Matrix inversion techniques

This allows us to represent partial differential equations as found in the balance equations using the collocation method. Equation (11.47) is a solution to a partial differential equation represented by a system of linear algebraic equations, formed by the interpolation coefficients, oij, and the operated radial functions. The interpolation coefficients are solved for using matrix inversion techniques to approximately satisfy the partial differential equation... 

We note that the RSOZ equations in Eqs. (G.lla)-(G.lld) are decoupled with respect to the wavenumber k and the angular index x- practice, the matrices and have finite dimensions due to a truncation of the rotationally invariant expansion in Eq. (G.8) at appropriate values of Zj and Z2. The RSOZ Eqs. (G.lla)-(G.lld) can thus be solved by standard matrix inversion techniques. 

Figure 5.14 EXAFS (top) and MEXAFS spectra (bottom) at the L2-edge of Pt in FejPt. Calculations for the ordered compound (full line), compared with the experimental data for the Feo.72Pto.28 (dashed line) (Ahlers 1998). The corresponding calculations for the scattering path operator thave been done using the matrix inversion technique for a cluster of 135 atoms in the XANES and 55 atoms in the EXAFS region, including the central absorber site. The effects of self-energy corrections (Fujikawa et al 1997 Mustre de Leon et al 1991) have been accounted for after calculating the spectra.

The matrix given in Equation 12.24 can of course be solved by any matrix inversion technique. Such techniques can be slow however (usually of the order of where p is the dimensionality of the matrix) and hence faster techniques have been developed to find the values of ak from the autocorrelation functions R k). In particular, it can be shown that the Levinson-Durbin recursion technique can solve Equation 12.28 in p time. For our purposes, analysis speed is really not an issue, and so we will forgo a detailed discussion of this and other related techniques. However, a brief overview of the technique is interesting in that it sheds light on the relationship between linear prediction and the all-pole tube model discussed in Chapter 11. 

Relations between the TCFIs and derivatives of pressure with respect to molar density can be written for any number of components, as in Section 1.1.6 in Chapter 1. Matrix inversion techniques can provide expressions for all of the pair TCFIs of the system. Section 1.2 in Chapter 1 gives the full relations for applications to pure, binary, and ternary systems. As shown in Section 1.2.3 in Chapter 1, there is also a set of relations for the derivatives in terms of the DCFI, which are somewhat simpler and more direct. There are two modeling objectives with these relations. One is to obtain a solution density at elevated pressures the other is to obtain the component partial molar volumes for the solution density variations with composition. The next section describes approaches that have been used for both objectives in a wide variety of pure and binary systems. 

This means that now a new value of,+,, T cannot be calculated only from values at time i as in Eq. (5.4-2) but that all the new values of T at t -(- At at all points must be calculated simultaneously. To do this an equation is written similar to Eq. (5.4-26) for each of the internal points. Each of these equations and the boundary equations are linear algebraic equations. These then can be solved simultaneously by the standard methods used, such as the Gauss-Seidel iteration technique, matrix inversion technique, and so on (G1,K1). 

Finite difference formulation. Suppose that analytieal solutions were not possible, and that recourse to computational methods was neeessary. Then, Equation 6-1, for example, would have to be approximated by writing discretized algebraic equations from node to node, and solving the eoupled equations using a matrix inversion technique. A simple way to introduee finite differences follows from Figure 6-2. First consider constant mesh widths Ar, and examine the Point A lying at the midpoint between successive indiees i-1 and i. It is clear that the first derivative dP(A)/dr = (Pi - Pi i)/Ar. Similarly,... 

This implies that once all the induced dipole moments are known, their energy in the total field reduces to just the induced dipole moments dotted into the electric field generated by the permanent charges. Methods for self-consistently calculating the induced moments include iterative/predictive methods and matrix inversion techniques.Expressions for the forces are given elsewhere. 

A comparison of the ccmputational efficioicy of stuping and matrix-inversion techniques has been carried out by Shimta and Luyt. For large columns in particular, it is shown that the former is much fester. 

Comparison of Stepping and General Complex Matrix Inversion Techniques in Calculating the Frequency Response of Binary Distillation Columns, I .C. Fund., 8 838 (1969). 

The electric field at an atom will have contributions from the charges and induced dipoles of all the other atoms in the system and so, there will a set of coupled equations of type 25, one for each MM atom, that must be solved if the induced dipoles are to be obtained. It turns out that the equations that result are linear and so can be solved by direct matrix inversion techniques for small systems or by iterative methods for larger cases. Once the dipoles are known the energy arising from the polarization term is calculated as ... 

To calculate the inverse of a matrix by this procedure is equally tedious and probably more work than solving a set of equations by the brute-force high-school technique. However, the procedure is readily converted into computer code and this is now the only way recommended for matrix inversions. 

Most of the asymmetric mixed-matrix membranes reported to date were prepared from concentrated mixed-matrix dopes via a phase inversion technique [69,... 

Kulprathipanja and coworkers reported the preparation of integrally skinned siUcaUte-1/cellulose acetate flat sheet asymmetric mixed-matrix membranes via phase inversion technique in 1992 [73]. The O2/N2 separation performance of these membranes was investigated. It was demonstrated that the separation factor of... 

One possible way to solve this problem is by obtaining a system of linear equations in a similar way as in Example 8.3. However, the equations must be organized carefully because the equations are coupled in the two directions. To avoid the complications given by the generation of the matrix for the linear system of equations, some iterative methodologies have been developed to solve for this type of problems. Such techniques solve the system of equations without the need of cumbersome matrix manipulation, such as LU-decomposition, matrix inversion, etc. [19]. 

Newton s method and quasi-Newton techniques make use of second-order derivative information. Newton s method is computationally expensive because it requires analytical first-and second-order derivative information, as well as matrix inversion. Quasi-Newton methods rely on approximate second-order derivative information (Hessian) or an approximate Hessian inverse. There are a number of variants of these techniques from various researchers most quasi-Newton techniques attempt to find a Hessian matrix that is positive definite and well-conditioned at each iteration. Quasi-Newton methods are recognized as the most powerful unconstrained optimization methods currently available. 

When the number of nodes is very large, an iterative technique may frequently yield a more efficient solution to the nodal equations than a direct matrix inversion. One such method is called the Gauss-Seidel iteration and is applied in the following way. From Eq. (3-31) we may solve for the temperature T, in terms of the resistances and temperatures of the adjoining nodes 7 as... 

Asymmetric membranes are usually produced by phase inversion techniques. In these techniques, an initially homogeneous polymer solution becomes thermodynamically unstable due to different external effects and the phase separates into polymer-lean and polymer-rich phases. The polymer-rich phase forms the matrix of the membrane, while the polymer-lean phase, rich in solvents and nonsolvents, fills the pores. Four main techniques exist to induce phase inversion and thus to prepare asymmetric porous membranes [85] (a) thermally induced phase separation (TIPS), (b) immersion precipitation (wet casting), (c) vapor-induced phase separation (VIPS), and (d) dry (air) casting. 

The second part of training radial basis function networks assumes that the number of basis functions, i.e., the number of hidden units, and their center and variability parameters have been determined. Then all that remains is to find the linear combination of weights that produce the desired output (target) values for each input vector. Since this is a linear problem, convergence is guaranteed and computation proceeds rapidly. This task can be accomplished with an iterative technique based on the perception training rule, or with various other numerical techniques. Technically, the problem is a matrix inversion problem ... 

A defining feature of the models discussed in the previous section, regardless of whether they are implemented via matrix inversion, iterative techniques, or predictive methods, is that they all treat the polarization response in each polarizable center using point dipoles. An alternative approach is to model the polarizable centers using dipoles of finite length, represented by a pair of point charges. A variety of different models of polarizability have used this approach, but especially noteworthy are the shell models frequently used in simulations of solid-state ionic materials. 

Usually, p is chosen to be a number between 4 and 10. In this way the system moves in the best direction in a restricted subspace. For this subspace the second-derivative matrix is constructed by finite differences from the stored displacement and first-derivative vectors and the new positions are determined as in the Newton-Raphson method. This method is quite efficient in terms of the required computer time, and the matrix inversion is a very small fraction of the entire calculation. The adopted basis Newton-Raphson method is a combination of the best aspects of the first derivative methods, in terms of speed and storage requirements, and the more costly full Newton-Raphson technique, in terms of introducing the most important second-de-... 

Using this approach, calibration can be performed knowing only the one component of interest in the system. Interfering compounds only have to be present, not quantified. They are implicitly modeled with this approach. The major implication of this technique is that application of the sensor array in remote environments is better facilitated. A restriction that this model imposes is that the number of sensors must be less than the number of calibration samples in order to perform the generalized inverse. This method usually has more error propagation due to the instability of the R matrix inversion. Collinearity plays an important role in this case. 

Simultaneous linear inhomogeneous equations can be solved with various techniques, including elimination, use of Cramer s formula, and by matrix inversion. 

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