scholarly journals Products of Irreducible Random Matrices in the (Max, +) Algebra

1997 ◽  
Vol 29 (2) ◽  
pp. 444-477 ◽  
Author(s):  
Jean Mairesse
Keyword(s):  
Stationary Regime ◽  
Recursive Equation ◽  
Periodic Regime ◽  
Jackson Networks ◽  
Matrix Vector Multiplication ◽  
The Matrix ◽  
Max Algebra ◽  
Matrix Vector ◽  
Stochastic Event Graphs ◽  
Stochastic Event

We consider the recursive equation x(n + 1)= A(n)⊗x(n), where x(n + 1) and x(n) are ℝk-valued vectors and A(n) is an irreducible random matrix of size k × k. The matrix-vector multiplication in the (max, +) algebra is defined by (A(n)⊗x(n))= maxj (Aij (n) + xj(n)). This type of equation can be used to represent the evolution of stochastic event graphs which include cyclic Jackson networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence {A(n), n ∈ ℕ} is i.i.d. or, more generally, stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matrices such that and the matrices have a unique periodic regime.

Products of Irreducible Random Matrices in the (Max, +) Algebra

1997 ◽  
Vol 29 (02) ◽  
pp. 444-477 ◽  
Author(s):  
Jean Mairesse
Keyword(s):  
Stationary Regime ◽  
Recursive Equation ◽  
Periodic Regime ◽  
Jackson Networks ◽  
Matrix Vector Multiplication ◽  
The Matrix ◽  
Max Algebra ◽  
Matrix Vector ◽  
Stochastic Event Graphs ◽  
Stochastic Event

We consider the recursive equation x(n + 1)= A(n)⊗x(n), where x(n + 1) and x(n) are ℝ k -valued vectors and A(n) is an irreducible random matrix of size k × k. The matrix-vector multiplication in the (max, +) algebra is defined by (A(n)⊗x(n))= maxj (Aij (n) + xj (n)). This type of equation can be used to represent the evolution of stochastic event graphs which include cyclic Jackson networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence {A(n), n ∈ ℕ} is i.i.d. or, more generally, stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matrices such that and the matrices have a unique periodic regime.

scholarly journals Improved Generalized Single-Source Tangential Equivalence Principle Algorithm with Contact-Region Modeling Method for Array Structures

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yao Han ◽  
Hanru Shao ◽  
Jianfeng Dong
Keyword(s):  
Equivalence Principle ◽  
Contact Region ◽  
Near Field ◽  
Single Source ◽  
Matrix Vector Multiplication ◽  
The Matrix ◽  
Array Structures ◽  
Fast Multipole Algorithm ◽  
Matrix Vector ◽  
Equivalence Principle Algorithm

An improved generalized single-source tangential equivalence principle algorithm (GSST-EPA) is proposed for analyzing array structures with connected elements. In order to use the advantages of GSST-EPA, the connected array elements are decomposed and computed by a contact-region modeling (CRM) method, which makes that each element has the same meshes. The unknowns of elements can be transferred onto the equivalence surfaces by GSST-EPA. The scattering matrix in GSST-EPA needs to be solved and stored only once due to the same meshes for each element. The shift invariant of translation matrices is also used to reduce the computation of near-field interaction. Furthermore, the multilevel fast multipole algorithm (MLFMA) is used to accelerate the matrix-vector multiplication in the GSST-EPA. Numerical results are shown to demonstrate the accuracy and efficiency of the proposed method.

Sparse matrix–vector multiplication

2020 ◽  
pp. 190-290
Author(s):  
Rob H. Bisseling
Keyword(s):  
Sparse Matrix ◽  
Sparse Matrices ◽  
Distributed Shared Memory ◽  
Sparsity Pattern ◽  
Matrix Vector Multiplication ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Vector ◽  
Memory Architectures ◽  
Shared Memory Architectures

This chapter introduces irregular algorithms and presents the example of parallel sparse matrix-vector multiplication (SpMV), which is the central operation in iterative linear system solvers. The irregular sparsity pattern of the matrix does not change during the multiplication, which may be repeated many times. This justifies putting a lot of effort into finding a good data distribution. The Mondriaan distribution of a sparse matrix is a useful non-Cartesian distribution that can be found by hypergraph-based partitioning. The Mondriaan package implements such a partitioning and also the newer medium-grain partitioning method. The chapter analyses the special cases of random sparse matrices and Laplacian matrices. It uses performance profiles and geometric means to compare different partitioning methods. Furthermore, it presents the hybrid-BSP model and a hybrid-BSP SpMV, which are aimed at hybrid distributed/shared-memory architectures. The parallel SpMV can be incorporated in applications, ranging from PageRank computation to artificial neural networks.

scholarly journals Solving a Modified Syndrome Decoding Problem using Integer Programming

2020 ◽  
Vol 15 (5) ◽  
Author(s):  
Vlad-Florin Dragoi ◽  
Pierre-Louis Cayrel ◽  
Brice Colombier ◽  
Dominic Bucerzan ◽  
Sorin Hoara
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Simplex Algorithm ◽  
Integer Vector ◽  
Matrix Vector Multiplication ◽  
Syndrome Decoding ◽  
The Matrix ◽  
Valid Solution ◽  
Matrix Vector

In this article, we model a variant of the well-known syndrome decoding problem as a linear optimization problem. Most common algorithms used for solving optimization problems, e.g. the simplex algorithm, fail to find a valid solution for the syndrome decoding problem over a finite field. However, our simulations prove that a slightly modified version of the syndrome decoding problem can be solved by the simplex algorithm. More precisely, the algorithm returns a valid error vector when the syndrome vector is an integer vector, i.e.,the matrix-vector multiplication, is realized over Z, instead of Fq.

scholarly journals Linear Transformation Recognition Using Radon Transform

2017 ◽  
Vol 1 (2) ◽  
pp. 40-47
Author(s):  
Fawaz Hjouj
Keyword(s):  
Image Processing ◽  
Linear Transformation ◽  
Radon Transform ◽  
General Linear ◽  
Regular Functions ◽  
Matrix Vector Multiplication ◽  
Special Cases ◽  
Radon Projections ◽  
The Matrix ◽  
Matrix Vector

Given two regular functions (images) f and g on R2 where g is formed from f by a general linear transformation, g(x) = f (Ax + b). We present a procedure to determine the transformation ‘parameters’ A and b using Radon projections of f and only two projections of g. We use these projections together with simple facts on matrix vector multiplication to recover the matrix A. The assumptions we have here are: f is nonnegative and A is nonsingular. Commonly used transformations in image processing such as rotation, scaling and others are special cases of our approach.

scholarly journals Matrix Mapping on Crossbar Memory Arrays with Resistive Interconnects and Its Use in In-Memory Compression of Biosignals

Micromachines ◽  
2019 ◽  
Vol 10 (5) ◽  
pp. 306 ◽  
Author(s):  
Yoon Kyeung Lee ◽  
Jeong Woo Jeon ◽  
Eui-Sang Park ◽  
Chanyoung Yoo ◽  
Woohyun Kim ◽  
...  
Keyword(s):  
Voltage Drop ◽  
Information Storage ◽  
Mapping Algorithm ◽  
Data Intensive ◽  
Matrix Vector Multiplication ◽  
The Matrix ◽  
Communication Devices ◽  
Matrix Mapping ◽  
Matrix Vector ◽  
The Relationship

Recent advances in nanoscale resistive memory devices offer promising opportunities for in-memory computing with their capability of simultaneous information storage and processing. The relationship between current and memory conductance can be utilized to perform matrix-vector multiplication for data-intensive tasks, such as training and inference in machine learning and analysis of continuous data stream. This work implements a mapping algorithm of memory conductance for matrix-vector multiplication using a realistic crossbar model with finite cell-to-cell resistance. An iterative simulation calculates the matrix-specific local junction voltages at each crosspoint, and systematically compensates the voltage drop by multiplying the memory conductance with the ratio between the applied and real junction potential. The calibration factors depend both on the location of the crosspoints and the matrix structure. This modification enabled the compression of Electrocardiographic signals, which was not possible with uncalibrated conductance. The results suggest potential utilities of the calibration scheme in the processing of data generated from mobile sensing or communication devices that requires energy/areal efficiencies.

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