A RECURSIVE METHOD FOR SOLVING HAPLOTYPE FREQUENCIES WITH APPLICATION TO GENETICS

2006 ◽  
Vol 04 (06) ◽  
pp. 1269-1285
Author(s):  
MICHAEL K. NG ◽  
ERIC S. FUNG ◽  
YIU-FAI LEE ◽  
WAI-KI CHING
Keyword(s):  
Recursive Algorithm ◽  
Recursive Method ◽  
Memory Requirement ◽  
Hapmap Data ◽  
Fast Solvers ◽  
Numerical Examples ◽  
Multiple Loci ◽  
Haplotype Frequencies ◽  
The Matrix ◽  
Large Linear System

Multiple loci analysis has become popular with the advanced developments in biological experiments. A lot of studies have been focused on the biological and the statistical properties of such multiple loci analysis. In this paper, we study one of the important computational problems: solving the probabilities of haplotype classes from a large linear system Ax = b derived from the recombination events in multiple loci analysis. Since the size of the recombination matrix A increases exponentially with respect to the number of loci, fast solvers are required to deal with a large number of loci in the analysis. By exploiting the nice structure of the matrix A, we develop an efficient recursive algorithm for solving such structured linear systems. In particular, the complexity of the proposed algorithm for the n loci problem is of O(n2n) operations and the memory requirement is of O(2n) locations for the 2n-by-2n matrix A. Numerical examples are given to demonstrate the effectiveness of our efficient solver. Finally, we apply our proposed method to analyze the haplotype classes for a set of single nucleotides polymorphisms (SNPs) from Hapmap data.

scholarly journals A New Iterative Algorithm for Solving a Class of Matrix Nearness Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Xuefeng Duan ◽  
Chunmei Li
Keyword(s):  
Iterative Algorithm ◽  
Projection Algorithm ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Von Neumann ◽  
Numerical Examples ◽  
Matrix Nearness Problem ◽  
Least Frobenius Norm Solution ◽  
Alternating Projection ◽  
The Matrix

Based on the alternating projection algorithm, which was proposed by Von Neumann to treat the problem of finding the projection of a given point onto the intersection of two closed subspaces, we propose a new iterative algorithm to solve the matrix nearness problem associated with the matrix equations AXB=E, CXD=F, which arises frequently in experimental design. If we choose the initial iterative matrix X0=0, the least Frobenius norm solution of these matrix equations is obtained. Numerical examples show that the new algorithm is feasible and effective.

The Elastic Field of an Elliptic Inclusion With a Slipping Interface

1986 ◽  
Vol 53 (1) ◽  
pp. 103-107 ◽  
Author(s):  
E. Tsuchida ◽  
T. Mura ◽  
J. Dundurs
Keyword(s):  
Closed Form ◽  
Infinite Series ◽  
Elastic Field ◽  
Elliptic Inclusion ◽  
Numerical Examples ◽  
Elastic Fields ◽  
Bonded Interface ◽  
The Matrix ◽  
Displacement Potentials ◽  
Uniform Expansion

The paper analyzes the elastic fields caused by an elliptic inclusion which undergoes a uniform expansion. The interface between the inclusion and the matrix cannot sustain shear tractions and is free to slip. Papkovich–Neuber displacement potentials are used to solve the problem. In contrast to the perfectly bonded interface, the solution cannot be expressed in closed form and involves infinite series. The results are illustrated by numerical examples.

Orthonormal Isotropic Vector Bases

1998 ◽  
Author(s):  
Namik Ciblak ◽  
Harvey Lipkin
Keyword(s):  
Recursive Algorithm ◽  
Computer Applications ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Orthonormal Bases ◽  
Isotropic Vector ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Deviatoric Stresses ◽  
Isotropic Vectors

Abstract Orthonormal bases of isotropic vectors for indefinite square matrices are proposed and solved. A necessary and sufficient condition is that the matrix must have zero trace. A recursive algorithm is presented for computer applications. The isotropic vectors of 3 × 3 matrices are solved explicitly. Deviatoric stresses in continuum mechanics, the existence of isotropic vectors (particularly in screw space), and stiffness synthesis by springs are shown to be related to the isotropic vector problem.

COUPLED MAGNETIC FIELD AND VISCOELASTICITY OF FERROGEL

2011 ◽  
Vol 03 (02) ◽  
pp. 259-278 ◽  
Author(s):  
YI HAN ◽  
WEI HONG ◽  
LEANN FAIDLEY
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Material Model ◽  
Finite Element Code ◽  
Equilibrium Thermodynamics ◽  
Numerical Examples ◽  
Rate Dependent ◽  
The Matrix ◽  
Specific Material ◽  
Soft Polymer

Composed of a soft polymer matrix and magnetic filler particles, ferrogel is a smart material responsive to magnetic fields. Due to the viscoelasticity of the matrix, the behaviors of ferrogel are usually rate-dependent. Very few models with coupled magnetic field and viscoelasticity exist in the literature, and even fewer are capable of reliable predictions. Based on the principles of non-equilibrium thermodynamics, a field theory is developed to describe the magneto-viscoelastic property of ferrogel. The theory provides a guideline for experimental characterizations and structural designs of ferrogel-based devices. A specific material model is then selected and the theory is implemented in a finite element code. Through numerical examples, the responses of a ferrogel in uniform and non-uniform magnetic fields are analyzed. The dynamic response of a ferrogel to cyclic magnetic fields is also studied, and the prediction agrees with our experimental results. In the reversible limit, our theory recovers existing models for elastic ferrogel, and is capable of capturing some instability phenomena.

scholarly journals Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Berna Bülbül ◽  
Mehmet Sezer
Keyword(s):  
Matrix Method ◽  
Numerical Approach ◽  
Duffing Equation ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Solution Function ◽  
Numerical Examples ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems.

A Discrete Element Method for the Analysis of Plane Elasto-Plastic Stress Problems

1966 ◽  
Vol 17 (1) ◽  
pp. 83-104 ◽  
Author(s):  
G. G. Pope
Keyword(s):  
Discrete Element Method ◽  
Plane Stress ◽  
Discrete Element ◽  
Stress Strain ◽  
Numerical Examples ◽  
Matrix Notation ◽  
The Matrix ◽  
Triangular Elements ◽  
Plastic Stress ◽  
Element Method

SummaryA procedure is developed for the analysis of plane stress problems when yielding occurs locally. The region is divided into triangular elements and the deformation is analysed on a step-by-step basis, using the matrix notation developed by Argyris. The simple expressions which are derived for the element properties are applicable with any stress-strain relations which are stable and time-independent. Simple numerical examples are given.

The Stress Field and Intensity Factor due to Crazes Formed at the Poles of a Spherical Inhomogeneity

1994 ◽  
Vol 61 (4) ◽  
pp. 803-808 ◽  
Author(s):  
Z. M. Xiao ◽  
K. D. Pae
Keyword(s):  
Stress Intensity Factor ◽  
Intensity Factor ◽  
Stress Field ◽  
Elastic Moduli ◽  
Superposition Principle ◽  
Numerical Examples ◽  
Equivalent Inclusion Method ◽  
Equivalent Inclusion ◽  
The Matrix ◽  
Good Agreement

The problem of two penny-shaped crazes formed at the top and the bottom poles of a spherical inhomogeneity has been investigated. The inhomogeneity is embedded in an infinitely extended elastic body which is under uniaxial tension. Both the inhomogeneity and the matrix are isotropic but have different elastic moduli. The analysis is based on the superposition principle of the elasticity theory and Eshelby’s equivalent inclusion method. The stress field inside the inhomogeneity and the stress intensity factor on the boundary of the craze are evaluated in the form of a series which involves the ratio of the radius of the penny-shaped craze to the radius of the spherical inhomogeneity. Numerical examples show the interaction between the craze and the inhomogeneity is strongly affected by the elastic properties of the inhomogeneity and the matrix. The conclusion deduced from the numerical results is in good agreement with experimental results given in the literature.

A logarithmic reduction algorithm for quasi-birth-death processes

1993 ◽  
Vol 30 (3) ◽  
pp. 650-674 ◽  
Author(s):  
Guy Latouche ◽  
V. Ramaswami
Keyword(s):  
Queueing Theory ◽  
Quadratic Convergence ◽  
Simple Algorithm ◽  
Rate Matrix ◽  
Computer Performance ◽  
Numerical Examples ◽  
Markov Chain Models ◽  
The Matrix ◽  
Logarithmic Reduction ◽  
Birth Death

Quasi-birth-death processes are commonly used Markov chain models in queueing theory, computer performance, teletraffic modeling and other areas. We provide a new, simple algorithm for the matrix-geometric rate matrix. We demonstrate that it has quadratic convergence. We show theoretically and through numerical examples that it converges very fast and provides extremely accurate results even for almost unstable models.

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