Convergence properties of systems of algebraic equations—explicit loops

Friedman, Francois; Ramirez, W. Fred

doi:10.1002/aic.690190323

Cited by 5 publications

(3 citation statements)

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“…Book and Ramirez (1976) developed methods of expressing all solution sequences which are acyclic for systems of equations without persistent iteration and all solution sequences with a minimum number of iterative variables in systems of equations with persistent iteration. Friedman and Ramirez (1973) show that convergence of explicit iterative loops by the methods of direct substitution is dominated by the order of solution of the equations in the loop. Convergence by direct substitution will occur on the reverse path if the forward path diverges and the reverse path exists.…”

Section: Structural Analysismentioning

confidence: 99%

Structural analysis and solution of systems of algebraic design equations

Book

Ramirez

1984

AIChE Journal

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A new method for expressing the structure of a system of equations is developed using a type of occurrence matrix entitled the functionality matrix. The functionality matrix indicates not only the Occurrence of variables in equations but also the functional form in which they occur. Since the difficulty of solving an equation for a variable is related to its functional form, analysis of the functionality matrix provides explicit information on the difficulty of solution of the equation.A methodology for the solution of design problems by digital computers is described. This methodology operates on the functionality matrix which describes the set of design equations. Algorithms using this methodology interact and guide the designer in an efficient selection of design variables and redundant equations. SCOPEDesign equations arc predominantly algebraic, often nonlinear, and commonly sparse in that each equation contains only a few of the system variables. The major difference between a simulation study and a design study is the parameters specified. The input, output, or combinations of input and output specifications may be known in a design study. Specified parameters may change from one solution to another. Equipment parameters and unspecified input and output parameters are the result of the calculations. All input and equipment parameters must be specified in a simulation study and the output parameters are calculated. The structure for simulation studies is set. This allows simulation studies to be formulated into a modular approach (Motard et al., 1975).Analysis of the characteristics of design problems demonstrates that a need exists for a methodology for analysis of systems of design equations. This methodology must be oriented a t the equation, not modular, level since the calculation path is not fixed for a design problem. This publication reports the development of a method for analyzing systems of equations which characterize design problems. Techniques for selecting design variables and redundant equations are given and methods for obtaining the solution to the system of equations efficiently are presented. CONCLUSIONS AND SIGNIFICANCEA new, flexible methodology is developed for the structural analysis and solution of large systems of algebraic equations. Application is made for the solution of design problems. The development which links the structural analysis with problem solution is the functionality matrix. The functionality matrix allows for the structural analysis and solution of systems of unordered, underconstraincd systems of equations which may contain redundant equations.Efficiency in the solution of large systems of algebraic problems is gained by the use of the solution methodology presented in this work. No ordering of the equations or logic is necessary in programming the solution of a system of equations. STRUCTURAL ANALYSISStructural analysis is the study of the interrelationships and interactions among the various components that form a system. The goal is the attainment of the simp...

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Section: Structural Analysismentioning

confidence: 99%

Structural analysis and solution of systems of algebraic design equations

Book

Ramirez

1984

AIChE Journal

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“…The values of some variables can be arbitrarily assumed (design variables), and the system must be solved for the remaining variables. One criterion for selecting design variables may be the ultimate degree of difficulty of solution of the resultant system as discussed recently by Edie and Westerberg (1971), Rainirez and Vestal ( 1972), and Friedman and Ramirez ( 1973).…”

Section: Future Developmentmentioning

confidence: 99%

Steady state chemical process simulation

1975

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In recent years process simulation by computer has become an accepted tool in the design and control of chemical processes, but there are still uncertainties and conflicting views about the real benefits relative to cost in both chemical industry and in education. This article surveys the literature of the last three years, exploring the answers to the question: How has chemical process simulation by computer fulfilled expectations in industry and education and what are the major areas of research identifiable at this time? It is concluded that process simulation remains a very dynamic field with a continuous expansion in the areas of application and in the number of users. New subjects requiring research are replacing older areas which have matured.

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“…In doing so, they developed the solution path and (unknowingly) selected a set of unknown variables for which the Jacobian was nonsingular. Researchers such as Lee & Rudd [1966], Himmelblau [1967], Christensen & Rudd [1969], Ramirez & Vestal [1972], Friedman & Ramirez [1973], Stadtherr et a1. [1974], Ramirez [1976, 19841, andRamirez [1989] contributed immensely to our understanding of equation management.…”

Section: Introductionmentioning

confidence: 99%

Innovative equation management techniques for implementation in kinematics and dynamics courses

Egelhoff

Blackketter²,

Glumbix³

FIE '98. 28th Annual Frontiers in Education Conference. Moving From 'Teacher-Centered' to 'Learner-Centered' Education. Confere

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The scope of this paper is the description of techniques and algorithms which engineers use in mathematically modeling physical systems. This work builds on recently developed algorithms that can be used to guide students in solving and understanding the mathematical modeling of physical systems. The techniques have been developed for general use in. a number of applications, especially in a Kinematics and Dynamics course. The techniques allow easy manipulation of equations, and systems of governing equations, in complex situations. The algorithms and techniques generally guarantee, and actually restrict, the student to speciji a design variables set in a manner that solutions are possible. This paper documents that the selection of design variables is d$$cult, anti-intuitive, and in some cases it is humanly impossible to generate all of the possible combinations of constraints.While currently the algorithms are implemented in a computer program, the techniques can also be applied to hand calculations. By way of demonstration, a Kinematics and Dynamics design sofiare program was developed which exemplifies the usefulness, and more importantly, the key benefits o f t h e developed algorithms. The intention is to show that the algorithms are applicable to any set of equations and can be used to enhance students problem solving abilities.

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Convergence properties of systems of algebraic equations—explicit loops

Cited by 5 publications

References 3 publications

Structural analysis and solution of systems of algebraic design equations

Structural analysis and solution of systems of algebraic design equations

Steady state chemical process simulation

Innovative equation management techniques for implementation in kinematics and dynamics courses

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