Linear algebraic system

Forsythe, G. E. and Meier, C. B., 1967. Computer Solution of Linear Algebraic Systems, Prentice Hall, Englewood Cliffs, NJ. 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

In CED, a number of different iterative solvers for linear algebraic systems have been applied. Two of the most successful and most widely used methods are conjugate gradient and multigrid methods. The basic idea of the conjugate gradient method is to transform the linear equation system Eq. (38) into a minimization problem... 

An iterative solution method for linear algebraic systems which damps the shortwave components of the iteration error very fast and, after a few iterations, leaves predominantly long-wave components. The Gauss-Seidel method [85] could be chosen as a suitable solver in this context. 

Solving the linear algebraic system (eqn. (7.5)) results in a linear interpolation equation given by... 

A linear algebraic system of rate equations for the fast species results, which can be solved a priori. Hence a strongly reduced (in the number of species to be treated) system is obtained. This concept originates from astrophysical applications and from Laser physics. It is in some instances also referred to as collisional-radiative approximation , for the fast species, lumped species concept , bundle-n method or intrinsic low dimensional manifold (ILDM) method in the literature. We refer to [9,12,13] for further references on this. 

We will get a non-trivial solution for these, if and only if the determinant of the associated matrix of the linear algebraic system given by above equations is zero i.e.,... 

The analysis of non-linear mechanisms and corresponding kinetic models are much more difficult than that of linear ones. The obvious difficulty in this case is the follows an explicit solution for steady-state reaction rate R can be obtained only for special non-linear algebraic systems of steady-state (or pseudo-steady-state) equations. In general case it is impossible to solve explicitly a system of non-linear steady-state (or pseudo-steady-state) equations. However, in the case of mass-action-law-model it is always possible to apply to this system a method of elimination of variables and reduce it to a polynomial in one variable [4], i.e., a polynomial in terms of the steady-state reaction rate. We refer a polynomial in the steady-state reaction as a kinetic polynomial. The idea of this polynomial was firstly emphasized in [5]. 

The degrees of freedom analysis DOF) allows the user to determine the variables needed to be specified to execute a simulation. In steady state simulation the degrees of freedom are the number of variables that must be assigned to solve the non-linear algebraic system describing the operational unit, Here we adopt the approach called variable-minus-equations, in which DOF is equal with the number of variables minus the number of independent equations ... 

The solution of Equations 12.38 and 12.39 (which is a linear algebraic system for the p/s) provides the instantaneous radical-type distribution. Once a procedure to explicitly calculate this distribution has been made available, it is possible to apply the pseudo-homopolymer approach by performing the following steps ... 

The linear algebraic system consists of 3n - 3 equations and 3n - 3 unknowns that can be solved to produce the optimal piecewise cubic spline. Press, Teukolsky, Vetterling, and Flannery describe a routine for cubic spline interpolation. " ... 

What about the solution of the linear algebraic systems. Direct solvers such as MA28 from the Harwell Library perform very well in ID. After testing a lot of different iterative schemes in 2D the BICGSTAB-algorithm [12] preconditioned by SSOR turned out to perform best for the current reaction-diffusion system. 

Error vector in multigrid method outline Unit vector with components e, in the i i = 1,2,3) directions Unit vector in the direction of incoming light in LDA Unit vector in the direction of scattered light in LDA Linear algebraic system source vector used in WRM Net force acting on a single particle (N)... 

Here, we have introduced some rather abstract concepts (vector spaces, linear transformations) to analyze the properties of linear algebraic systems. For a fuller theoretical treatment of these concepts, and their extension to include systems of differential equations, consult Naylor Sell (1982). 

Here, we solve aboundary value problem from fluidmechanics numerically by converting it into a linear algebraic system. As this example makes clear, it is sometimes possible to reduce greatly the computational burden of elimination when the matrix is banded, i.e., aU nonzero elements are found near the principal diagonal. 

We wish to solve this problem numerically using the method of finite differences for a uniform grid of points Xj,j = 1,2,..., in the interior of the domain, as was done in the flow example for a single field. We compute numerically the field values at each point, (p Xj) = (pj and if xj) = fj, by solving through elimination the resulting linear algebraic system. 

In the former option, the nser supplies a sparse matrix S whose sparsity pattern (location of nonzero elements) matches that of the Jacobian. That is, even though the Jacobian may be difficult to compute analytically, the user can at least specify that only a small subset of Jacobian elements are known to be nonzero, fsolve can use this information to reduce the computational burden and memory requirement when generating an approximate Jacobian. With JacobMult , the user supplies the name of a routine that returns the product of the Jacobian matrix with an input vector. The usefulness of this option will become clearer after our discussion of iterative methods for solving linear algebraic systems in Chapter 6. 

This yields the following linear algebraic system for xf +i- 9+ l... 

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