Sparse systems

Freed and Jortner226 have reworked the formal theory of radiationless transitions described in this paper. They carefully account for the difference between distinguishable and indistinguishable levels, and allow for variable coupling of the sparse system to the dense system of states. Of course, only certain vibrational modes in the dense manifold have the appropriate symmetries to couple to the sparse manifold and thereby contribute to the radiationless transition. Freed and Jortner take this into account in the fashion in which the zero-order manifolds of the molecule are classified. 

Minimizing Round-Off Error in Direct Solution of Large Sparse Systems of Linear Equations... 

Sparse systems encountered in practical problems are characterized by a very large value of n (several hundreds) and the number of non-zero elements in matrix A is small, usually less than 5%. When solving such a system, advantage of its sparseness can be taken in two ways 1. By storage of only the nonzero elements of A and 2. By making arithmetical operations with these elements only. 

The solution of a sparse system of equations can be carried out in three stages 1. Partitioning, 2. Reordering or "tearing", and 3. Numerical solution. Stages 1 and 2 contain only logical operations and their objective is to obtain a system which can be solved faster and/or with smaller round-off error propagated. 

The improved algorithm for solving sparse systems of linear equations follows ... 

The round-off error propagation associated with the use of Shacham and Kehat s direct method for the solution of large sparse systems of linear equations is investigated. A reordering scheme for reducing error propagation is proposed as well as a method for iterative refinement of the solution. Accurate solutions for linear systems, which contain up to 500 equations, have been obtained using the proposed method, in very short computer times. 

Shacham, M. Kehat, E., "A Direct Method for the Solution of Large Sparse Systems of Linear Equations", Comp. J., 1976, 1 ( 0, 353. 

The new coefficient matrix is symmetric as M lA can be written as M 1/2AM 1/Z. Preconditioning aims to produce a more clustered eigenvalue structure for M A and/or lower condition number than for A to improve the relevant convergence ratio however, preconditioning also adds to the computational effort by requiring that a linear system involving M (namely, Mz = r) be solved at every step. Thus, it is essential for efficiency of the method that M be factored very rapidly in relation to the original A. This can be achieved, for example, if M is a sparse component of the dense A. Whereas the solution of an n X n dense linear system requires order of 3 operations, the work for sparse systems can be as low as order n.13-14... 

For stiff differential equations, the backward difference algorithm should be preferred to the Adams-Moulton method. The well-known code LSODE with different options was published in 1980 s by Flindmarsh for the solution of stiff differential equations with linear multistep methods. The code is very efficient, and different variations of it have been developed, for instance, a version for sparse systems (LSODEs). In the international mathematical and statistical library, the code of Hindmarsh is called IVPAG and DIVPAG. 

In Equation-Oriented (EO) approach all the modelling equations are assembled in a large sparse system producing Non-linear Algebraic Equations (NAE) in steady state simulation, and stiff Differential Algebraic Equations (DAE) in dynamic simulation. Thus, the solution is obtained by solving simultaneously all the modelling equations. 

In this chapter, we discuss the opposition effects for sparse media and closely packed systems of particles. Section 2 summarizes the basic equations describing the scattering of light by systems of spherical particles. Specific differences in the description of light scattering by closely packed and by sparse systems of particles are discussed. In section 3, equations for the reflection matrix of a layer of sparse medium are given. Numerical examples illustrate considerable dependence of the opposition effects on microscopic... 

An example of the power of this method in structural elucidation is shown for the sesquiterpene lactone acetyhsomontanohde (Fig. 5.77). The complete carbon skeleton of the molecule can be traced directly in the spectrum, with breakdown occurring only when heteroatom linkages arise. Unfortunately, it is rare indeed to have sufficient sample quantities at hand to consider employing this technique, and still it remains of limited application in its original form. Nevertheless, in very proton-sparse systems, it may present the only option as in the identification of the nitration product of 5.11a. The product was to... 

Multicommand operation in a sparse system The operator makes several stops (more than two) in a system but considerably fewer than the number of pick aisles. For a single block of aisles, called a ladder structure (see Figure 7), an adaptation of the traveling salesman problem may be applied to the routing (Ratliff and Rosenthal 1983). When there are cross-aisles or multiple blocks of aisle, heuristic algorithms are available (Kees 2000). 

Literature-based examples and industrial case studies are collected in Chapter 5. Implementation tricks and useful functions to handle very large and sparse systems with/without parallel computing are introduced. 

A novel approach was recently introduced to the BzzMath library (Manenti et al., 2009). A new DAE solver was created for very sparse systems corrupted by a few unstructured elements, as they typically occur in the process control field. The implementation of this algorithm is now part of... 

In the process of Gaussian elimination (with, say partial pivoting), applied to a sparse system, some zero entries... 

Schubert s updating formula is particularly useful in the case of large-scale ( ) sparse systems, since it preserves Jacobian sparsity and executes only the calculations for nonzero derivatives. 

The BzzNonLinearSystemSparse class has several constructors that allow sparse systems of different kinds to be solved. 

Replace orthogonal projection by evaluating the nuD space N with the procedure described in the next section that exploits stable Gauss factorization of a system in which the variables are separated into m dependent variables, xj, and n m independent variables, x . This is the optimal choice for large-scale sparse systems when the equations are preventively normalized and the dependent variables are carefully selected. 

To build the Jacobian, the function that calculates the residuals must be invoked only n-c times, where nc is the number of variable groups (4 in this case), nc is usually much smaller than the number of variables n-v for sparse systems. 

In an object from the BzzConstrainedMinimization class with sparse systems, two BzzVectorint objects, nlH and nlP, are used to indicate which variables are really nonlinear in the nonlinear equality or inequality constraints. They are necessary to efficiently update the Jacobians and the Hessians of the functions of constraints. 

When investigating a model of a chemical plant or process, one of the most important tasks is to determine the influence of model parameters like operation conditions or geometric dimensions on performance and dynamics. Because in most cases a large number of parameters has to be examined, an efficient tool for the determination of parameter dependencies is required. Continuation methods in conjunction with the concepts of bifurcation theory have proved to be useful for the analysis of nonlinear systems and are increasingly used in chemical engineering science. They offer the possibility to compute steady states or periodic solutions directly as a function of one or several parameters and to detect changes in the qualitative behaviour of a system like the appearance or disappearance of multiple steady states. In this paper, numerical methods for the continuation of steady states and periodic solutions for large sparse systems with arbitrary structural properties are presented. The application of this methods to models of chemical processes and the problems which arise in this context are discussed for the example of a special type of catalytic fixed bed reactor, the so-called circulation loop reactor. 

The predictor/corrector algorithm in Diva includes a stepsize control in order to minimize the number of predictor and corrector steps. Finally, the continuation package contains methods for the computation of the dominating eigenvalues of DAEs. This allows a stability analysis of the steady state solutions and a detection of local bifurcations for large sparse systems. As the continuation method is embedded into a dynamic simulator, the user has the opportunity to switch interactively from continuation to time integration. This allows additional investigations of transient behaviour or domains of attraction with the same simulation tool[2]. 

Error Reduction in Classic Iterative Methods. Iterative methods for the solution of large sparse systems of equations have been presented here. These methods produce, by iteration, a sequence of approximations to the required solution, which converge to the solution. This process progressively reduces the error related to each approximation. A given approximation is then accepted as the solution when the deviation from the previous approximation (or some norm of it) is smaller than a predefined threshold. Therefore, an analysis of the error expressed in Eq. [30] as a function of the iteration number (or of the required computer time, because the number of operations per iteration is constant) can provide a useful indication of the solver performance. 

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