Nonlinear simultaneous equations

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

The computational procedure can now be explained with reference to Fig. 19. Starting from points Pt and P2, Eqs. (134) and (135) hold true along the c+ characteristic curve and Eqs. (136) and (137) hold true along the c characteristic curve. At the intersection P3 both sets of equations apply and hence they may be solved simultaneously to yield p and W for the new point. To determine the conditions at the boundary, Eq. (135) is applied with the downstream boundary condition, and Eq. (137) is applied with the upstream boundary condition. It goes without saying that in the numerical procedure Eqs. (135) and (137) will be replaced by finite difference equations. The Newton-Raphson method is recommended by Streeter and Wylie (S6) for solving the nonlinear simultaneous equations. In the specified-time-... 

For zeroth-order reaction steps, we still have linear equations, but for other orders of kinetics we have nonlinear simultaneous equations, which generally have no closed-form analytical solutions. We must solve these sets of equations numerically to find C/i(t), Cfi(r), and Cc(t). 

C.G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comp. 19 (1965) 577. 

The Matlab program given in Figure 2.19 solves these nonlinear simultaneous equations for given reactor volume, temperature, and feeds. Figure 2.20 gives results... 

The other solution species are formed from this "basis" by a series of chemical reactions, and their concentrations can be expressed through the use of equilibrium constants, in terms of the concentration of the chosen "basis". The resulting set of nonlinear simultaneous equations consists of as many unknowns as there are elements and can be solved by conventional niamerical methods. The "free energy minimization" method utilizes only free energy criteria for chemical equilibria making no distinction among the constituent species and is essentially a constraint non-linear minimization problem. A number of search methods have... 

Broyden, C. G., A Class of Methods for Solving Nonlinear Simultaneous Equations, Math Comput., V. 19, p. 577 (1965). 

When Step 6 of the systematic approach is complete, we have a mathematical problem of solving several nonlinear simultaneous equations. This job is often formidable, tedious, and time consuming unless a suitable computer program is available or approximations can be found that decrease the number of unknowns and equations. In this section, we consider in general terms how equations describing equilibrium relationships can be simplified by suitable approximations. 

Several software packages are available for solving multiple nonlinear simultaneous equations rigorously. Three such programs are Mathcad, Mathematica, and Excel. 

Bird, R. B., Stewart, W. E., Lightfoot, E. N., Transport Phenomena, 2nd edn. Wiley New York, 2002. Broyden, C. G., A class of methods for solving nonlinear simultaneous equations. Math. Comput. 

We will now illustrate the way that equation (2.70) could be solved as part of a stiff integration package. The solution relies partly on using the New-ton-Raphson technique for solving nonlinear simultaneous equations, the principles of which will now be explained. We may describe a system of n nonlinear. 

Solving nonlinear simultaneous equations in a process model iterative method... 

Given that the boundary pressures p, Pi. Pi and pt are either input variables or state variables (or explicitly derivable from the model s states), we have in equations (2.84) to (2.90) a set of seven nonlinear simultaneous equations in the seven unknowns )V 2, ( 23, VV24, W45, W46, p2 and p4. We can in this case reduce the order of the problem easily by substituting for the flows into equations (2.89) and (2.90) to give... 

But we are still left with two nonlinear simultaneous equations in the two pressure unknowns p2 and p4. 

A good subroutine for solving nonlinear simultaneous equations may be provided within the overall simulation package, or it may be necessary for the modeller to introduce such a routine himself. Commercial software is available if not already provided within the simulation package. Further detail on iterative methods for solving implicit equations is given in Chapter 18, Section 18.5, which includes a discussion on how to speed up convergence in flow networks. 

An alternative method of solving nonlinear simultaneous equations within a simulation is based on the properties of equation (2.93). Since the vector function, g, is constant (at zero) throughout all time, it follows that its time differential is also zero at all times ... 

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). 

The exact method for calculating compressible flow summarized in Section 6.4 and illustrated in Section 6.7 requires the iterative solution of a set of highly nonlinear simultaneous equations. However, it is shown in Appendix 2 that it is possible to use the results produced by the exact method as the basis for an approximate method of calculation that will... 

The assumption that the flows in the network are in a continuously evolving steady state brings the benefit that the model is rendered much less stiff thereby. Furthermore, the resulting simultaneous equations may be solved explicity in simple flow networks. In more complex networks, however, the resulting set of nonlinear, simultaneous equations requires a more sophisticated approach in order to bring about the desired savings in model execution time. This chapter considers both... 

An alternative method of solving the implicit equations associated with complex flow networks is to use the Method of Referred Derivatives discussed in Section 2.11 of Chapter 2. The general set of nonlinear, simultaneous equations... 

Equation (24.81) is a system of kt nonlinear, simultaneous equations in (k2 pi) unknowns. If there are more equations than unknowns, i.e. ki > (ki - - p ), then the possibility of inconsistency arises, and a solution may not be possible. If the number of equations equals the number of unknowns, i.e. k = (k2 -I- pi), then any solution will be unique. If there are more unknowns than equations, i.e. k < (k2 + p ), then the best that can be done is to define the k unknowns in terms of the remaining (k2 + Pi) — < i unknowns. The last possibility is almost certainly the one we will meet, and although the solution appears weak at first glance, nevertheless it will prove sufficient for our purposes. 

These pi +k2 + k nonlinear simultaneous equations in Pi -f 2 + unknowns may be solved at each timestep using a nonlinear equation solver of the type discussed in Chapter 2, Section 2.10 or by the Method of Referred Derivatives, discussed in Section 2.11 and in Chapter 18, Sections 18.7 to 18.9. The application of the latter method will be discussed further in the next section. 

Equations (A6.15) and (A6.18), taken together with the auxiliary equations (A6.16) and (A6.17) represent two nonlinear simultaneous equations in the two unknowns, A /A, and pi/por. the latter at the point of minimum calculated efficiency. The fact that a solution is possible, albeit iterative, demonstrates that the necessary procedure for calculating the divergence ratio has been developed. The minimum calculated nozzle efficiency will now match the minimum measured efficiency. 

Finding the roots of a system of nonlinear simultaneous equations like those in Eq. (2.18) can be problematic. No solution is guaranteed, and convergence and numeric stability can be dependent on the initial solution guess as well as other factors. Experience with Eq. (2.18) indicates that a reasonable initial guess for / is provided by setting f( = 0 Vi on the RHS of Eq. (2.17) to obtain... 

We note that the RISM equations, a large set of stiff, nonlinear simultaneous equations, must be solved for all the conformations sampled. However, we employ our robust algorithm [13] that is over two orders of magnitude faster than the conventional one. As a result, the ensemble-averaged structure of the solvent, which is in equilibrium with the pep-... 

The values of Q, R, Wi, and W2 chosen above may not be optimal. The optimal values can be determined by solving the four nonlinear simultaneous equations, shown in Appendix 2. 

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