Search results for "Diagonal matrix"
showing 10 items of 19 documents
Bicausative matrices to measure structural change: Are they a good tool?
1999
The causative-matrix method to analyze temporal change assumes that a matrix transforms one Markovian transition matrix into another by a left multiplication of the first matrix; the method is demand-driven when applied to input-output economics. An extension is presented without assuming the demand-driven or supply-driven hypothesis. Starting from two flow matrices X and Y, two diagonal matrices are searched, one premultiplying and the second postmultiplying X, to obtain a result the closer as possible to Y by least squares. The paper proves that the method is deceptive because the diagonal matrices are unidentified and the interpretation of results is unclear. Keywords : Input-Output ; Ch…
Numerical experiments with a parallel fast direct elliptic solver on Cray T3E
1997
A parallel fast direct O(N log N) solver is shortly described for linear systems with separable block tridiagonal matrices. A good parallel scalability of the proposed method is demonstrated on a Cray T3E parallel computer using MPI in communication. Also, the sequential performance is compared with the well-known BLKTRI-implementation of the generalized. cyclic reduction method using a single processor of Cray T3E.
On the hyperbolicity of certain models of polydisperse sedimentation
2012
The sedimentation of a polydisperse suspension of small spherical particles dispersed in a viscous fluid, where particles belong to N species differing in size, can be described by a strongly coupled system of N scalar, nonlinear first-order conservation laws for the evolution of the volume fractions. The hyperbolicity of this system is a property of theoretical importance because it limits the range of validity of the model and is of practical interest for the implementation of numerical methods. The present work, which extends the results of R. Burger, R. Donat, P. Mulet, and C.A. Vega (SIAM Journal on Applied Mathematics 2010; 70:2186–2213), is focused on the fluxes corresponding to the …
The tensor of interaction of a two-level system with an arbitrary strain field
2007
The interaction between two-level systems (TLS) and strain fields in a solid is contained in the diagonal matrix element of the interaction hamiltonian, $\delta$, which, in general, has the expression $\delta=2[\gamma]:[S]$, with the tensor $[\gamma]$ describing the TLS ``deformability'' and $[S]$ being the symmetric strain tensor. We construct $[\gamma]$ on very general grounds, by associating to the TLS two objects: a direction, $\hat\bt$, and a forth rank tensor of coupling constants, $[[R]]$. Based on the method of construction and on the invariance of the expression of $\delta$ with respect to the symmetry transformation of the solid, we conclude that $[[R]]$ has the same structure as …
A branch and bound algorithm for the matrix bandwidth minimization
2008
In this article, we first review previous exact approaches as well as theoretical contributions for the problem of reducing the bandwidth of a matrix. This problem consists of finding a permutation of the rows and columns of a given matrix which keeps the non-zero elements in a band that is as close as possible to the main diagonal. This NP-complete problem can also be formulated as a labeling of vertices on a graph, where edges are the non-zero elements of the corresponding symmetrical matrix. We propose a new branch and bound algorithm and new expressions for known lower bounds for this problem. Empirical results with a collection of previously reported instances indicate that the propose…
Robust and Efficient IMEX Schemes for Option Pricing under Jump-Diffusion Models
2013
We propose families of IMEX time discretization schemes for the partial integro-differential equation derived for the pricing of options under a jump diffusion process. The schemes include the families of IMEX-midpoint, IMEXCNAB and IMEX-BDF2 schemes. Each family is defined by a convex parameter c ∈ [0, 1], which divides the zeroth-order term due to the jumps between the implicit and explicit part in the time discretization. These IMEX schemes lead to tridiagonal systems, which can be solved extremely efficiently. The schemes are studied through Fourier stability analysis and numerical experiments. It is found that, under suitable assumptions and time step restrictions, the IMEX-midpoint fa…
Six Matrix Adjustment Problems Solved by Some Fundamental Theorems on Biproportion
2011
After defining biproportion (or RAS) rigorously, we recall two fundamental theorems: unicity of biproportion (any biproportional algorithm leads to the same solution than biproportion, which turns biproportion into a mathematical tool as indisputable than proportion), ineffectiveness of separability (premultiplying or post multiplying the initial matrix by a diagonal matrix does not change the biproportional solution) and its corollary (it is equivalent to do a separable modification of the initial matrix or to do a proportional change of each biproportional factors). We then apply these theorems to show immediately that: i) no difficulties are encountered when solving the biproportional pr…
On the implementation of weno schemes for a class of polydisperse sedimentation models
2011
The sedimentation of a polydisperse suspension of small rigid spheres of the same density, but which belong to a finite number of species (size classes), can be described by a spatially one-dimensional system of first-order, nonlinear, strongly coupled conservation laws. The unknowns are the volume fractions (concentrations) of each species as functions of depth and time. Typical solutions, e.g. for batch settling in a column, include discontinuities (kinematic shocks) separating areas of different composition. The accurate numerical approximation of these solutions is a challenge since closed-form eigenvalues and eigenvectors of the flux Jacobian are usually not available, and the characte…
Time-harmonic elasticity with controllability and higher-order discretization methods
2008
The time-harmonic solution of the linear elastic wave equation is needed for a variety of applications. The typical procedure for solving the time-harmonic elastic wave equation leads to difficulties solving large-scale indefinite linear systems. To avoid these difficulties, we consider the original time dependent equation with a method based on an exact controllability formulation. The main idea of this approach is to find initial conditions such that after one time-period, the solution and its time derivative coincide with the initial conditions.The wave equation is discretized in the space domain with spectral elements. The degrees of freedom associated with the basis functions are situa…
Tridiagonal preconditioning for Poisson-like difference equations with flat grids: Application to incompressible atmospheric flow
2011
AbstractThe convergence of many iterative procedures, in particular that of the conjugate gradient method, strongly depends on the condition number of the linear system to be solved. In cases with a large condition number, therefore, preconditioning is often used to transform the system into an equivalent one, with a smaller condition number and therefore faster convergence. For Poisson-like difference equations with flat grids, the vertical part of the difference operator is dominant and tridiagonal and can be used for preconditioning. Such a procedure has been applied to incompressible atmospheric flows to preserve incompressibility, where a system of Poisson-like difference equations is …