Relevant bibliographies by topics / Julia set in a matrix algebra

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  1. Journal articles
  2. Dissertations / Theses
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  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Julia set in a matrix algebra'

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Published: 7 September 2021
Last updated: 18 February 2022

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Journal articles on the topic "Julia set in a matrix algebra"

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Rojas-Rodriguez, Jose C., Ana Y. Aguilar-Bustos, and Eusebio Bugarin. "An O(N) Algorithm for the Computation of the Centroidal Dynamics with Application in the Postural Balance of a Humanoid Robot Using Whole Body Control." International Journal of Humanoid Robotics 18, no. 03 (2021): 2150010. http://dx.doi.org/10.1142/s0219843621500109.

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In this paper, we introduce an [Formula: see text] algorithm for the computation of the centroidal momentum matrix (CMM) and its time derivative using spatial algebra and expressed with Lie algebra operators. The proposed algorithm is applied to the postural balance of a humanoid robot using whole body control with quadratic programming. The employed tasks only require the CMM and its time derivative without the need of the joint space inertia matrix and the Coriolis terms reducing this way the computational cost of the controller. Finally, four simulation scenarios programmed in Julia are considered where several perturbations for the balance of the robot have been taken into account and according to the tracking graphs of the center of mass, centroidal momentum and the trajectories of the center of pressure it is concluded that the performance of the proposed algorithm is satisfactory.
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Zhang, Weijian, and Nicholas J. Higham. "Matrix Depot: an extensible test matrix collection for Julia." PeerJ Computer Science 2 (April 6, 2016): e58. http://dx.doi.org/10.7717/peerj-cs.58.

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Matrix Depot is a Julia software package that provides easy access to a large and diverse collection of test matrices. Its novelty is threefold. First, it is extensible by the user, and so can be adapted to include the user’s own test problems. In doing so, it facilitates experimentation and makes it easier to carry out reproducible research. Second, it amalgamates in a single framework two different types of existing matrix collections, comprising parametrized test matrices (including Hansen’s set of regularization test problems and Higham’s Test Matrix Toolbox) and real-life sparse matrix data (giving access to the University of Florida sparse matrix collection). Third, it fully exploits the Julia language. It uses multiple dispatch to help provide a simple interface and, in particular, to allow matrices to be generated in any of the numeric data types supported by the language.
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Devaney, Robert L., and Daniel M. Look. "Symbolic dynamics for a Sierpinski curve Julia set." Journal of Difference Equations and Applications 11, no. 7 (2005): 581–96. http://dx.doi.org/10.1080/10236190412331334473.

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Rivera-Letelier, Juan. "On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets." Fundamenta Mathematicae 170, no. 3 (2001): 287–317. http://dx.doi.org/10.4064/fm170-3-6.

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Domínguez, P., A. Hernández, and G. Sienra. "Residual Julia set for functions with the Ahlfors' Property." Journal of Difference Equations and Applications 20, no. 7 (2014): 1019–32. http://dx.doi.org/10.1080/10236198.2014.884084.

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Rudhito, Marcellinus Andy, Sri Wahyuni, Ari Suparwanto, and Frans Susilo. "Matriks atas Aljabar Max-Plus Interval." Jurnal Natur Indonesia 13, no. 2 (2012): 94. http://dx.doi.org/10.31258/jnat.13.2.94-99.

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This paper aims to discuss the matrix algebra over interval max-plus algebra (interval matrix) and a method tosimplify the computation of the operation of them. This matrix algebra is an extension of matrix algebra over max-plus algebra and can be used to discuss the matrix algebra over fuzzy number max-plus algebra via its alpha-cut.The finding shows that the set of all interval matrices together with the max-plus scalar multiplication operationand max-plus addition is a semimodule. The set of all square matrices over max-plus algebra together with aninterval of max-plus addition operation and max-plus multiplication operation is a semiring idempotent. As reasoningfor the interval matrix operations can be performed through the corresponding matrix interval, because thatsemimodule set of all interval matrices is isomorphic with semimodule the set of corresponding interval matrix,and the semiring set of all square interval matrices is isomorphic with semiring the set of the correspondingsquare interval matrix.
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Aspenberg, Magnus. "Rational Misiurewicz maps for which the Julia set is not the whole sphere." Fundamenta Mathematicae 206 (2009): 41–48. http://dx.doi.org/10.4064/fm206-0-3.

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Hawkins, Jane. "A family of elliptic functions with Julia set the whole sphere." Journal of Difference Equations and Applications 16, no. 5-6 (2010): 597–612. http://dx.doi.org/10.1080/10236190903257859.

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Hireš, Máté, Monika Molnárová, and Peter Drotár. "Robustness of Interval Monge Matrices in Fuzzy Algebra." Mathematics 8, no. 4 (2020): 652. http://dx.doi.org/10.3390/math8040652.

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Max–min algebra (called also fuzzy algebra) is an extremal algebra with operations maximum and minimum. In this paper, we study the robustness of Monge matrices with inexact data over max–min algebra. A matrix with inexact data (also called interval matrix) is a set of matrices given by a lower bound matrix and an upper bound matrix. An interval Monge matrix is the set of all Monge matrices from an interval matrix with Monge lower and upper bound matrices. There are two possibilities to define the robustness of an interval matrix. First, the possible robustness, if there is at least one robust matrix. Second, universal robustness, if all matrices are robust in the considered set of matrices. We found necessary and sufficient conditions for universal robustness in cases when the lower bound matrix is trivial. Moreover, we proved necessary conditions for possible robustness and equivalent conditions for universal robustness in cases where the lower bound matrix is non-trivial.
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FISHBACK, PAUL E., and MATTHEW D. HORTON. "QUADRATIC DYNAMICS IN MATRIX RINGS: TALES OF TERNARY NUMBER SYSTEMS." Fractals 13, no. 02 (2005): 147–56. http://dx.doi.org/10.1142/s0218348x05002787.

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We describe the quadratic dynamics in certain three-component number systems, which like the complex numbers, can be expressed as rings of real matrices. This description is accomplished using the properties of the real quadratic family and its various first- and second-order phase and parameter derivatives. We demonstrate that the fundamental dichotomy of defining the Mandelbrot set either in terms of filled Julia sets or in terms of the orbit of the origin extends to these ternary number systems.
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Dissertations / Theses on the topic "Julia set in a matrix algebra"

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Baptista, Alexandra Cristina Ferros dos Santos Nascimento. "Sistemas dinâmicos discretos em álgebras." Doctoral thesis, Universidade de Évora, 2012. http://hdl.handle.net/10174/14751.

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Neste trabalho é feito o estudo de sistemas dinâmicos discretos em álgebras de matrizes. Este tema é explorado recorrendo a várias ferramentas da álgebra linear, com o objectivo de tirar partido da estrutura algébrica do espaço. É estudada a aplicação quadrática matricial, tomando uma matriz como parâmetro, aliando as propriedades algébricas à teoria das aplicações quadráticas escalares já existente, no caso real e complexo. São exploradas diversas características da dinâmica, tais como, a existência de ciclos comutativos e não-comutativos, a sua estabilidade, entre outras. São estudadas possíveis generalizações para o caso matricial das noções de conjunto de Mandelbrot e de conjunto de Julia. Os resultados atingidos são aplicados ao estudo da dinâmica da aplicação quadrática em diferentes álgebras hipercomplexas. É explorada a iteração quadrática no conjunto das matrizes estocásticas simétricas; as conclusões ilustram o comportamento do sistema dinâmico discreto definido no espaço das cadeias de Markov reversíveis; ABSTRACT: In this work we study discrete dynamical systems in matrix algebras. This subject is explored using different tools of linear algebra, in order to take advantage of the algebraic structure of the space. It is studied the iteration of a quadratic family in the algebra of real matrices, with a parameter matrix, combining the properties of the algebraic theory with the theory of the quadratic map in the real and complex cases. Several characteristics of the dynamics are explored, such as, the existence of commutative and non-commutative cycles, its stability, among others. Possible generalizations of the Mandelbrot set and Julia set are considered and studied. The results obtained are applied to the study of the quadratic dynamic in different hypercomplex algebras. Quadratic iteration is explored in the set of symmetric stochastic matrices; the findings illustrate the behavior of the discrete dynamical system on the space of reversible Markov chains.
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Books on the topic "Julia set in a matrix algebra"

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Linear And Multilinear Algebra And Function Spaces: (Contemporary Mathematics). American Mathematical Society, 2020.

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Bisseling, Rob H. Parallel Scientific Computation. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198788348.001.0001.

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This book explains how to use the bulk synchronous parallel (BSP) model to design and implement parallel algorithms in the areas of scientific computing and big data. Furthermore, it presents a hybrid BSP approach towards new hardware developments such as hierarchical architectures with both shared and distributed memory. The book provides a full treatment of core problems in scientific computing and big data, starting from a high-level problem description, via a sequential solution algorithm to a parallel solution algorithm and an actual parallel program written in the communication library BSPlib. Numerical experiments are presented for parallel programs on modern parallel computers ranging from desktop computers to massively parallel supercomputers. The introductory chapter of the book gives a complete overview of BSPlib, so that the reader already at an early stage is able to write his/her own parallel programs. Furthermore, it treats BSP benchmarking and parallel sorting by regular sampling. The next three chapters treat basic numerical linear algebra problems such as linear system solving by LU decomposition, sparse matrix-vector multiplication (SpMV), and the fast Fourier transform (FFT). The final chapter explores parallel algorithms for big data problems such as graph matching. The book is accompanied by a software package BSPedupack, freely available online from the author’s homepage, which contains all programs of the book and a set of test programs.
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Book chapters on the topic "Julia set in a matrix algebra"

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"Matrix algebra." In The Finite Element Method Set. Elsevier, 2005. http://dx.doi.org/10.1016/b978-075066431-8.50220-4.

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ZIENKIEWICZ, O., R. TAYLOR, and J. ZHU. "Matrix algebra." In The Finite Element Method Set. Elsevier, 2005. http://dx.doi.org/10.1016/b978-075066431-8/50220-4.

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Eshel, Gidon. "Matrix Properties, Fundamental Spaces, Orthogonality." In Spatiotemporal Data Analysis. Princeton University Press, 2011. http://dx.doi.org/10.23943/princeton/9780691128917.003.0003.

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This chapter provides an introduction to linear algebra. Topics covered include vector spaces, matrix rank, fundamental spaces associated with A ɛ ℝM×N, and Gram–Schmidt orthogonalization. In summary, an M × N matrix is associated with four fundamental spaces. The column space is the set of all M-vectors that are linear combinations of the columns. If the matrix has M independent columns, then the column space is ℝM; otherwise the column space is a subspace of ℝM. Also in ℝM is the left null space, the set of all M-vectors that the matrix’s s transpose maps to the zero N-vector. The row space is the set of all N-vectors that are linear combinations of the rows. If the matrix has N independent rows, then the row space is ℝN; otherwise, the row space is a subspace of ℝN. Also in ℝN is the null space, the set of all N-vectors that the matrix maps to the zero M-vector.
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Rodman, Leiba. "Vector spaces and matrices: Basic theory." In Topics in Quaternion Linear Algebra. Princeton University Press, 2014. http://dx.doi.org/10.23943/princeton/9780691161853.003.0003.

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This chapter covers the basics on the vector space of columns with quaternion components, matrix algebra, and various matrix decompositions. The real and complex representations of quaternions are extended to vectors and matrices. Various matrix decompositions are studied; in particular, Cholesky factorization is proved for matrices that are hermitian with respect to involutions other than the conjugation. A large part of this chapter is devoted to numerical ranges of quaternion matrices with respect to conjugation as well as with respect to other involutions. Finally, a brief exposition is given for the set of quaternion subspaces, understood as a metric space with respect to the metric induced by the gap function.
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Kulik, Boris, Alexander Fridman, and Alexander Zuenko. "Logical Inference and Defeasible Reasoning in N-tuple Algebra." In Diagnostic Test Approaches to Machine Learning and Commonsense Reasoning Systems. IGI Global, 2013. http://dx.doi.org/10.4018/978-1-4666-1900-5.ch005.

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This chapter examines the usage potential of n-tuple algebra (NTA) developed by the authors as a theoretical generalization of structures and methods applied in intelligence systems. NTA supports formalization of a wide set of logical problems (abductive and modified conclusions, modelling graphs, semantic networks, expert rules, etc.). This chapter mostly focuses on implementation of logical inference and defeasible reasoning by means of NTA. Logical inference procedures in NTA can include, besides the known logical calculus methods, new algebraic methods for checking correctness of a consequence or for finding corollaries to a given axiom system. Inference methods consider (above feasibility of certain substitutions) inner structure of knowledge to be processed, thus providing faster solving of standard logical analysis tasks. Matrix properties of NTA objects allow decreasing the complexity of intellectual procedures. As for making databases more intelligent, NTA can be considered as an extension of relational algebra to knowledge processing.
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Mussardo, Giuseppe. "Form Factors and Correlation Functions." In Statistical Field Theory. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198788102.003.0019.

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At the heart of a quantum field theory are the correlation functions of the various fields. In the case of integrable models, the correlators can be expressed in terms of the spectral series based on the matrix elements on the asymptotic states. These matrix elements, also known as form factors, satisfy a set of functional and recursive equations that can exactly solved in many cases of physical interest. Chapter 19 covers general properties of form factors, Faddeev–Zamolodchikov algebra, symmetric polynomials, kinematical and bound state poles, the operator space and kernel functions, the stress-energy tensor and vacuum expectation values and the Ising model in a magnetic field.
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Alimey, Fred John, Libing Bai, and Yuhua Cheng. "Tensor Based Finite Element Model for the Calculation of Leakage Field in Magnetic Flux Leakage Testing." In Studies in Applied Electromagnetics and Mechanics. IOS Press, 2020. http://dx.doi.org/10.3233/saem200019.

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Magnetic flux leakage (MFL) testing is a widely used electromagnetic nondestructive testing (ENDT) method, which has the ability to detect both surface and sub-surface defects in conductive materials. One of its best features is its ability to mathematically model field leakage from the defect area in a magnetized material. In this paper, we propose an optimized FEM model using geometrical weighted tensor (TBFEM), for the calculation of leakage field in MFL. This model using the Einstein’s convention eliminates the bulky nature of traditional FEM based on its matrix algebra formation allowing for easy implementation and fast calculations. The proposed model achieves this by reducing the set of matrix equations into a single equation using suffixes which can then be solved with regular mathematical operations.
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Doveton, John H. "Compositional Analysis of Mineralogy." In Principles of Mathematical Petrophysics. Oxford University Press, 2014. http://dx.doi.org/10.1093/oso/9780199978045.003.0009.

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Formation lithologies that are composed of several minerals require multiple porosity logs to be run in combination in order to evaluate volumetric porosity. In the most simple solution model, the proportions of multiple components together with porosity can be estimated from a set of simultaneous equations for the measured log responses. These equations can be written in matrix algebra form as: . . . CV = L . . . where C is a matrix of the component petrophysical properties, V is a vector of the component unknown proportions, and L is a vector of the log responses of the evaluated zone. The equation set describes a linear model that links the log measurements with the component mineral properties. Although porosity represents the proportion of voids within the rock, the pore space is filled with a fluid whose physical properties make it a “mineral” component. If the minerals, their petrophysical properties, and their proportions are either known or hypothesized, then log responses can be computed. In this case, the procedure is one of forward-modeling and is useful in situations of highly complex formations, where geological models are used to generate alternative log-response scenarios that can be matched with actual logging measurements in a search for the best reconciliation between composition and logs. However, more commonly, the set of equations is solved as an “inverse problem,” in which the rock composition is deduced from the logging measurements. Probably the earliest application of the compositional analysis of a formation by the inverse procedure applied to logs was by petrophysicists working in Permian carbonates of West Texas, who were frustrated by complex mineralogy in their attempts to obtain reliable porosity estimates from logs, as described by Savre (1963). Up to that time, porosities had been commonly evaluated from neutron logs, but the values were excessively high in zones that contained gypsum, caused by the hydrogen within the water of crystallization. The substitution of the density log for the porosity estimation was compromised by the occurrence of anhydrite as well as gypsum.
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Wang, Yingxu, Jason Huang, and Jingsheng Lei. "The Formal Design Models of a Universal Array (UA) and its Implementation." In Advances in Abstract Intelligence and Soft Computing. IGI Global, 2013. http://dx.doi.org/10.4018/978-1-4666-2651-5.ch017.

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Arrays are one of the most fundamental and widely applied data structures, which are useful for modeling both logical designs and physical implementations of multi-dimensional data objects sharing the same type of homogeneous elements. However, there is a lack of a formal model of the universal array based on it any array instance can be derived. This paper studies the fundamental properties of Universal Array (UA) and presents a comprehensive design pattern. A denotational mathematics, Real-Time Process Algebra (RTPA), allows both architectural and behavioral models of UA to be rigorously designed and refined in a top-down approach. The conceptual model of UA is rigorously described by tuple- and matrix-based mathematical models. The architectural models of UA are created using RTPA architectural modeling methodologies known as the Unified Data Models (UDMs). The physical model of UA is implemented using linear list that is indexed by an offset pointer of elements. The behavioral models of UA are specified and refined by a set of Unified Process Models (UPMs). As a case study, the formal UA models are implemented in Java. This work has been applied in a number of real-time and nonreal-time systems such as compilers, a file management system, the real-time operating system (RTOS+), and the ADT library for an RTPA-based automatic code generation tool.
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Pawlowsky-Glahn, Vera, and Richardo A. Olea. "Regionalized compositions." In Geostatistical Analysis of Compositional Data. Oxford University Press, 2004. http://dx.doi.org/10.1093/oso/9780195171662.003.0008.

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In this chapter we set the rationale for the analysis of regionalized compositions. Required definitions for nonregionalized compositions are extended to vector random functions and necessary concepts from the theory of regionalized variables are related to vector random functions that form a composition. In order to avoid continually repeating references to literature, the reader is referred especially to the works of Matheron (1971) and Aitchison (1986), on which the following developments are based. Here the exposition is very concise; its purpose is basically to introduce terminology and notation. Proofs analogous to those of the nonregionalized case are omitted, for the most part. In general they can be derived directly from the corresponding definitions. For concepts of matrix algebra required by this work, refer to Kemény (1984) and Golub and Van Loan (1989). There are many excellent textbooks that treat concepts of probability theory and multivariate statistics. We have used mainly the books by Fahrmeir and Hamerle (1984) and Krzanowski (1988), and others have served as complementary bibliography, e.g., Feller (1968), Kendall and Stuart (1979), Kendall et al. (1983), Kres (1983), Stuart and Ord (1987), Johnson et al. (1994), and Kotz et al. (2000). A similar situation holds for the foundations of univariate geostatistics; refer to David (1977), Journel and Huijbregts (1978), Rendu (1978), Clark (1979), Isaaks and Srivastava (1989), Samper-Calvete and Carrera-Ramírez (1990), Cressie (1991), Goovaerts (1997), Chilès and Delfiner (1999), and Olea (1999). Treatments of multivariate geostatistics are found in Matheron (1979), François-Bongarçon (1981), Carr et al. (1985), and Wackernagel (1998). We base our presentation mainly on Journel and Huijbregts (1978) and Deutsch and Journel (1998), but also on Myers (1982), in which the matrix formulation of cokriging is given. Geostatistical terminology conforms, as far as possible, to that found in the Geostatistical Glossary and Multilingual Dictionary, compiled by members of the 1984-1989 IAMG Committee on Geostatistics and edited by R. A. Olea (1991).
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Conference papers on the topic "Julia set in a matrix algebra"

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Gao, Jinsong, Kenneth W. Chase, and Spencer P. Magleby. "Comparison of Assembly Tolerance Analysis by the Direct Linearization and Modified Monte Carlo Simulation Methods." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0047.

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Abstract Two methods for performing statistical tolerance analysis of mechanical assemblies are compared: the Direct Linearization Method (DLM), and Monte Carlo simulation. A selection of 2-D and 3-D vector models of assemblies were analyzed, including problems with closed loop assembly constraints. Closed vector loops describe the small kinematic adjustments that occur at assembly time. Open loops describe critical clearances or other assembly features. The DLM uses linearized assembly constraints and matrix algebra to estimate the variations of the assembly or kinematic variables, and to predict assembly rejects. A modified Monte Carlo simulation, employing an iterative technique for closed loop assemblies, was applied to the same problem set. The results of the comparison show that the DLM is accurate if the tolerances are relatively small compared to the nominal dimensions of the components, and the assembly functions are not highly nonlinear. Sample size is shown to have great influence on the accuracy of Monte Carlo simulation.
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Ge, Q. J., and B. Ravani. "Computation of Spatial Displacements From Redundant Geometric Features." In ASME 1992 Design Technical Conferences. American Society of Mechanical Engineers, 1992. http://dx.doi.org/10.1115/detc1992-0156.

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Abstract This paper follows a previous one on the computation of spatial displacements (Ravani and Ge, 1992). The first paper dealt with the problem of computing spatial displacements from a minimum number of simple features of points, lines, planes, and their combinations. The present paper deals with the same problem using a redundant set of the simple geometric features. The problem for redundant information is formulated as a least squares problem which includes all simple features. A Clifford algebra is used to unify the handling of various feature information. An algorithm for determining the best orientation is developed which involves finding the eigenvector associated with the least eigenvalue of a 4 × 4 symmetric matrix. The best translation is found to be a rational cubic function of the best orientation. Special cases are discussed which yield the best orientation in closed form. In addition, simple algorithms are provided for automatic generation of body-fixed coordinate frames from various feature information. The results have applications in robot and world model calibration for off-line programming and computer vision.
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