threshold. The contrast threshold is the contrast of a target when an observer is just able to detect its presence. The spatial frequency difference threshold is the smallest difference in spatial frequency that permits two grating targets to be distinguished. The model investigated was for observation with a fixed visual field size of one-dimensional, time-invariant sinusoidal grating targets. A mathematical development indicated that a linear model could be used to represent the human spatial frequency difference threshold function. The model was implemented using an electro-optical hardware system which consisted of a charge-coupled device video camera, a frame grabber, and a personal computer. Because of its similarities to the structure of the eye, linear response, and ability to acquire digital image data, a charge coupled device array video camera and frame grabber were used to simulate the eye. The action of the neural pathways and visual cortex was simulated by Fourier transform computations on the camera and frame grabber output. The experimental data from the electro-optical hardware system agreed with the theoretical models for both parameters.

The convergence of solutions to the discrete or sampled time linear quadratic regulator problem and associated Riccati equation for infinite dimensional systems to the solutions to the corresponding continuous time problem and equation, as the length of the sampling interval (the sampling rate) tends toward zero (infinity) is established. Both the finite and infinite time horizon problems are studied. In the finite time horizon case, strong continuity of the operators which define the control system and performance index together with a stability and consistency condition on the sampling scheme are required. For the infinite time horizon problem, in addition, the sampled systems must be stabilizable and detectable, uniformly with respect to the sampling rate. Classes of systems for which this condition can be verified are discussed. Results of numerical studies involving the control of a heat/diffusion equation, a hereditary of delay system, and a flexible beam are presented and discussed.

We present a stability analysis framework for the general class of discrete-time linear switching systems for which the switching sequences belong to a regular language. They admit arbitrary switching systems as special cases. Using recent results of X. Dai on the asymptotic growth rate of such systems, we introduce the concept of multinorm as an algebraic tool for stability analysis. We conjugate this tool with two families of multiple quadratic Lyapunov functions, parameterized by an integer T >= 1, and obtain converse Lyapunov Theorems for each. Lyapunov functions of the first family associate one quadratic form per state of the automaton defining the switching sequences. They are made to decrease after every T successive time steps. The second family is made of the path-dependent Lyapunov functions of Lee and Dullerud. They are parameterized by an amount of memory (T-1) >= 0. Our converse Lyapunov theorems are finite. More precisely, we give sufficient conditions on the asymptotic growth rate of a stable system under which one can compute an integer parameter T >= 1 for which both types of Lyapunov functions exist. As a corollary of our results, we formulate an arbitrary accurate approximation scheme for estimating the asymptotic growth rate of switching systems with constrained switching sequences.

Research has been conducted in two major problem areas: Applications of Random Differential Equations to Engineering Science. Wave Propagation in Turbulent Media and Random Linear Hyperbolic Systems. These two area are not disjoint. In fact the second problem area is a follow-up study on problems in the first one with emphasis on wave propagation in turbulent media. We have investigated several turbulence-related problems arising from engineering science. They include problems in wave propagation through turbulent media, turbulent transport theory, the surface roughness effect on hydrodynamic lubrication and wave scattering, and stability of elastic structures under random loading or with imperfections. In the theoretical aspect, the research is mainly concerned with the development of new methodology in solving differential equations with random coefficients, examination of the existing closure approximations with regards to their validity and possible improvements. Certain related mathematical questions are also studied.

In this paper, we present a class of iterative descent methods for solving large, sparse, nonsymmetric systems of linear equations whose coefficient matrices have positive-definite symmetric parts. Such problems commonly arise from the discretization of non-self-adjoint elliptic partial differential equations. The methods we consider are modelled after the conjugate gradient method. They require no estimation of parameters and their rate of convergence appears to depend on the spectrum of A rather than ATA. Their convergence can also be accelerated by preconditioning techniques.

In this paper, we present a class of iterative descent methods for solving large, sparse, nonsymmetric systems of linear equations whose coefficient matrices have positive-definite symmetric parts. Such problems commonly arise from the discretization of non-self-adjoint elliptic partial differential equations. The methods we consider are modelled after the conjugate gradient method. They require no estimation of parameters and their rate of convergence appears to depend on the spectrum of A rather than ATA. Their convergence can also be accelerated by preconditioning techniques.

In this paper, we present a class of iterative descent methods for solving large, sparse, nonsymmetric systems of linear equations whose coefficient matrices have positive-definite symmetric parts. Such problems commonly arise from the discretization of non-self-adjoint elliptic partial differential equations. The methods we consider are modelled after the conjugate gradient method. They require no estimation of parameters and their rate of convergence appears to depend on the spectrum of A rather than ATA. Their convergence can also be accelerated by preconditioning techniques.

The solution of a linear system of equations Ax=b is often computed using variations of Gaussian elimination. A major problem with these methods is the large storage requirement to compute the factorization of matrices that arise in practice. We focus on the Cholesky method for factoring banded, symmetric, positive definite matrices, which arise in finite-difference and finite-element simulations. We develop and analyze methods for using secondary storage in cases where there is not enough primary memory to compute the band Cholesky factorization. Many computer configurations used for scientific computation allow for control over the parallel execution of I/O and computation. For several of the secondary storage methods, we present storage and I/O schemes that allow I/O to be overlapped with computation. We derive conditions under which the factorization is compute-bound, i.e., all I/O between the initial input and the final output is completely hidden behind concurrent computation. Further, we show that the amount of memory needed to achieve compute-boundedness is independent of the size and bandwidth of the matrix. Thus, for a given processor speed and I/O rate, a constant amount of memory is sufficient to keep the processor busy during the band Cholesky factorization.