Jacobian evaluation

Some methods adopt the Jacobian matrix J of the system f since its elements caimot be provided analytically in certain circumstances, it is necessary to approximate them numerically. Conceptually, there are no further difficulties in doing so, since they are the first derivatives of functions with respect to the different variables. However, one numerical problem does raise its head large round-off errors are obtained in one small step, whereas the first derivatives are spoiled by higher order derivatives if a large step is adopted. 

In practice, a satisfactory step for the variable is the product of the same variable (or a reasonable value when it is zero) and the square root of the macheps (Buzzi-Ferraris and Manenti, 2010a). 

If the f i of the nonlinear system can be expanded in terms of a Taylor series  

The following iterative procedure represents the elementary formulation of Newton s method  

Consequently, Newton s method has the following search direction  

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The first term on the right-hand side is the product of the physical problem s current Jacobian matrix and the sensitivity-coefficient matrix (i.e., the dependent variable). Assuming that the underlying physical problem (i.e., Eq. 15.58) is solved by implicit methods, the Jacobian evaluation is already part of the solution algorithm. The second term, which is the matrix that describes the explicit dependence of f on the parameters, must be evaluated to form the sensitivity equation. Note that all terms on the right-hand side are time dependent, as are the sensitivity coefficients S(t). 

The most costly operation in the Newton process is the Jacobian evaluation. In quasi-Newton methods, the same Jacobian is maintained during a few iterations, or approximations to J(x) are generated with increasing accuracy as the iterations proceed. 

Hence, the Jacobian evaluated at E2 has a positive eigenvalue (denoted by in the previous discussion of the eigenvalues of this Jacobian). 

The usual acceptance criterion for volume changes must be multiplied by the ratio of the Jacobians evaluated in the trial and original states. 

The resulting system of equations, [23], can be solved using linear algebra techniques. We rewrite this system in matrix form by first defining the column vector w and the matrix J (the Jacobian evaluated at steady state) as... 

Equation (7.84) does not allow the unequivocal evaluation of all the components of the Jacobian when the number of equations is wy > 1. In this case, ny — 1 additional conditions are necessary. In 1965, Broyden proposed choosing the conditions to be added to equation (7.84) in order to keep the product of the Jacobian evaluated in x and in Xj+i and an orthogonal vector to Ax invariant Generally, for any given vector qj with... 

Number of Newton method applications 13 Number of Quasi Newton method applications 73 Number of analytical Jacobian evaluations 0 Number of numerical Jacobian evaluations 13 Number of Gradient searches 0 Number of Gauss factorizations 7 Number of LQ factorizations 13 Number of linear system solutions 172... 

Table 5.1 Comparison of performance results for the implicit Euler scheme applied to different formulations of the pendulum problem. NFE number of function evaluations, NJC number of Jacobian evaluations, NIT average number of Newton iterations, ep,ey and ey. absolute errors in pi(4),ui(4), A(4), resp.

Table 6.4 Effort for the solution of the wheel suspension. (NFE Number of function evaluation, IJC Number of Jacobian decompositions, IMC Number of Jacobian evaluations, NRED number of rejected steps, NST number of integration steps, CPU normalized CPU time)...

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. 

Unlike the other alternative methods, analytical expressions of partial derivatives are required and the Jacobian must be evaluated in the Newton-Raphson method. These requirements sometimes prove to be the undoing when the method is applied to complicated equations. Brown (B12) has developed a modification to the Newton-Raphson method, which requires only some of the partial derivatives to be calculated. We have tested Brown s method on our sample problems but have found that it actually required more computing time than the unmodified Newton-Raphson method. 

The advantage of this method is that it avoids both the evaluation of partial derivatives and the inversion of the Jacobian. To start the iterations, an initial estimate H0 is also required an identity matrix is frequently used for this purpose. 

To evaluate the Jacobian matrix, we need to compute values for dmq/dnw, dmq/dnii, and dmq/dmp. For the Kt and Freundlich models,... 

At each step in the Newton-Raphson iteration, we evaluate the residual functions and Jacobian matrix. We then calculate a correction vector as the solution to the matrix equation... 

Similar substitution into Equations 16.10-16.12 gives masses of the basis entries at the end of a time step, Equations 16.13-16.14 yields the residual functions, and Equations 16.18-16.21 gives the entries in the Jacobian matrix. In evaluating the Jacobian, the derivatives dr /dnw and dr /dm, can be obtained by differentiating the appropriate rate law (Eqn. 17.9, 17.12, or 17.21), as discussed in Appendix 4, or their values determined just as efficiently by finite differences. 

Figure 26. The proposed workflow of structural kinetic modeling Rather than constructing a single kinetic model, an ensemble of possible models is evaluated, such that the ensemble is consistent with available biological information and additional constraints of interest. The analysis is based upon a (thermodynamically consistent) metabolic state, characterized by a vector S° and the associated flux v° v(S°). Since based only on the an evaluation of the eigenvalues of the Jacobian matrix are evaluated, the approach is (computationally) applicable to large scale system. Redrawn and adapted from Ref. 296.

The elements of the matrix A have the units of inverse time and determine the state at which the Jacobian is to be evaluated. Each element of A is at least in principle experimentally accessible and does not hinge upon a specific mathematical representation of any biochemical rate equations. 

Once the elements of the matrix 6% are specified, the Jacobian matrix of the metabolic network can be evaluated. A more detailed discussion, including a thermodynamically consistent parameterization, is given in Section VIII.E. 

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. 

Once the parametric representation of the Jacobian is obtained, the possible dynamics of the system can be evaluated. As detailed in Sections VILA and VII.B, the Jacobian matrix and its associated eigenvalues define the response of the system to (small) perturbations, possible transitions to instability, as well as the existence of (at least transient) oscillatory dynamics. Moreover, by taking bifurcations of higher codimension into account, the existence of complex dynamics can be predicted. See Refs. [293, 299] for a more detailed discussion. 

For any arbitrary metabolic network, the Jacobian matrix can be decomposed into a sum of three fundamental contributions A term M eg that relates to allosteric regulation. A term M in that relates to the kinetic properties of the network, as specified by the dissociation and Michaelis Menten parameters. And, finally, a term that relates to the displacement from thermodynamic equilibrium. We briefly evaluate each contribution separately. 

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