Linear simultaneous

It can readily be appreciated that in the absence of any knowledge regarding the differential polarizability tensor, it is a difficult exercise to obtain precise information from the Raman measurements. However, if r is known, Equations (19) are six linear simultaneous Equations in the six quantities (ot2I0N0) 1, P)0o> P220) PIoo> P420 and... 

The most convenient way to set up and solve the equations is to use a spreadsheet but any of the standard procedures and programs available for the solution of linear simultaneous equations can be used Westlake (1968), Mason (1984). 

Most proprietary spreadsheets include a routine for the inversion of matrices and the solution of sets of linear simultaneous equations. By using cell references, with cell copying and cell pointing, it is a simple procedure to set up the split fraction matrices... 

Hence, as is often stated, the determination of the normal coordinates is equivalent to the successful search for a matrix L that diagonalizes the product GF via a similarity transformation. This system of linear, simultaneous homogeneous equations can be written in the form... 

The corrected parameters are used to calculate a new A matrix and F vector, new corrections are calculated and the process is repeated until the calculated corrections are essentially zero. As will be shown later, a total of four parameters must be specified in order to determine a unique solution for the E and C numbers, because we are not dealing with linear simultaneous equations. The following parameters were held fixed and not allowed to vary iodine Ea = 1.00 iodine Ca = 1.00 DMA Eb = T32 diethyl sulfide Cb=7.40. These latter two parameters where selected to yield a solution close to the earlier one 39). 

Accdg to von Stein Alster (Ref 41), accurate determination of isochoric adiabatic flame temp of an expl often involves a series of tedious calcns of the equilibrium of compn of the expln products at several temps. Calcg the expln product compn at equilibrium is a tedious process for it.requires the soln of a number of non-linear simultaneous equations by a laborious iterative procedure. Damkoehler Edse (Ref 11) developed a graphical procedure and Wintemitz (Ref 26a) improved it. by transforming it into its algebraic equivalent. Unfortunately both.methods proved less useful with.hetorogeneous equilibria which. contain solid carbon... 

Practically concurrent with the growing utilization of automatic computers in industry has been the expanding use of various spectrometric methods for routine laboratory analyses. This coincidence is not entirely due to chance, because it seems doubtful that these analytical methods could have achieved their present wide acceptance had it not been practical for computers to do the calculations associated with them. These calculations are quite similar in the several applications, involving the solution of a system of linear simultaneous equations for each analysis. 

This set of linear simultaneous equations is a mathematical model of the blending operation. Given the volumes of the components, vh and the... 

The Stefan-Maxwell diffusion velocities are, in general, solved from a set of K (linear) simultaneous equations. Equation 12.170, with the A th equation equation replaced by Eq. 12.171, can be rewritten in matrix form as... 

Matrices are useful in dealing with systems of linear equations. The set of n linear simultaneous equations (1.80) can be written as the following matrix equation ... 

As with single-channel scattering (see section 3.2), the resulting set of linear simultaneous equations can be written as... 

This is a rather obvious and natural extension, to two open channels, of the set of linear simultaneous equations for a single open channel, equation (3.53). The formulation can be readily extended to accommodate more open channels. Once all the matrix elements have been calculated, the determination of the variational K-matrix proceeds in a similar manner to that described previously for single-channel scattering. Further details were given by Armour and Humberston (1991). 

From symmetry and conservation, there are three linear simultaneous results for the off-diagonal elements of ss ... 

Set up four linear, simultaneous equations represented in matrix form as A.X = B, where fljj is the weight fraction of oxide component i in phase j,. Vj is the weight percentage of phase j in the clinker, and Aj is the corrected weight percentage of oxide component i in the clinker ... 

This paper proposes a system of 10 non-linear, simultaneous differential equations (Table I) tdiich upon further development and validation, may serve as a comprehensive model for predicting steady state, vertical profiles of chemical parameters in the sulfide dominated zones of marine sediments. The major objective of the model is to predict the vertical concentration profiles of H2S, hydrotriolite (FeS) and p3nrite (FeS2). As with any model there are a number of assumptions involved in its construction that may limit its application. In addition to steady state, the major limiting assumptions of this model are the assumptions that the sediment is free of CaC03, that the diffusion coefficients of all dissolved sulfur species are equivalent and that dissolved oxygen does not penetrate into the zone of sulfate reduction. 

To make the most efficient use of computer storage, and to give a quick response time, the efficient sparse matrix solution algorithm developed by D. J. Gunn (1977) (1982) is used in program MM3, but any suitable procedure for the solution of linear simultaneous equations can be used. 

Solve Equation 3 for w = u/ r ) for each x at the end of the current time-step (Equation 3 becomes a set of linear simultaneous equations). 

Since there are 4 such equations for J shells, we have a system of 4 J non-linear simultaneous equations in 4J unknowns. Equation (14) can be rewritten in the form ... 

This is a system of a homogeneous linear simultaneous equations in the a unknown quantities Kn, , K o- Written out in full, these equations are... 

Equation (2.98) represents a set of k linear simultaneous equations in the k unknowns, dzifdt, which are calculated by reference to the derivatives of the state variables and the derivatives of the inputs - hence the name Method of Referred Derivatives. 

Using the Method of Referred Derivatives, it is possible to integrate the vector dz/dt in the same way as the vector dx/dt. Thus this method replaces the need to solve a set of nonlinear, simultaneous equations at each timestep by the simpler requirement of solving a set of linear, simultaneous equations, followed by integration of the resultant time-differentials from a feasible initial condition, z(0). 

Write a computer program that uses Gaussian elimination to solve a system of n linear, simultaneous, inhomogeneous equations in n unknowns, where n 10. Test it on a couple of examples. 

This approximation resolves the computational difficulty encountered in the direct exact formulation that requires repeated computations of the solution of linear simultaneous algebraic equations and determinants of the matrices with huge dimensions. The efficiency in the approximated expansion is gained by the appreciation that the conditioning information can be truncated within one period of the system only. For linear systems, the expressions for the reduced-order likelihood function p(yi, yj, - - -, yNp W, C) and the conditional PDFs p(.yn 0, yn-Np, yn-Np+1, , y -i, C) are available since they are Gaussian and the correlation functions are known in closed forms regardless of the stationarity of the response. For stationary response, the method is very efficient in the sense that evaluation of all the conditional PDFs p(ynW, yn-Np,yn-Np+i,, y -i, C) requires the inverse and determinant of two relatively small matrices only. 

---