A New Algorithm for Computing the Actions of Trigonometric and Hyperbolic Matrix Functions

A new algorithm is derived for computing the actions $f(tA)B$ and $f(tA^{1/2})B$, where $f$ is cosine, sinc, sine, hyperbolic cosine, hyperbolic sinc, or hyperbolic sine function. $A$ is an $n\times n$ matrix and $B$ is $n\times n_0$ with $n_0 \ll n$. $A^{1/2}$ denotes any matrix square root of $A$ and it is never required to be computed. The algorithm offers six independent output options given $t$, $A$, $B$, and a tolerance. For each option, actions of a pair of trigonometric or hyperbolic matrix functions are simultaneously computed. The algorithm scales the matrix $A$ down by a positive integer $s$, approximates $f(s^{-1}tA)B$ by a truncated Taylor series, and finally uses the recurrences of the Chebyshev polynomials of the first and second kind to recover $f(tA)B$. The selection of the scaling parameter and the degree of Taylor polynomial are based on a forward error analysis and a sequence of the form $\|A^k\|^{1/k}$ in such a way the overall computational cost of the algorithm is optimized. Shifting is used where applicable as a preprocessing step to reduce the scaling parameter. The algorithm works for any matrix $A$ and its computational cost is dominated by the formation of products of $A$ with $n\times n_0$ matrices that could take advantage of the implementation of level-3 BLAS. Our numerical experiments show that the new algorithm behaves in a forward stable fashion and in most problems outperforms the existing algorithms in terms of CPU time, computational cost, and accuracy.Comment: 4 figures, 16 page

The optimisation of the estimating and tendering process in warship refit - a case study

The optimisation of a tendering process for warship refit contracts is presented. The tendering process, also known as the pre-contract award process (PCA), involves all the activities needed to be successfully awarded a refit contract. Process activities and information flows have been modelled using Integrated Definition Language IDEF0 and a Dependency Structure Matrix (DSM) with optimisation performed via a Genetic Algorithm (DSM-GA) search technique. By utilising this approach the process activities were re-sequenced in such an order that the number and size of rework cycles were reduced. The result being a 57% reduction in a criterion indicating 're-work' cycles

Assembly and Disassembly Planning by using Fuzzy Logic & Genetic Algorithms

The authors propose the implementation of hybrid Fuzzy Logic-Genetic Algorithm (FL-GA) methodology to plan the automatic assembly and disassembly sequence of products. The GA-Fuzzy Logic approach is implemented onto two levels. The first level of hybridization consists of the development of a Fuzzy controller for the parameters of an assembly or disassembly planner based on GAs. This controller acts on mutation probability and crossover rate in order to adapt their values dynamically while the algorithm runs. The second level consists of the identification of theoptimal assembly or disassembly sequence by a Fuzzy function, in order to obtain a closer control of the technological knowledge of the assembly/disassembly process. Two case studies were analyzed in order to test the efficiency of the Fuzzy-GA methodologies