); and Sándor Kemény is professor at the same institution (email: kemeny@mail.bme.hu ). The authors wish to thank Mr. Richárd Király for his preliminary work. The authors are grateful to the Associate Editor of STCO and the unknown reviewers for their helpful suggestions.
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ABSTRACT AND KEY WORDSUnfortunately many of the numerous algorithms for computing the cdf and noncentrality parameter of the noncentral F and beta distributions can return completely incorrect results as demonstrated in the paper by examples. Existing algorithms are scrutinized and those parts that involve numerical difficulties are identified. As a result, a pseudo code is presented in which all the known numerical problems are resolved. This pseudo code can be easily implemented in programming language C or FORTRAN without understanding the complicated mathematical background. Symbolic evaluation of a finite and closed formula is proposed to compute exact cdf values. This approach makes it possible to check quickly and reliably the values returned by professional statistical packages over an extraordinarily wide parameter range without any programming knowledge. This research was motivated by the fact that a most useful table for calculating the size of detectable effects for ANOVA tables contains suspicious values in the region of large noncentrality parameter values compared to the values obtained by Patnaik's 2-moment central-F approximation. The reason is identified and the corrected form of the table for ANOVA purposes is given. The accuracy of the approximations to the noncentral-F distribution is also discussed.
A feedback arc set of a directed graph
G
is a subset of its arcs containing at least one arc of every cycle in
G
. Finding a feedback arc set of minimum cardinality is an NP-hard problem called the
minimum feedback arc set problem
. Numerically, the minimum set cover formulation of the minimum feedback arc set problem is appropriate as long as all simple cycles in
G
can be enumerated. Unfortunately, even those sparse graphs that are important for practical applications often have Ω (2
n
) simple cycles. Here we address precisely such situations: An exact method is proposed for sparse graphs that enumerates simple cycles in a lazy fashion and iteratively extends an incomplete cycle matrix. In all cases encountered so far, only a tractable number of cycles has to be enumerated until a minimum feedback arc set is found. The practical limits of the new method are evaluated on a test set containing computationally challenging sparse graphs, relevant for industrial applications. The 4,468 test graphs are of varying size and density and suitable for testing the scalability of exact algorithms over a wide range.
The need of reliably solving systems of nonlinear equations often arises in the everyday practice of chemical engineering. In general, standard methods cannot provide theoretical guarantee for convergence to a solution, cannot reliably find multiple solutions, and cannot prove non-existence of solutions. Interval methods provide tools to overcome these problems, thus achieving reliability. To the authors' best knowledge, computation of distillation columns with interval methods have not yet been considered in the literature. This paper presents significant enhancements compared to a previously published interval method of the authors. The proposed branch-and-prune algorithm is guaranteed to converge, and is fairly general at the same time. If no solution exists then this information is provided by the method as a result. Power of the suggested method is demonstrated by solving, with guaranteed convergence, even the MESH equations of a 22 stage extractive distillation column with a ternary mixture.
Topical heading: Separations
Computing the steady state of multistage counter-current processes like distillation, extraction, or absorption is the equivalent to finding solutions for large scale non-linear equation systems. The conventional solution techniques are fast and efficient if a good estimation is available but are prone to fail, and do not provide information about the reason for the failure. This is the main motive to apply reliable methods in solving them.
Topical heading: Separations
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