By Guido Buzzi-Ferraris, Flavio Manenti
This 3rd e-book in a collection of 4 functional courses is an engineer's better half to utilizing numerical equipment for the answer of advanced mathematical difficulties. the mandatory software program is supplied when it comes to the freeware mathematical library BzzMath that's built and maintained via the authors. the current quantity specializes in optimization and nonlinear structures resolution. The ebook describes numerical tools, leading edge thoughts and methods which are all applied in a well-established, freeware library. every one of those convenient courses allows the reader to take advantage of and enforce general numerical instruments for his or her paintings, explaining the idea at the back of a few of the features and challenge solvers, and showcasing functions in different medical and engineering fields. quite a few examples, pattern codes, courses and purposes are proposed and mentioned. The publication teaches engineers and scientists how one can use the newest and strongest numerical tools for his or her day-by-day paintings.
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Additional resources for Nonlinear Systems and Optimization for the Chemical Engineer: Solving Numerical Problems
It is worth remarking that, if the function is monotone, it is possible to ﬁnd a rough interval of uncertainty and, once detected, the solution can be obtained using very efﬁcient methods, although the interval of uncertainty is wide. 7 Stop Criteria In all problems solved using iterative methods, it is important to deﬁne a criterion to stop the evaluations either when a reasonable approximation of the solution is achieved or when convergence is impossible. There are various possibilities where the speciﬁc problem of function rootﬁnding is concerned and it is advisable for us to analyze some of them here.
5 Use of a Multiprocessor Machine with a Known Interval of Uncertainty If an interval of uncertainty is known and some processors are available to ﬁnd the root, Bolzano’s method can effectively be coupled with a function approximation method. There are various strategies when more than a single processor, nP > 1, are available. The following is the one adopted in the BzzMath library. Suppose we have nP available processors. The ﬁrst one is used to calculate the function in the point selected by the efﬁcient method being used, whereas the remaining nP À 1 processors evaluate the function in the points opportunely placed in the two new subintervals, created by the splitting of the interval of uncertainty.
Publsihed 2014 by Wiley-VCH Verlag GmbH & Co. KGaA. 42 2 One-Dimensional Optimization special one-dimensional methods, which are dedicated to multidimensional problems, are usually adopted instead (Chapter 3). Optimization can mean minimization or maximization problems. By inverting the sign of the objective function, we can solve either one or other problem. For example, the maximum of the function GðtÞ ¼ Àt3 þ 2t2 þ 3t þ 5 ð2:3Þ in the interval tA ¼ 0, tB ¼ 4 corresponds to the minimum of the function in the same interval: F ðtÞ ¼ t3 À 2t2 À 3t À 5 ð2:4Þ For the sake of clarity, however, we will refer to minimization problems only.
Nonlinear Systems and Optimization for the Chemical Engineer: Solving Numerical Problems by Guido Buzzi-Ferraris, Flavio Manenti