... America).Chapter 16. Integration of Ordinary Differential Equations16.0 IntroductionProblems involving ordinarydifferential equations (ODEs) can always bereduced to the study of sets of first-order differential ... 1973,Computational Methods inOrdinaryDifferential Equations(New York: Wiley).Lapidus, L., and Seinfeld, J. 1971,Numerical Solution ofOrdinaryDifferential Equations(NewYork: Academic ... 708Chapter 16. Integration ofOrdinaryDifferential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C)...
... 1973,Computational Methods inOrdinaryDifferential Equations(New York: Wiley).Lapidus, L., and Seinfeld, J. 1971,Numerical Solution ofOrdinaryDifferential Equations(NewYork: Academic ... discussion of the pitfalls in constructing a good Runge-Kutta code is given in [3].Here is the routine for carrying out one classical Runge-Kutta step on a set of n differential equations. You input ... derive from this basic712Chapter 16. Integration ofOrdinaryDifferential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C)...
... Minimizing lateness/tardiness• Minimizing makespan• Maximizing system/machine utilization• Minimizing WIP (work in process)• Maximizing throughput• Minimizing average flow time• Minimizing ... routings. A routing is selected such that the following goals are achieved:• Minimizing number of blocked machines.• Minimizing total processing time.• Minimizing number of processing stepsFour ... Simulation of scheduling rules in a flexiblemanufacturing system using fuzzy logic, IEA-AIE96 Ninth International Conference on Industrialand Engineering Applicationof Artificial Intelligence...
... methods.free_vector(ytemp,1,n);free_vector(ak6,1,n);free_vector(ak5,1,n);free_vector(ak4,1,n);free_vector(ak3,1,n);free_vector(ak2,1,n);}Noting that the above routines are all in single precision, don’t be too greedy in specifying eps. Thepunishment forexcessive greediness is interestingand worthyofGilbertand Sullivan’sMikado: ... + H, zn)](16.3.2)714Chapter 16. Integration ofOrdinaryDifferential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) ... 1971,Numerical Initial Value Problems inOrdinaryDifferential Equations(EnglewoodCliffs, NJ: Prentice-Hall), Chapter 2. [2]Shampine, L.F., and Watts, H.A. 1977, in Mathematical Software III,...
... 722Chapter 16. Integration ofOrdinaryDifferential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) ... hmin) nrerror("Step size too small in odeint");h=hnext;}nrerror("Too many steps in routine odeint");}CITED REFERENCES AND FURTHER READING:Gear, C.W. 1971,Numerical Initial ... H/n (16.3.1) In principle, one could use the modified midpoint method in its own right as an ODEintegrator. In practice, the method finds its most important application as a part of the more powerful...
... remind you once again that scaling of the variables is often crucial forsuccessful integration ofdifferential equations. The scaling “trick” suggested in the discussion following equation (16.2.8) ... extrapolate eachcomponent of a vector of quantities.728Chapter 16. Integration ofOrdinaryDifferential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... encountered in practice, is discussed in §16.7.)726Chapter 16. Integration ofOrdinaryDifferential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright...
... 734Chapter 16. Integration ofOrdinaryDifferential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) ... isa particular class of equations that occurs quite frequently in practice where you can gainabout a factor of two in efficiency by differencing the equations directly. The equations aresecond-order ... FURTHER READING:Stoer, J., and Bulirsch, R. 1980,Introduction to Numerical Analysis(New York: Springer-Verlag),§7.2.14. [1]Gear, C.W. 1971,Numerical Initial Value Problems inOrdinary Differential...
... as in the original Bulirsch-Stoer method.The starting point is an implicit form of the midpoint rule:yn+1− yn−1=2hfyn+1+ yn−12(16.6.29)738Chapter 16. Integration ofOrdinaryDifferential ... methods have been, we think, squeezed740Chapter 16. Integration ofOrdinaryDifferential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... Example of an instability encountered in integrating a stiff equation (schematic). Hereit is supposed that the equation has two solutions, shown as solid and dashed lines. Although the initialconditions...
... methods have been, we think, squeezed752Chapter 16. Integration ofOrdinaryDifferential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... methods in all applications. We are willing, however, to be corrected.CITED REFERENCES AND FURTHER READING:Gear, C.W. 1971,Numerical Initial Value Problems inOrdinaryDifferential Equations(EnglewoodCliffs, ... method .In functional iteration, we take some initial guess for yn+1, insert it into the right-handside of (16.7.2) to get an updated value of yn+1, insert this updated value back intothe...
... risk of being left behind.Simply, CRM is a high-tech way of gathering mou-ntains of information about customers, then usingit to make customers happy—or at least a source of more business. ... onbehalf of human users to perform laborious androutine tasks such as locating and accessing neces-sary information, resolving inconsistencies in theretrieved information, filtering away irrelevantand ... irrelevantand unwanted information, and integrating infor-mation from heterogeneous information sources. In order to execute tasks on behalf of a businessprocess, computer application, or an individual,agents...
... other measures of individual in- come uncertainty: the income quintile share ratio (income of the 20% richest to income of 20% poorest), the percentage of working households in the risk of poverty ... thoughthe decline in the lending rate, higher uncertainty deters individuals from taking loans,and the share of household credit in GDP declines from 14.3% to 13.4%. If individualuncertainty is low, ... reflects the life- cycle profiles of income and consumption. In initial periods,when income is relatively low, individuals are taking loans as they expect that theirincome will increase in the future....
... stability of the following func-tional equation f (mx + ny)=(m + n)f(x + y)2+(m − n)f(x − y)2(1) in various spaces, which was introduced and investigated in [13].2. Preliminaries In this ... there exists a mapping A : X ® Y satisfying the following:(1) A is a fixed point of J, that is,Axm=1mA(x)(5)for all x Î X. The mapping A is a unique fixed point of J in the set= ... there exists a mapping A : X ® Y satisfying the following:(1) A is a fixed point of J, that is,Axm=1mA(x)(12)for all x Î X. The mapping A is a unique fixed point of J in the set=...