A Survey of Continuous Karnik-Mendel Algorithms and Their Generalizations

Karnik–Mendel (KM) algorithms are important tools for type-2 fuzzy logic. This survey chapter summarizes some extensions of continuous Karnik–Mendel algorithms. It is shown that the solution of KM algorithms can be transformed into the solution of root-finding problems, and that the iteration formula in KM algorithms is equivalent to the Newton-Raphson root-finding method in numerical analysis. New iteration formulas are summarized that accelerate the convergence speed and it is shown that numerical integration methods can be used to improve computation accuracy. This chapter demonstrates that properties and structures of KM algorithms can be understood and improved with the techniques from numerical analysis.

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