Algebraic System

The algebraic system given as Equation (2.111) represents the working equation of the 6 method. On the basis of this equation a global set is derived and solved to obtain the unknowns at time level n + using the known values at time level n. 

Forsythe, G. E. and Meier, C. B., 1967. Computer Solution of Linear Algebraic Systems, Prentice Hall, Englewood Cliffs, NJ. 

Petzold, L. R. A Description of DASSL A Differential-Algebraic System Solver, Sandia National Laboratory Report SAND82-8637 also in Stepleman, R. S. et al., eds. IMACS Trans, on Scientific Computing, vol. 1, pp. 65-68. 

Differential-Algebraic Systems Sometimes models involve ordinary differentia equations subject to some algebraic constraints. For example, the equations governing one equihbrium stage (as in a distillation column) are... 

If one carries out the analogous calculation one finds that the ultimate algebraic system for the determination of A0 and B0 is... 

Nakhleh Krajcik, 1994) macroscopic system microscopic system symbolic system algebraic system... 

A sutmnaty of the above shows various terms used for eaeh type of representation first (maero level, maeroscopic level, macroscopic world), second (sub-micro level, microscopic level, submicro level, submicroscopic level, molecular world, atomic world), and third (symbolic level, sy mbolic world, representational chemistry, algebraic system). In onr view, the system of terminology shonld be both as brief as possible and avoid any possible ambiguities of meaning. Conseqnently, sub-micro and snb-microscopic fall foul of our first criterion for they perhaps imply that snch a level can be seen through an optical microscope. For those reasons, we have decided to nse macro, submicro, symbolic for the individual types and triplet relationship to cover all three. The triplet relationship is a key model for chemical edncation. However, the authors in this book have been fiee to decide for themselves which conventions to use. Nevertheless, it is our intention to promote the terms macro, submicro, symbolic in all subsequent work and to discuss the value of the triplet relationship in chemical education. 

Differential/Algebraic System Solver", Proc. of the IMACS World Congress, Montreal, August 8-13, 1982... 

Following these procedures, we are led to a system of algebraic equations, thereby reducing numerical solution of an initial (linear) differential equation to solving an algebraic system. 

Within esqjlicit schemes the computational effort to obtain the solution at the new time step is very small the main effort lies in a multiplication of the old solution vector with the coeflicient matrix. In contrast, implicit schemes require the solution of an algebraic system of equations to obtain the new solution vector. However, the major disadvantage of explicit schemes is their instability [84]. The term stability is defined via the behavior of the numerical solution for t —> . A numerical method is regarded as stable if the approximate solution remains bounded for t —> oo, given that the exact solution is also bounded. Explicit time-step schemes tend to become unstable when the time step size exceeds a certain value (an example of a stability limit for PDE solvers is the von-Neumann criterion [85]). In contrast, implicit methods are usually stable. 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

In CED, a number of different iterative solvers for linear algebraic systems have been applied. Two of the most successful and most widely used methods are conjugate gradient and multigrid methods. The basic idea of the conjugate gradient method is to transform the linear equation system Eq. (38) into a minimization problem... 

An iterative solution method for linear algebraic systems which damps the shortwave components of the iteration error very fast and, after a few iterations, leaves predominantly long-wave components. The Gauss-Seidel method [85] could be chosen as a suitable solver in this context. 

The optimality criteria based on which the conditions for the next experiment are determined are the same for dynamic and algebraic systems. However, for a dynamic system we determine the conditions not just of the next measure-... 

The above procedure can be modified somewhat to minimize the overall computational effort. For example, during the computation of the Information Indices in Step 1, matrix Anew can also be computed and thus an early estimate of the best grid point can be established. In addition, as is the case with algebraic systems, the optimum conditions are expected to lie on the boundary of the operability region. Thus, in Steps 2, 3 and 4 the investigation can be restricted only to the grid points which lie on the boundary surface indicated by the preliminary estimate. As a result the computation effort can be kept fairly small. 

Taking these effects into account, internal pore diffusion was modeled on the basis of a wax-filled cylindrical single catalyst pore by using experimental data. The modeling was accomplished by a three-dimensional finite element method as well as by a respective differential-algebraic system. Since the Fischer-Tropsch synthesis is a rather complex reaction, an evaluation of pore diffusion limitations... 

My purpose in this paper is to provide a broad introduction to the capabilities of MACSYMA It is my hope that this information will create new users of Computer Algebra systems by showing what one might expect to gain by using them and what one will lose by not using them ... 

The user can explore extremely complex problems that cannot be solved in any other manner. This capability is often thought of as the major use of Computer Algebra systems. However, one should not lose sight of the fact that MACSYMA is often used as an advanced calculator to perform everyday symbolic and numeric problems. It also complements conventional tools such as reference tables or numeric processors. 

Notice that KACSYKA has obtained the roots analytically and that numeric approximations have not been made. This demonstrates a fundamental difference between a Computer Algebra system and an ordinary numeric equation solver, namely the ability to obtain a solution without approximations. 1 could have given KACSYKA a "numeric" cubic equation in X by specifying numeric values for A and B. KACSYKA then would have solved the equation and given the numeric roots approximately or exactly depending upon the specified command. 

This algebraic system is diagrammed in Figure 1.2. The input to the system is the independent variable x. The output from the system is the dependent variable y. The transform that relates the output to the input is the well defined mathematical relationship given in Equation 1.1. The mathematical equation transforms a given value of the input, x, into an output value, y. If x = 0, then y = 2. If x = 5, then y = 7, and so on. In this simple system, the transform is known with certainty. 

A system variable is defined as a quantity or quality associated with the system that may assume any value from a set containing more than one value. In the algebraic system described previously, x is an input variable it can assume any... 

Yeast and fruit are input variables in the wine-making process. In the case of yeast, the amount of a given strain could be varied, or the particular type of yeast could be varied. If the variation is of extent or quantity (e.g., the use of one ounce of yeast, or two ounces of yeast, or more) the variable is said to be a quantitative variable. If the variation is of type or quality (e.g., the use of Saccharomyces cerevisiae, or Saccharomyces ellipsoideus, or some other species) the variable is said to be a qualitative variable. Thus, yeast could be a qualitative variable (if the amount added is always the same, but the type of yeast is varied) or it could be a quantitative variable (if the type of yeast added is always the same, but the amount is varied). Similarly, fruit added in the wine-making process could be a qualitative variable or a quantitative variable. In the algebraic system, x is a quantitative variable. 

In the algebraic system discussed previously, x is a factor its value determines what the particular result y will be. Yeast and fruit are factors in the wine-making process the type and amount of each contributes to the alcohol content and flavor of the final product. 

Let us reconsider two of the systems discussed previously, looking first at the algebraic system, y = x + 2. In this system, the experimenter does not have any direct control over the response y, but does have an indirect control on y through the system by direct control of the factor x. If the experimenter wants y to have the value 7, the factor X can be set to the value 5. If y is to have the value -3, then by setting x = -5, the desired response can be obtained. 

Just as there are many different types of systems, there are many different types of transforms. In the algebraic system pictured in Figure 1.2, the system transform is the algebraic relationship y = a + 2. In the wine-making system shown in Figure 1.3, the transform is the microbial colony that converts raw materials into a Hnished wine. Transforms in the chemical process industry are usually sets of chemical reactions that transform raw materials into finished products. 

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