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Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
planar array pipeline when a data dependency or other conflict exists. pipeline processor a processor that executes more than one instruction at a time, in pipelined fashion. pipelined bus See split transaction. pitch factor in an electric machine, the ratio of the fractional pitch in electrical degrees to the full pitch, also in electrical degrees. pivoting when applying Gaussian elimination to solve a set of simultaneous linear equations, the natural solution order is sometimes varied. The process of varying the natural solution order is termed pivoting. Pivoting is used to avoid fill-in and to maintain the accuracy of a solution. pixel contraction of "picture element:" each sample of a digital image, i.e., a square or rectangular area of size x × y of constant intensity, located at position (k x, l y) of the image plane. Also called pel. pixel adjacency the property of pixels being next to each other. The adjacency of pixels is ambiguous and is defined several ways. Pixels with four-adjacency or four-connected pixels share an edge. Pixels with eight-adjacency or eight-connected pixels share an edge or a corner. See chain code, connectivity, pixel. pixel density a parameter that specifies how closely the pixel elements are spaced on a given display, usually a color display. PLA See programmable logic array.
Exponential Time Differencing Schemes for Fuel Depletion and Transport in Molten Salt Reactors: Theory and Implementation
Published in Nuclear Science and Engineering, 2022
Zack Taylor, Benjamin S. Collins, G. Ivan Maldonado
Many matrix exponential algorithms rely on solving systems of linear equations, and the choice of linear solvers affects the algorithms’ accuracy. The half-lives and microscopic cross sections for nuclides can vary significantly, causing the magnitude coefficients in the transition matrix to vary from zero to (Ref. 16). For example, radioactive decay results in half-lives that range from s to billions of years.17 Many iterative solvers have difficulty dealing with the rounding errors introduced by the coefficients, and the resulting system will also have extremely small and large eigenvalues. Iterative solvers that rely on Krylov subspace methods become disadvantageous for solving such systems because of the spectral properties of the matrix.16 To achieve a high order of accuracy and stability, direct solvers are chosen over iterative ones. Such solvers include SuperLU or sparse Gaussian elimination with partial pivoting.
Characteristic Chemical Time Scales for Reactive Flow Modeling
Published in Combustion Science and Technology, 2021
Eva-Maria Wartha, Markus Bösenhofer, Michael Harasek
For the chemical time scale definitions IETS and EVTS (see sections 3.8 and 3.9) the eigenvalues, and for EVTS also the eigenvectors, for a real nonsymmetric matrix have to be computed. The Jacobian matrix of the reaction system is in general neither sparse nor symmetric. The numerical calculation is usually done by first transforming the matrix into Hessenberg form and applying the eigenvalue search to this simplified matrix. The transformation can be conducted by a sequence of Householder transformations or by an elimination method analogous to Gaussian elimination with pivoting. Algorithmic details are presented in (Press et al. 2007; Wilkinson and Reinsch 1971). This algorithm needs approximately operations (Press et al. 2007).
Design of experiments for steady-state system identification with applications in genetic and business network modelinG
Published in Journal of Industrial and Production Engineering, 2020
Cenny Taslim, Theodore T. Allen, Mario Lauria, Shih-Hsien Tseng
We begin with a simple example involving six genes to illustrate the proposed simulation and estimation procedures. Table 5(a) shows an A matrix generated using the random resilient matrices (RRM) method. Table 5(b) shows the qualitative system connections associated with the generated A matrix. The D-optimal experimental design used is in Table 3(a). Table 5(c) shows one set of simulated concentrations (run 1) derived from adding random errors with standard deviations σεj = 1.0 for all j = 1, …, 6. Table 5(d) shows the estimated or “recovered” A matrix derived using straightforward least squares estimation in Equation (7) and numerical inversion using Gaussian elimination with partial pivoting. The empirical sum squared estimation errors (SEE) for the derived Aest are 0.66. The remaining parts (e) and (f) are based on the second set of random errors (run 2) resulting in a much higher SEE equal to 350.5.