Nonlinear Jacobian evaluation

We have stated that we do not in general know the number or even the existence of solutions to a nonlinear algebraic system. This is true however, it is possible to identify points at which the existence properties of the system change through locating bifurcation points i.e., choices of parameters at which the Jacobian, evaluated at the solution, is singular. 

Despite the obvious correspondence between scaled elasticities and saturation parameters, significant differences arise in the interpretation of these quantities. Within MCA, the elasticities are derived from specific rate functions and measure the local sensitivity with respect to substrate concentrations [96], Within the approach considered here, the saturation parameters, hence the scaled elasticities, are bona fide parameters of the system without recourse to any specific functional form of the rate equations. Likewise, SKM makes no distinction between scaled elasticities and the kinetic exponents within the power-law formalism. In fact, the power-law formalism can be regarded as the simplest possible way to specify a set of explicit nonlinear functions that is consistent with a given Jacobian. Nonetheless, SKM seeks to provide an evaluation of parametric representation directly, without going the loop way via auxiliary ad hoc functions. 

Before moving on to real Rosenbrock methods, consider again (4.66). The left-hand side contains a term in fy if we are dealing with a system of odes, this is called the Jacobian of the system. It is often constant, evaluable in advance. It will be seen in Chap. 9 that unless the diffusion problem has nonlinear concentration terms (for example from higher-order homogeneous reactions), the Jacobian is constant. If not, it must be evaluated at every step. 

The 78 equality constraints in the complete model were thus reduced to 6 nonlinear equations as the genetic algorithm, NSGA-II-aJG is not effective in handling multiple equality constraints. Its inadequateness was also observed even when the equations had been reduced to 6 equations. Hence, the Broyden s update and finite-difference Jacobian function (DNEQBF) of the IMSL Library was embedded in the objective evaluation to solve the nonlinear equations 10.1 to 10.6. 

If the Jacobian is evaluated numerically, it is not convenient to increment variables one at a time and to perform a call to the nonlinear system. 

Buzzi-Ferraris and Manenti (2014, Vol. 3) have shown that the Jacobian of a nonlinear system can be calculated by simultaneously varying several variables when the Jacobian is sparse. If Equation 2.240 is adopted to evaluate a Jacobian matrix, which is supposed to be ftill, then the vector is the null array except for position k where the element is equal to 1. In this case, the system is called N times to evaluate the derivatives of the functions with respect to the N variables. Consider the sparse Jacobian matrix shown in Figure 2.11, where the symbol x represents a nonzero element. 

The geometric interpretation of (7.22) is that the quantity A fi 2 measures the distance of Xi from the solution Xs. With reference to nonlinear systems, equation (7.22) is a measure of the distance of Xi from the solution of the linearized system where A = J represents the Jacobian matrix evaluated in x . 

If Newton s method is adopted to solve the nonlinear system (see the following sections), then the Jacobian matrix Jj has already been factorized to solve the linear system produced by the method itself. The evaluation of the merit function W (x) in the two points, x and xj+i, is therefore straightforward and manageable. 

Every new iteration requires the evaluation of the Jacobian matrix. If the Jacobian is evaluated numerically, this means that the nonlinear system (7.1)... 

With the information available, the object can self-dedde regarding derivative updating and re-evaluation, without any support decision or operation from the user. In any case, when some variables are nonlinear while others are linear within certain equations, it is possible and useful to inform the object that the Jacobian must be updated only with respect to certain specific variables. 

In the previous example, only the variables 4, 5, and 6 are nonlinear. After having evaluated the Jacobian with respect to all the variables and having collected it in an object A from the BzzMatrixSparseLocked dass, the code sample to... 

Nonlinear constraints may include linear and nonlinear terms. Only the nonlinear ones enter the calculations of the Hessians or in a new evaluation of the Jacobians. 

As previously commented, the standard method for solving equations is Newton s method. But this requires the calculation of a Jacobian matrix at each iteration. Even assuming that accurate derivatives can be calculated, this is frequently the most time-consuming activity for some problems, especially if nested nonlinear procedures are used. On the other hand, we can also consider the class of quasi-Newton methods where the Jacobian is approximated based on differences in x and/(x), obtained from previous iterations. Here, the motivation is to avoid evaluation of the Jacobian matrix. 

Performs nonlinear regression using the Gauss-Newton estimation method. The jc-data is given as x, while the y-data is given as y. The function, FUN, that is to be fitted must be written as an m-file. It will take three arguments the coefficient values, x, and y (in this order). The function should be written to allow for matrix evaluatitni. The initial guess is specified in bataO. The vector beta contains the estimated values of the coefficients, the vector r contains the residuals, and covb is the estimated covariance matrix for the problem. J is the Jacobian matrix evaluated with the best estimate for the parameters. 

The values of the functions Uf, V ,and Wf have to be calculated at discrete points and 10 elements with 6 collocation points per element were found to suffice. Rather than solve (15) and (16) separately it was found best to use collocation on these also, using four radial collocation points. This gives 459 simultaneous nonlinear algebraic equations which were solved by a modified Newton-Raphson Iteration. The modifications were the use of under-relaxation and less frequent evaluation of the Jacobian. 

In order to solve the set of ns nonlinear equations (4.3.3a) Newton s method is applied. Again, like in the multistep case, Newton s method is implemented in a simplified manner, which saves evaluations (and decompositions) of the Jacobian With J being an approximation to the Jacobian... 

Comparing after 2 periods, a,tt = 4.0 s, the absolute error in position Cp, in velocity 6v and in the Lagrange multipliers we see that for all step sizes the index-2 formulation and the explicit ODE formulation (ssf) give the best results while the index-3 and index-1 approach may even fail. It can also be seen from this table that the effort for solving the nonlinear system varies enormously with the index of the problem. In this experiment the nonlinear system was solved by Newton s method, which was iterated until the norm of the increment became less than 10 . If this margin was not reached within 10 iterates the Jacobian was updated. The process was regarded as failed, if even with an updated Jacobian this bound was not met. We see, that the index-3 formulation requires the largest amount of re-evaluations of the Jacobian (NJC) and fails for small step sizes. It is a typical property of... 

Method of Solution The Marquardt method using the Gauss-Newton technique, described in Sec. 7.4.4, and the concept of multiple nonlinear regression, covered in Sec. 7.4.5, have been combined together to solve this example. Numerical differentiation by forward finite differences is used to evaluate the Jacobian matrix defined by Eq. (7.164). 

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