Linear equation system

With constant steps and v = n eq.(5) can be described as linear equation system presented in matrix form ... 

Constant steps are not necessary, but they simplify the matrix g of eq.(6). Eq.(5) and eq.(6) respectively show the relationship between input and output signal for discrete signal processing. It is given by a linear equation system, which can easily be solved. 

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... 

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-... 

One sees that the ion flow caused by a gas is proportional to the partial pressure. The linear equation system can be solved only for the special instance where m = g (square matrix) it is over-identified for m> g. Due to unavoidable measurement error (noise, etc.) there is no set of overall ion flow Ig (partial pressures or concentrations) which satisfies the equation system exactly. Among all the conceivable solutions it is now necessary to identify set 1 which after inverse calculation to the partial ion flows 1, will exhibit the smallest squared deviation from the partial ion currents i actually measured. Thus ... 

It seems that we have to invert a matrix in each iteration, which takes roughly n3 operations. However, rewrite as a linear equation system in a correction d ... 

It may appear as if this is no great improvement, since finding a solution to a linear equation system with direct methods requires about n3 operations, about half as many as the inversion. However, the solution of the linear equation system can be accomplished by iterative methods where, in each step, some product jv is formed. Superficially, this cuts down, the number of operations, but still requires the Jacobian to be computed and stored. However, for a very large class of important problems, such a product can be efficiently computed without the need of precalculating or storing the Jacobian. 

After this digression, let us return to the linear equation system, repeated here for convenience ... 

The most prominent of these methods is probably the second order Newton-Raphson approach, where the energy is expanded as a Taylor series in the variational parameters. The expansion is truncated at second order, and updated values of the parameters are obtained by solving the Newton-Raphson linear equation system. This is the standard optimization method and most other methods can be treated as modifications of it. We shall therefore discuss the Newton-Raphson approach in more detail than the alternative methods. 

Let us consider a linear equation system, which we write it in the form ... 

By substituting the right side of Equation (15) into Equation (14) for each h, a linear equation system is obtained, the lith equation of which is ... 

The above equation evaluates to a tridiagonal linear equation system, after some arrangement,... 

Discrete dipole approximation. For particles with complex shape and/or complex composition, presently the only viable method for calculating optical properties is the discrete dipole approximation (DDA). This decomposes a grain in a very big number of cubes that are ascribed the polarizability a according to the dielectric function of the dust material at the mid-point of a cube. The mutual polarization of the cubes by the external field and the induced dipoles of all other dipoles is calculated from a linear equations system and the absorption and scattering efficiencies are derived from this. The method is computationally demanding. The theoretical background and the application of the method are described in Draine (1988) and Draine Flatau (1994). 

The system (3.3) is called underdetermined if < L. The system (3.3) is called overdetermined if > L. Very often in geophysical applications, we work with an overdetermined system wherein the number of observations exceeds the number of model parameters. At the same time, in many situations it may be necessary to work with an underdetermined system. We will examine both types of linear equation systems below. 

Local vertex invariants (LOVIs) obtained as the solutions of a linear equation system defined as ... 

Different sets of LOVIs can be obtained by different choices of matrices and vectors defining the linear equation system several combinations were studied on linear alkanes (Table M-6). 

Closely related to MPR descriptors are local vertex invariants called graph potentials denoted by U,- [Golender et al., 1981 Ivanciuc et ai, 1992]. They are calculated as the solutions of a linear equation system defined as ... 

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. 

When the extinction is measured at different frequencies/, this equation becomes a linear equation system, which can be solved for Cpp and q2(x). The key for the calculation of the particle-size distribution is the knowledge of the related extinction cross section K as a function of the dimensionless size parameter c = iTOifk. For spherical particles K can be evaluated directly from the acoustic scattering theory. A more general approach is an empirical method using measurements on reference instruments as input. 

The stable, implicit method from Crank and Nicolson can be used without this restriction. A generalisation of (2.286) delivers the tridiagonal linear equation system... 

The zones 1,2,... m shall have given temperatures and the zones m + 1, m + 2,... n shall have stipulated heat flows. The linear equation system for the radiosities becomes... 

In order to illustrate the technical problem with the help of the simplest mathematical formalism, and for the sake of simplicity, we first assume that the sensors are linear and that their responses are independent for each investigated chemical species. Therefore the Ay quantities are only calibration constants. From the basic algebra of linear equation systems, it then follows that one needs N independent equations to solve the equation system (1). Therefore, the number M of different sensors has to be larger than or equal to the number TV of chemical species, i.e.,... 

In the work of Lindborg et al [119], the resulting linear equation systems were solved with preconditioned Krylov subspace projection methods [166]. The Poisson equation was solved by a conjugate gradient (CG)-solver, while the other transport equations were solved using a bi-conjugate gradient (BCG)-solver which can handle also non-symmetric equations systems. The solvers were preconditioned with a Jacobi preconditioner. 

The most commonly adopted procedure for optimizing the variational parameters in (24) is the non-linear Newton-Raphson procedure . The energy expression is expanded to second order in the parameters S and T. By assuming the first-order derivatives of this expression to be zero, a linear equation system is obtained in S and... 

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