Solution of linear equation system

This matrix formulation may be used in the iterative procedure by replacing the inner cycle with the solution of linear equation system of eq.(51) (Coitino et al., 1995a). However, this approach could be too cumbersome a more interesting application is the direct minimization of the free energy functional. We need to make a digression here. 

Hestenes and Stiefel (1952) explained their approach for the solution of linear equation systems with sparse symmetric positive definite matrices ... 

Solution of Linear Equation Systems after we have just obtained ... 

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... 

Let us estimate the correspondent left eigenvector f (a vector row). The eigenvalue is known, hence it is easy to do just by solution of linear equations. This system of —1 equations is ... 

Precise analytical solutions of the equation system (5.22) and (5.23) may be found comparatively simply for weak excitation when the approximation ry7K, Tp/TK,u3/-fK,uJs/TK — 0 is valid. In this case the system of equations (5.22), (5.23) is of much simpler form [303]. Examples of such solutions may be found in [96, 133, 303]. For strong excitation, when the interaction between the molecular ensemble and light becomes non-linear, whilst the above parameters still remain smaller than unity, the solution for polarization moments may be obtained in the form of an expansion over the powers of these parameters. Finally, at excitation by very strong irradiation, when non-linearity is considerable, the determination of polarization moments fq and numerical methods for solving Eqs. (5.22) and (5.23). 

The idea is satisfactory solely for problems in which constraints play only a marginal role and are small in number. Alternatively, it is preferable to view the problem as a solution of an equation system with a parametric method (Chapter 14) rather than as an optimization problem. This is particularly true in the case of many linear constraints. 

Solution of Systems of Linear Equations. Systems of linear equations were studied shortly after the introduction of variables (examples existed in early Babylonia). Solving a system of linear equations involves determining whether a solution exists and then using either a direct or an iterative method to find the solution. The earliest iterative solutions of linear system were developed by Gauss, and newer iterative algorithms are stiU published. Many of the problems of science are expressed as systems of linear equations, such as balancing chemical equations, and are solved when the linear system is solved. 

The Finite Element Method (FEM), which means method of elements with limited size, is a powerful tool for numerical solutions of mechanical problems of elastic and plastic materials. The basis is the calculation of linear equation systems by a computer. The system to calculate, i.e. structme, is divided into fitting elements... 

Linear equation systems of the form Ax = b are conventional in practical calculation problems. For solution of such equation systems by pocket calculator, one can benefit from Cramer s formula in case of two or three unknowns. The advantage in this solution method is, for example, that keying and calculation can be performed in a simple and systematic manner. 

Table 2.3 illustrates how the number of operations required by Cramer s rule increases as the value of n increases. Forn = 3, a total of 51 multiplications and divisions are needed. However, when n = 10, this number climbs to 359,251,210. For this reason, Cramer s rule is rarely used for systems with n > 3, The Gauss elimination, Gauss-Jordan reduction, and Gauss-Seidel methods, to be described in the next three sections of this chapter, are much more efficient methods of solution of linear equations than Cramer s rule. 

These equations reduce to a 3 x 3 matrix Ricatti equation in this case. In the appendix of [20], the efficient iterative solution of this nonlinear system is considered, as is the specialization of the method for linear and planar molecules. In the special case of linear molecules, the SHAKE-based method reduces to a scheme previously suggested by Fincham[14]. 

C. Lanczos. Solution of systems of linear equations by minimized iterations. J. Res. Nat. Bureau Standards, 49 33-53, 1952. 

Ire boundary element method of Kashin is similar in spirit to the polarisable continuum model, lut the surface of the cavity is taken to be the molecular surface of the solute [Kashin and lamboodiri 1987 Kashin 1990]. This cavity surface is divided into small boimdary elements, he solute is modelled as a set of atoms with point polarisabilities. The electric field induces 1 dipole proportional to its polarisability. The electric field at an atom has contributions from lipoles on other atoms in the molecule, from polarisation charges on the boundary, and where appropriate) from the charges of electrolytes in the solution. The charge density is issumed to be constant within each boundary element but is not reduced to a single )oint as in the PCM model. A set of linear equations can be set up to describe the electrostatic nteractions within the system. The solutions to these equations give the boundary element harge distribution and the induced dipoles, from which thermodynamic quantities can be letermined. 

Iterative improvement of the solution of systems of linear equations... 

Because of the work involved in solving large systems of simultaneous linear equations it is desirable that only a small number of us be computed. Thus the gaussian integration formulas are useful because of the economy they offer. See references on numerical solutions of integral equations. 

The coefficients CK for a solution to the Schrodinger equation (Eq. II. 1) may now be determined by the variation principle (Eq. II.7) which leads to an infinite system of linear equations... 

General equations of momentum and energy balance for dispersed two-phase flow were derived by Van Deemter and Van Der Laan (V2) by integration over a volume containing a large number of elements of the dispersed phase. A complete system of solutions of linearized Navier-Stokes equations... 

If the matrix A is positive definite, i.e. it is symmetric and has positive eigenvalues, the solution of the linear equation system is equivalent to the minimization of the bilinear form given in Eq. (64). One of the best established methods for the solution of minimization problems is the method of steepest descent. The term steepest descent alludes to a picture where the cost function F is visualized as a land-... 

The solution of a system of linear equations depends on certain condi-dons, viz. 

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