System equations

Many different manipulations of these equations have been used to obtain solutions. As discussed by King (1971), many of the older approaches work in terms of V/L, which has the disadvantage of being unbounded and which, in the classical implementation, leads to poorly convergent iterative calculations. A preferable arrangement of this equation system for solution is based on the ratio V/F, which must lie between 0 and 1. If we substitute in Equation (7-1) for L from Equation (7-2) and for y from Equation (7-4), and then divide by F, we obtain... 

The equation systems representing equilibrium separation calculations can be considered multidimensional, nonlinear objective functions... 

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. 

The equation system of eq.(6) can be used to find the input signal (for example a crack) corresponding to a measured output and a known impulse response of a system as well. This way gives a possibility to solve different inverse problems of the non-destructive eddy-current testing. Further developments will be shown the solving of eq.(6) by special numerical operations, like Gauss-Seidel-Method [4]. 

Here t. is the intrinsic lifetime of tire excitation residing on molecule (i.e. tire fluorescence lifetime one would observe for tire isolated molecule), is tire pairwise energy transfer rate and F. is tire rate of excitation of tire molecule by the external source (tire photon flux multiplied by tire absorjDtion cross section). The master equation system (C3.4.4) allows one to calculate tire complete dynamics of energy migration between all molecules in an ensemble, but tire computation can become quite complicated if tire number of molecules is large. Moreover, it is commonly tire case that tire ensemble contains molecules of two, tliree or more spectral types, and experimentally it is practically impossible to distinguish tire contributions of individual molecules from each spectral pool. 

Note that in equation system (2.64) the coefficients matrix is symmetric, sparse (i.e. a significant number of its members are zero) and banded. The symmetry of the coefficients matrix in the global finite element equations is not guaranteed for all applications (in particular, in most fluid flow problems this matrix will not be symmetric). However, the finite element method always yields sparse and banded sets of equations. This property should be utilized to minimize computing costs in complex problems. 

Let us suppose that we can convert the n x coefficient matrix in equation system (6.5) into an upper triangular form as... 

Weltin, E. A Numerical Method to Galculate Equilibrium Goncentrations for Single-Equation Systems, /. Chem. Educ. 1991, 68, 486M87. 

In order to eompute the time response of a dynamie system, it is neeessary to solve the differential equations (system mathematieal model) for given inputs. There are a number of analytieal and numerieal teehniques available to do this, but the one favoured by eontrol engineers is the use of the Laplaee transform. 

If a 33 = 0, we have a i3 = 0, and the function 03 is then a linear combination of the functions 0X and 0 2 and should be omitted in the orthogonalization process, which is here simply accomplished by means of the Gaussian elimination technique developed for solving equation systems. The connection between the matrices a and a may be written in the form ... 

The integration of this equation system was carried out numerically on a computer, after the partial pressures had been expressed as functions of... 

Equations, systems of Bairstow method,87 Newton-Raphson method, 86 Equilibrium... 

The approach has proven to be advantageous in comparison with the conventional" method. Because the reduced equation is a special form of the Reynolds equation, a full numerical solution over the entire computation domain, including both the hydrodynamic and the contact areas, thus obtained through a unified algorithm for solving one equation system. In this way, both hydrod5mamic and con-... 

This chapter describes a DML model proposed by the authors, based on the expectation that the Reynolds equation at the ultra-thin film limit would yield the same solutions as those from the elastic contact analysis. A unified equation system is therefore applied to the entire domain, which gives rise to a stable and robust numerical procedure, capable of predicting the tribological performance of the system through the entire process of transition from full-film to boundary lubrication. 

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... 

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 above method is the well-known Gauss-Newton method for differential equation systems and it exhibits quadratic convergence to the optimum. Computational modifications to the above algorithm for the incorporation of prior knowledge about the parameters (Bayessian estimation) are discussed in detail in Chapter 8. 

In this section we first present an efficient step-size policy for differential equation systems and we present two approaches to increase the region of convergence of the Gauss-Newton method. One through the use of the Information Index and the other by using a two-step procedure that involves direct search optimization. 

The proposed step-size policy for differential equation systems is fairly similar to our approach for algebraic equation models. First we start with the bisection rule. We start with g=l and we keep on halving it until an acceptable value, pa, has been found, i.e., we reduce p until... 

The main difference with differential equation systems is that every evaluation of the objective function requires the integration of the state equations, In this section we present an optimal step size policy proposed by Kalogerakis and Luus (1983b) which uses information only at g=0 (i.e., at k ) and at p=pa (i.e., at... 

The TDE moisture module (of the model) is formulated from three equations (1) the water mass balance equation, (2) the water momentum, (3) the Darcy equation, and (4) other equations such as the surface tension of potential energy equation. The resulting differential equation system describes moisture movement in the soil and is written in a one dimensional, vertical, unsteady, isotropic formulation as ... 

In case (a) each species can be calibrated and evaluated independently from the other. In this fully selective case, the following equation system corresponds to the matrix A in Eq. (6.66) ... 

A system of equations where the first unknown is missing from all subsequent equations and the second unknown is missing from all subsequent equations is said to be in echelon form. Every set or equation system comprised of linear equations can be brought into echelon form by using elementary algebraic operations. The use of augmented matrices can accomplish the task of solving the equation system just illustrated. 

Matrix A is termed the matrix of the equation system . The matrix formed by [A c] is termed the augmented matrix . For this problem the augmented matrix is given as ... 

Thus matrix operations provide a simplified method for solving equation systems as compared to elementary algebraic operations for linear equations. 

By multiplying [A] x [B] we can calculate the two basic equation systems to use in solving this problem as ... 

Now let us see what happens when we use pure, unadulterated matrix power to solve this equation system, such that... 

In the previous chapter [1] we promised a discussion of an easier way to solve equation systems - the method of determinants [2], To begin, given an X2/2 matrix [A] as... 

To use determinants to solve a system of linear equations, we look at a simple application given two equations and two unknowns. For the equation system... 

We can also solve for j82 by algebraic manipulation of the equation system. Elimination of the (3l term is accomplished by multiplying the first equation by Akll and the second equation by Akn and subtracting the results, dividing by the common term, and lastly, by converting both the numerator and the denominator to determinants, finally arriving at equation 6-12. 

---