Solving linear equations Newtons method

Given a function/(x) that is continuous and has a continuous derivative and also, given a starting value of x  

Value of X4 is exact to the 6th decimal. A larger number of analyses may be required if the function does not converge rapidly. 

Process engineering and design using Visual Basic 

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Quasilinearization is a technique where nonlinear differential equations are solved by obtaining a sequence of solutions to related linear equations. The method somewhat resembles a generalized Newton-Raphson method. The most important development in the quasilinearization technique was the use of the maximum operation to prove that the representation of the original nonlinear equations by a sequence of linear equations converges to the nonlinear equation. This result is due to Bellman, who also used the concept in his development of dynamic programming (Bellman, 1957). 

The above equation represents a set of p nonlinear equations which can be solved to obtain koutput vector around the trajectory xw(t). Kalogerakis and Luus (1983b) showed that when linearization of the output vector is used, the quasilinearization computational algorithm and the Gauss-Newton method yield the same results. 

The above unconstrained estimation problem can be solved by a small modification of the Gauss-Newton method. Let us assume that we have an estimate kw of the parameters at the j iteration. Linearization of the model equation and the constraint around kw yields,... 

At this point we can summarize the steps required to implement the Gauss-Newton method for PDE models. At each iteration, given the current estimate of the parameters, ky we obtain w(t,z) and G(t,z) by solving numerically the state and sensitivity partial differential equations. Using these values we compute the model output, y(t k(i)), and the output sensitivity matrix, (5yr/5k)T for each data point i=l,...,N. Subsequently, these are used to set up matrix A and vector b. Solution of the linear equation yields Ak(jH) and hence k°M) is obtained. The bisection rule to yield an acceptable step-size at each iteration of the Gauss-Newton method should also be used. 

When the Gauss-Newton method is used to estimate the unknown parameters, we linearize the model equations and at each iteration we solve the corresponding linear least squares problem. As a result, the estimated parameter values have linear least squares properties. Namely, the parameter estimates are normally distributed, unbiased (i.e., (k )=k) and their covariance matrix is given by... 

Like Newton s method, the Newton-Raphson procedure has just a few steps. Given an estimate of the root to a system of equations, we calculate the residual for each equation. We check to see if each residual is negligibly small. If not, we calculate the Jacobian matrix and solve the linear Equation 4.19 for the correction vector. We update the estimated root with the correction vector,... 

In this section we consider how Newton-Raphson iteration can be applied to solve the governing equations listed in Section 4.1. There are three steps to setting up the iteration (1) reducing the complexity of the problem by reserving the equations that can be solved linearly, (2) computing the residuals, and (3) calculating the Jacobian matrix. Because reserving the equations with linear solutions reduces the number of basis entries carried in the iteration, the solution technique described here is known as the reduced basis method. ... 

K, K 2, and K3 are the equilibrium constants for the formation of hydrogen molecule, H2S and H20 gases respectively from the atomic elements. The equations for each of the atomic elements form simultaneous non-linear equations which can be solved for example by Newton s method, starting with very small initial values of the number of each atomic and molecular species, i.e. 10-8. 

The condition number of the Hessian matrix of the objective function is an important measure of difficulty in unconstrained optimization. By definition, the smallest a condition number can be is 1.0. A condition number of 105 is moderately large, 109 is large, and 1014 is extremely large. Recall that, if Newton s method is used to minimize a function/, the Newton search direction s is found by solving the linear equations... 

A set of nonlinear equations can be solved by combining a Taylor series linearization with the linear equation-solving approach discussed above. For solving a single nonlinear equation, h(x) = 0, Newton s method applied to a function of a single variable is the well-known iterative procedure... 

The Newton-Raphson method consists in solving simultaneously the conservation and mass action equations. Because of its simplicity and rather fast convergence, it is well-fitted to sets of non-linear equations in several unknowns, as described in Chapter 3. 

X2° = X30 = 0 assumed to be known exactly. The only observed variable is = x. Jennrich and Bright (ref. 31) used the indirect approach to parameter estimation and solved the equations (5.72) numerically in each iteration of a Gauss-Newton type procedure exploiting the linearity of (5.72) only in the sensitivity calculation. They used relative weighting. Although a similar procedure is too time consuming on most personal computers, this does not mean that we are not able to solve the problem. In fact, linear differential equations can be solved by analytical methods, and solutions of most important linear compartmental models are listed in pharmacokinetics textbooks (see e.g., ref. 33). For the three compartment model of Fig. 5.7 the solution is of the form... 

In this formula, p. stands for gTd This is certainly an eigensystem equation. However, we must add a small correction to H, and moreover, this correction is not known in advance. It turns out that this correction can be left out, unless it is important that the linear equation is exactly solved. This is not necessary if the object is to find a good step for a macroiteration. Moreover, it turns out that, in such a context, the discrepancy introduced between this method and the exact NR-steps has the same asymptotic dependence as the error. Therefore, the method is still a second-order method with this modification, and there is no way to say a priori that this method is better or worse than the exact NR-iterations. This method is called the augmented Hessian (AH-)method. It is seen to be equivalent to a Newton-Raphson using a shifted hessian. This can be very advantageous, since this shift tends to keep the step down, and to keep the shifted hessian positive definite, when one is far from a solution. The size of... 

The most prominent of these methods is probably the second order Newton-Raphson approach, where the energy is expanded as a Taylor series in the variational parameters. The expansion is truncated at second order, and updated values of the parameters are obtained by solving the Newton-Raphson linear equation system. This is the standard optimization method and most other methods can be treated as modifications of it. We shall therefore discuss the Newton-Raphson approach in more detail than the alternative methods. 

Computational aspects on the Newton-Raphson procedure When constructing methods to solve the system of linear equations (4 22) one should be aware of the dimension of the problem. It is not unusual to have Cl expansion comprising 104 - 106 terms, and orbital spaces with more than two hundred orbitals. In such calculations it is obviously not possible to explicitly construct the Hessian matrix. Instead we must look for iterative algorithms... 

The expressions derived for the EOS and the chemical potential of component i in a binary mixture were used to model the phase equilibria of binary mixtures. A set of non-linear equations was obtained and solved by the use of a Newton s method. 

Class II Methods. The methods of Class II are those that use the simultaneous Newton-Raphson approach, in which all the equations are linearized by a first order Taylor series expansion about some estimate of the primitive variables. In its most general form, this expansion includes terms arising from the dependence of the thermo-physical property models on the primitive variables. The resulting system of linear equations is solved for a set of iteration variable corrections, which are then applied to obtain a new estimate. This procedure is repeated until the magnitudes of the corrections are sufficiently small. 

For a given value of R, eqs. (29), (30), and (31) can be solved readily for Pj and p° using the Newton-Raphson method. It is easy to arrange the linearized equations so that the coefficient matrix is upper triangular with only one non-null column above the diagonal. The solution therefore does not add appreciably to the computational load of the inside loop. 

These equations are linear and can be solved by a linear equation solver to get the next reference point (ah, A21). Iteration is continued until a solution of satisfectory precison is reached. Of course, a solution may not be reached, as illustrated in Fig. L,6c, or may not be reached because of round-off or truncation errors. If the Jacobian matrix [see Eq. (L.ll) below] is singular, the linearized equations may have no solution or a whole family of solutions, and Newton s method probably will fail to obtain a solution. It is quite common for the Jacobian matrix to become ill-conditioned because if ao is far from the solution or the nonlinear equations are badly scaled, the correct solution will not be obtained. 

Brent s and Brown s methods are variations of Newton s method that improve convergence. The calculation of the elements in in Eq. (L.ll) and the solving of the linear equations are intermingled. Each row of Jt is obtained as needed using the latest information available. Then one more step in the solution of the linear equations is executed. Brown s method is an extension of Gaussian elimination Brent s method is an extension of QR factorization. Computer codes are generally implemented by using numerical approximations for the partial derivatives in J. ... 

This nonlinear system can be solved by using the Newton-Raphson method, or any other non-linear equation solver. However, this is not the best way since a very good initial guess is needed in order to ensure convergence. 

Equation 15 was used as a constraint with a value between 12 and 13 for Z (n-decane conversion), during optimization of the reaction variables, using a Non-linear Quasi-Newton search method with tangential extrapolation for estimates, forward differencing for estimation of partial derivatives, a tolerance of 0.05 and precision of 0.0005. The search was also constrained by boundary conditions 1 to -1 for the reaction variables x, and solved for maximization of Y . 

A wide variety of iterative solution procedures for solving nonlinear algebraic equations has appeared in the literature. In general, these procedures make use of equation partitioning in conjunction with equation tearing and/or linearization by Newton-Raphson techniques, which are described in detail by Myers and Seider. The equation-tearing method was applied in Section 7.4 for computing an adiabatic flash. 

Mass balance of solid Mass balance of water Mass balance of air Momentum balance for the medium Internal energy balance for the medium The resulting system of Partial Differential Equations is solved numerically. Finite element method is used for the spatial discretization while finite differences are used for the temporal discretization. The discretization in time is linear and the implicit scheme uses two intermediate points, t and t between the initial 1 and final t limes. Finally, since the problems are nonlinear, the Newton-Raphson method has been adopted following an iterative scheme. 

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