Deep-pipelined FPGA implementation of ellipse estimation for eye tracking

1. Linear systems: Introduction to Linear Systems. Gauss Elimination. Numerical Solutions. 2. Vectors: Vector Operations. Dot Product. Span. Linear Independence. The Product [Ax]. Cross Product. Lines, Planes, and Hyperplanes. 3. Matrices: Matrix Operations. Matrix Inverse. Elementary and Invertible Matrices. LU Factorization. 4. Vector spaces: Subspaces of [R to the n Power]. Vector Spaces. Linear Independence Bases. Dimension. Coordinate Vectors and Change of Basis. Rank and Nullity. Applications to Coding Theory. 5. Linear transformations: Matrix Transformations. Linear Transformations. Kernel and Range. The Matrix of a Linear Transformation. The Algebra of Linear Transformations. 6. Determinants: Determinants Cofactor Expansion. Properties of Determinants. The Adjoint Cramer's Rule. Determinants with Permutations. 7. Eigenvalues and eigenvectors: Eigenvalues and Eigenvectors. Diagonalization. Approximations of Eigenvalues and Eigenvectors. Applications to Dynamical Systems. Applications to Markov Chains. 8. Dot and inner products: Orthogonal Sets and Matrices. Orthogonal Projections Gram-Schmidt Process. The QR Factorization. Least Squares. Orthogonalization of Symmetric Matrices. Quadratic Forms and Conic Sections. The Singular Value Decomposition (SVD). Inner Products. Appendices: Answers to selected exercises Each chapter concludes with Applications, Mini-Projects, and Computer Exercises.

Determining the fodder composition is one of the important things to be done in animal raising because it may affect production. The process of determining the fodder composition is difficult to do because there are many things that must be considered at the same time, for example, the necessity to fulfill the nutrient needs while minimizing the total cost of the feed ingredients used. Evolutionary algorithms are often used to optimize the composition of animal feed with a random initial value. In this study, the use of numerical methods such as Cramer's Rule, Gauss-Elimination and Gauss-Jordan method is used as a solution for determining the initial value in evolutionary algorithms. The initial value which calculated using these three methods is the coefficient values that describe the amount of feed ingredients used in mixing fodder. The results showed that Cramer's Rule is better than Gauss-Elimination and Gauss-Jordan method with the difference in value of 7 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−13</sup> .

The calculus of building energy consumption is a demanding task because multiple factors must be considered during experimentation. Additionally, the definition of the model and the experiments is complex because the problem is multidisciplinary. When we face complex models and experiments that require a considerable amount of computational resources, the application of solutions is imperative to reduce the amount of time needed to define the model and the experiments and to obtain the answers. In this paper, we first address the definition and the implementation of an environmental model that describes the behavior of a building from a sustainability point of view and enables the use of several simulations and calculus engines in a cosimulation scenario. Second, we define a distributed experimental framework that enables us to obtain results in an accurate amount of time. This methodology has been applied to the energy consumption calculation, but it can also be applied to other modeling problems that usually require a considerable amount of resources by reducing the amount of time needed to perform modeling, implementation, verification, and experimentation.

CHAPTER FOURTEEN - MATRICES AND DETERMINANTS

CHAPTER TEN - MATRICES AND DETERMINANTS

In the previous installment, we looked at some generalpurpose linear solvers—some approaches to solving systems of simultaneous linear algebraic equations for cases in which there are no distinctive features offered by the coefficient matrix used in the description of the system. We concentrated on Gaussian elimination and LU decomposition. The computational effort required to solve an n-square system via either of those techniques is O(n3), although, as was pointed out, solution via LU decomposition requires only O(n2) arithmetic operations after the factorization is complete. Thus, solving a system of linear equations for more than one right-hand side (i.e., for more than one forcing vector) enjoys significant computational economy when LU decomposition is employed instead of Gaussian elimination.

Preface. Conventions and Notations. 1 An Introduction To Maple. 1.1 The Commands . 1.2 Programming. 2 Linear Systems of Equations and Matrices. 2.1 Linear Systems of Equations. 2.2 Augmented Matrix of a Linear System and Row Operations. 2.3 Some Matrix Arithmetic. 3 Gauss-Jordan Elimination and Reduced Row Echelon Form. 3.1 Gauss-Jordan Elimination and rref. 3.2 Elementary Matrices. 3.3 Sensitivity of Solutions to Error in the Linear System. 4 Applications of Linear Systems and Matrices. 4.1 Applications of Linear Systems to Geometry. 4.2 Applications of Linear Systems to Curve Fitting. 4.3 Applications of Linear Systems to Economics. 4.4 Applications of Matrix Multiplication to Geometry. 4.5 An Application of Matrix Multiplication to Economics. 5 Determinants, Inverses and Cramer s Rule. 5.1 Determinants and Inverses from the Adjoint Formula. 5.2 Determinants by Expanding Along Any Row or Column . 5.3 Determinants Found by Triangularizing Matrices. 5.4 LU Factorization. 5.5 Inverses from rref. 5.6 Cramer s Rule. 6 Basic Linear Algebra Topics. 6.1 Vectors. 6.2 Dot Product. 6.3 Cross Product. 6.4 Vector Projection. 7 A Few Advanced Linear Algebra Topics. 7.1 Rotations in Space. 7.2 Rolling a Circle Along a Curve. 7.3 The TNB Frame. 8 Independence, Basis and Dimension for Subspaces of Rn. 8.1 Subspaces of Rn. 8.2 Independent and Dependent Sets of Vectors in Rn. 8.3 Basis and Dimension for Subspaces of Rn. 8.4 Vector Projection onto a Subspace of Rn. 8.5 The Gram-Schmidt Orthonormalization Process. 9 Linear Maps from Rn to Rm. 9.1 Basics About Linear Maps. 9.2 The Kernel and Image Subspaces of a Linear Map. 9.3 Composites of Two Linear maps and Inverses. 9.4 Change of Bases for the Matrix Representation of a Linear Map. 10 The Geometry of Linear and Affine Maps. 10.1 The Effect of a Linear Map on Area and Arclength in Two Dimensions. 10.2 The Decomposition of Linear Maps into Rotations, Reflections and Rescalings in R2. 10.3 The Effect of Linear Maps on Volume, Area and Arclength in R3. 10.4 Rotations, Reflections and Rescalings in Three Dimensions. 10.5 Affine Maps. 11 Least Squares Fits and Pseudoinverses. 11.1 Pseudoinverse to a Non-Square Matrix and Almost Solving an Overdetermined Linear System. 11.2 Fits and Pseudoinverses. 11.3 Least Squares Fits and Pseudoinverses. 12 Eigenvalues and Eigenvectors. 12.1 What Are Eigenvalues and Eigenvectors, and Why Do We Need Them? 12.2 Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as well as the Exponential of a Matrix. 12.3 Applications of the Diagonalizability of Square Matrices. 12.4 Solving a Square First Order Linear. System of Differential Equations ... 12.5 Basic Facts About Eigenvalues and Eigenvectors, and Diagonalizability. 12.6 The Geometry of the Ellipse Using Eigenvalues and Eigenvectors. 12.7 A Maple Eigen-Procedure. Bibliography. Indices. Keyword Index. Index of Maple Commands and Packages.

Direct solvers for dense systems of linear equations commonly use partial pivoting to ensure numerical stability. However, pivoting can introduce significant performance overheads, such as synchronization and data movement, particularly on distributed systems. To improve the performance of these solvers, we present an alternative to pivoting in which numerical stability is obtained through additive updates. We implemented this approach using SLATE, a GPU-accelerated numerical linear algebra library, and evaluated it on the Summit supercomputer. Our approach provides better performance (up to 5-fold speedup) than Gaussian elimination with partial pivoting for comparable accuracy on most of the tested matrices. It also provides better accuracy (up to 15 more digits) than Gaussian elimination with no pivoting for comparable performance.

In this work, we present a dynamic memory allocation technique for a novel, horizontally partitioned memory subsystem targeting contemporary embedded processors with a memory management unit (MMU). We propose to replace the on-chip instruction cache with a scratchpad memory (SPM) and a small minicache. Serializing the address translation with the actual memory access enables the memory system to access either only the SPM or the minicache. Independent of the SPM size and based solely on profiling information, a postpass optimizer classifies the code of an application binary into apageableand acacheablecode region. The latter is placed at a fixed location in the external memory and cached by the minicache. The former, the pageable code region, is copied on demand to the SPM before execution. Both the pageable code region and the SPM are logically divided into pages the size of an MMU memory page. Using the MMU's pagefault exception mechanism, a runtime scratchpad memory manager (SPMM) tracks page accesses and copies frequently executed code pages to the SPM before they get executed. In order to minimize the number of page transfers from the external memory to the SPM, good code placement techniques become more important with increasing sizes of the MMU pages. We discuss code-grouping techniques and provide an analysis of the effect of the MMU's page size on execution time, energy consumption, and external memory accesses. We show that by using the data cache as a victim buffer for the SPM, significant energy savings are possible. We evaluate our SPM allocation strategy with fifteen applications, including H.264, MP3, MPEG-4, and PGP. The proposed memory system requires 8% less die are compared to a fully-cached configuration. On average, we achieve a 31% improvement in runtime performance and a 35% reduction in energy consumption with an MMU page size of 256 bytes.

This paper outlines the parallelisation and vectorisation methods we have used to port a LU decomposition library to the Xeon Phi co-processor. We ported a LU factorisation algorithm, which utilizes the Gaussian elimination method to perform the decomposition, using Intel LEO directives, OpenMP 4.0 directives, Intel's Cilk array notation, and vectorisation directives. We compare the performance achieved with these different methods, investigate the cost of data transfer on the overall time to solution, and analyse the impact of these optimization and parallelisation techniques on code running on the host processors as well. The results show that performance can be improved on the Xeon Phi by optimising the memory operations, and that Cilk array notation can benefit this benchmark on standard processors but do not have the same impact on the Xeon Phi co-processor. We have also demonstrated cases where the Xeon Phi will compute our implementations faster than we can run them on a node of a HPC system, and that our implementations are not as efficient as the LU factorisation implemented in the mkl library.

This paper intends to show the impact of compiler optimization flags and the variable's precision on direct methods to solve linear systems. The chosen six methods are simple direct methods, so our work could be a study for new researchers in this field. The methods are LU decomposition, LDU decomposition, Gaussian Elimination, Gauss-Jordan Elimination, Cholesky decomposition, and QR decomposition using the Gram-Schmidt orthogonalization process. Our study showed a huge difference in time between single- and double-precision in all methods, but the error encountered in the single-precision was not so high. Also, the best flags to these methods were the `-O3' and the `-Ofast' ones.

This note is concerned with the accuracy of the solution of nearly uncoupled Markov chains by a direct method based on the LU decomposition. It is shown that plain Gaussian elimination may fail in the presence of rounding errors. A modification of Gaussian elimination with diagonal pivoting and correction of small pivots is proposed and analyzed. It is shown that the accuracy of the solution is affected by two condition numbers associated with aggregation and the coupling respectively.

A GENERALIZED REDUCED GRADIENT ALGORITHM FOR SOLVING LARGE-SCALE DISCRETE-TIME NONLINEAR OPTIMAL CONTROL PROBLEMS

Application of the method of weighted residuals to mixed (split) boundary value problems. Triple series equations

In Gaussian elimination and LU factorization we solve a linear system by transforming it to triangular form. These are not the only kind of transformations that can be used for such a task. Matrices with orthonormal columns, called unitary matrices can be used to reduce a square matrix to upper triangular form and more generally a rectangular matrix to upper triangular (also called upper trapezoidal) form. This lead to a decomposition of a rectangular matrix known as a QR decomposition and a reduced form which we refer to as a QR factorization. The QR decomposition and factorization will be used in later chapters to solve least squares- and eigenvalue problems.