Matrix vector difference equation

Equation (8.76) is called the matrix vector difference equation and can be used for the recursive discrete-time simulation of multivariable systems. 

This means that onee A is known, it ean be multiplied into several b veetors to generate a solution set x = A b for each b vector. It is easier and faster to multiply a matrix into a vector than it is to solve a set of simultaneous equations over and over for the same coefficient matrix but different b vectors. 

The models discrete in space and continuous in time as well as those continuous in space and time, led many times to non-linear differential equations for which an analytical solution is extremely difficult or impossible. In order to solve the equations, simplifications, e.g. linearization of expressions and assumptions must be carried out. However, if this is not sufficient, one must apply numerical solutions. This led the author to a major conclusion that there are many advantages of using Markov chains which are discrete in time and space. The major reason is that physical models can be presented in a unified description via state vector and a one-step transition probability matrix. Additional reasons are detailed in Chapter 1. It will be shown later that this presentation coincides also with the fact that it yields the finite difference equations of the process under consideration on the basis of which the differential equations have been derived. 

It should be emphasized that the matrix representation becomes possible due to the Euler integration of the differential equations, yielding appropriate difference equations. Thus, flow systems incorporating heat and mass transfer processes as well as chennical reactions can easily be treated by Markov chains where the matrix P becomes "automatic" to construct, once gaining enough experience. In addition, flow systems are presented in unified description via state vector and a one-step transition probability matrix. 

An effective approximation to Equation 1 is obtained by segmenting the water body of interest into n volume elements of volume Vj and representing the derivatives in Equation 1 by differences. Let V be the n X n diagonal matrix of volumes V, A, the n X n matrix of dispersive and advective transport terms SPy the n vector of source terms SPjy averaged over the volume Vjy- and P, the n vector of concentrations P which are the concentrations in the volumes. Then the finite difference equations can be expressed as a vector differential equation... 

The Russians, according to their report at the Geneva Conference [18], employ a method of matrix factorization of the difference equations written in vector matrix form to solve the energy groups simultaneously. They write yjfc+i — + Ckyk-i = —Pk and reduce this system to a system of... 

The discretization scheme, which leads to an error 0 h ) for second-order differential equations (without first derivative) with the lowest number of points in the difference equation, is the method frequently attributed to Nu-merov [494,499]. It can be efficiently employed for the transformed Poisson Eq. (9.232). In this approach, the second derivative at grid point Sjt is approximated by the second central finite difference at this point, corrected to order h, and requires values at three contiguous points (see appendix G for details). Finally, we obtain tri-diagonal band matrix representations for both the second derivative and the coefficient function of the differential equation. The resulting matrix A and the inhomogeneity vector g are then... 

The computation of the matrices appearing in the formulae above is not necessary. An efficient computation avoids both the matrix inversion and matrix multiplications, replacing them with the solution of systems of equations and matrix-vector multiplications. Apart from the calculation of inertia terms and applied forces, the bulk of the computational effort in all formulations is constituted by the computation of the Cholesky factor of the matrix G. Once the factor is computed, formulations 1 and 3 require the solution of a single equation with matrix G, the others require two. For some systems it may be advisable to organise the computations in a different manner, solving instead equations with the indefinite system matrix... 

This equation determines a rank-1 matrix, and the eigenvector of its only one nonzero eigenvalue gives the direction dictated by the nonadiabatic couphng vector. In the general case, the Hamiltonian differs from Eq.(l), and the Hessian matrix has the form... 

Thus, for example, where x represents the mean of the data for which equation 1-1 describes the distribution, there is a corresponding quantity X that represents in matrix notation the fact that for each of the axes shown in Figure 1-1, each datum has a value, and therefore the collection of data has a mean value along each dimension. This quantity represented as a list of the set of means along all the different dimensions is called a vector, and is represented as X (as opposed to x, an individual mean). 

Let us also define a matrix Bcolumn vector is obtained by stacking the vector b /nrn ri (p p) under the vector bj /nrn, f2 - (p p). These vectors, for different n, Qk, are then placed side-by-side thereby generating a square matrix B 7111" whose dimensions are the total number of LHSFs (channels) used. The coupled hyperradial equation satisfied by this matrix has the form... 

Here 4 is the target state vector at time index k and Wg contains two random variables which describe the unknown process error, which is assumed to be a Gaussian random variable with expectation zero and covariance matrix Q. In addition to the target dynamic model, a measurement equation is needed to implement the Kalman filter. This measurement equation maps the state vector t. to the measurement domain. In the next section different measurement equations are considered to handle various types of association strategies. 

The vector nk describes the unknown additive measurement noise, which is assumed in accordance with Kalman filter theory to be a Gaussian random variable with zero mean and covariance matrix R. Instead of the additive noise term nj( in equation (20), the errors of the different measurement values are assumed to be statistically independent and identically Gaussian distributed, so... 

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