Nonlinear Jacobian matrix

Steady-state solutions are found by iterative solution of the nonlinear residual equations R(a,P) = 0 using Newton s methods, as described elsewhere (28). Contributions to the Jacobian matrix are formed explicitly in terms of the finite element coefficients for the interface shape and the field variables. Special matrix software (31) is used for Gaussian elimination of the linear equation sets which result at each Newton iteration. This software accounts for the special "arrow structure of the Jacobian matrix and computes an LU-decomposition of the matrix so that qu2usi-Newton iteration schemes can be used for additional savings. 

The authors describe the use of a Taylor expansion to negate the second and the higher order terms under specific mathematical conditions in order to make any function (i.e., our regression model) first-order (or linear). They introduce the use of the Jacobian matrix for solving nonlinear regression problems and describe the matrix mathematics in some detail (pp. 178-181). 

Fig. 4.4. Comparison of the computing effort, expressed in thousands of floating point operations (Aflop), required to factor the Jacobian matrix for a 20-component system (Nc = 20) during a Newton-Raphson iteration. For a technique that carries a nonlinear variable for each chemical component and each mineral in the system (top line), the computing effort increases as the number of minerals increases. For the reduced basis method (bottom line), however, less computing effort is required as the number of minerals increases.

As a consequence, the gradient of the objective function and the Jacobian matrix of the constraints in the nonlinear programming problem cannot be determined analytically. Finite difference substitutes as discussed in Section 8.10 had to be used. To be conservative, substitutes for derivatives were computed as suggested by Curtis and Reid (1974). They estimated the ratio /x of the truncation error to the roundoff error in the central difference formula... 

Joris and Kalitventzeff (1987) proposed a classification procedure for nonlinear systems, which is based on row and column permutation of the occurrence matrix corresponding to the Jacobian matrix of the linearized model. 

The nonlinearity of (11.8) comes from the terms 4>(z°(t)) and y (r) 2 (r) involved in the nonlinear negative feedback regulation. What we want to do is replace these nonlinear terms by a linear function in the vicinity of the equilibrium state (x, y, z ). This involves writing the Jacobian matrix of the linearized system (cf. Appendix A) ... 

Numerical calculation has been carried out using a software interface which is based on the so-called "Method of lines" (14). Gear s backward difference formulas (15) are used for the time integration. A modified Newton s method with the internally generated Jacobian matrix is utilized to solve the nonlinear equations. ... 

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. 

Determine the parameter values bj andZ>2 by using the data given in Example 9.1 and the nonlinear least squares method. Recall that in Example 9.1 we needed the elements of the Jacobian matrix 7 (see equation (9.142)). In this case, integrate simultaneously the time dependent sensitivity coefficients (i.e., the Jacobi matrix elements dyfdb and dyjdb2 ) and the differential equations. The needed three differential equations can be developed by taking the total derivative (as shown below) of the right hand side of equation (9.149) which we call h ... 

These equations show that the sensitivity coefficients are the elements of the Jacobian matrix. For models that are nonlinear in the dependent variables we may not have analytical expressions for the sensitivity coefficients. In this case, the... 

Solution To obtain the linearization about (x, y ) = (0,0), we can either compute the Jacobian matrix directly from the definition, or we can take the following shortcut. For any system with a fixed point at the origin, x and y represent deviations from the fixed point, since m = x-x = x and v = y-y = y, hence we can linearize by simply omitting nonlinear terms in x and y. Thus the linearized system is X = -y, y-x. The Jacobian is... 

The computational domain is the unit square in u and v, and this was divided into a 15 x 15 mesh i.e., 225 elements, and 16 x 16 = 256 nodes, so 256 basis functions and 256 residual equations. The Jacobian matrix was banded with a total bandwidth of 35. The first solution computed was the minimal surface, for which the initial estimate was an hyperbolic paraboloid. The nonlinear system of residual equations was solved by Newton iteration on a Cyber 124, each iteration using about 1 second cpu time. For nearly all the surfaces calculated, the mesh was an even mesh over the entire unit square. However, for the surfaces just near the close-packed spheres (CPS) limit, the nodes were evenly spaced in the u-direction but placed as follows in the i -direction i = 0,1/60,1/30,0.05,0.075,0.1,0.15,0.2,0.3,0.4,0.5,0.6,0.7,... 

As might be apparent, this method is tedious and quickly becomes cumbersome. Other methods, such as determining the rank of the Jacobian matrix (Jacquez, 1996) or Taylor series expansion for nonlinear compartment models (Chappell, Godfrey, and Vajda, 1990), are even more difficult and cumbersome to perform. There... 

Alternatively, a nonlinear analogy to the HAT matrix in a linear model can be developed using the Jacobian matrix, instead of x. In this case,... 

For the multivariable, nonlinear system, the time constants are the negative reciprocals of the non-zero eigenvalues of the Jacobian matrix, J, which are the roots of the equation ... 

Numerical methods used to solve a system of ODEs are widely available in computational libraries and through texts such as Numerical Recipes. Certain considerations arise in the use of these standard techniques for nonlinear systems, particularly in models of chemical systems, which often consist of systems of stiff equations that require special care. Stiff equations are characterized by the presence of widely differing time scales, which leads to eigenvalues of the Jacobian matrix differing by many orders of magnitude. 

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)... 

In optimization problems, the Hessian is only occasionally ill-conditioned at the function minimum. In the solution of nonlinear equations systems, the Jacobian matrix may become singular when the gradient of the merit function approaches zero in correspondence with the minimum of the same function. 

For numerical computation, the Jacobian matrix is best obtained numerically. Usually, it takes about a few iterations to converge the above nonlinear equation. 

Herein, A Tu( /) is an approximation to the Jacobian matrix of the residual r = f — But at a time point tj < tk. The method avoids - because of the linearly-implicit structure - the solution of nonlinear systems, a clear structural advantage. As opposed to BDF-methods the one-step nature of our scheme allows an easy change of the computational grids after each basic time step. 

Jacobian matrix containing the derivatives of the nonlinear model with respect to... 

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. 

A couple of precautions are in order. Systems of nonlinear equations may have many solutions. Depending on the initial guess, x°, the Newton-Raphson method may converge to different solutions. In that case, it is wise to make the best initial guesses possible and use physical reasoning in interpreting the solution. Also, the Jacobian matrix may become singular as the solution is approached. If this occurs, solution by the Newton-Raphson technique may be impossible, and other, nonderivative methods should be used. 

Unlike in linear regression where exact results can be obtained under the stated assumptions, in nonlinear regression the results are only approximate. Furthermore, there do not exist nice matrix-based solutions for the various parameters. This section provides a convenient summary of the useful equations for nonlinear regression. In general, to compute the approximate confidence intervals for a nonlinear regression problem, the final grand Jacobian matrix, J, can be used in place of A and J in place of in the linear regression formulae. 

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 unknowns are taken in a large vector in the order [Rq, Pq, Co,ifo,Ri,Pi,Ci, ifi,..., RN,PN,CN,ifNV but are lumped into the vector of four-point vectors U = [Ri, Pi, Ci, ifiY, i = 0... N, to prepare for the block-tridiagonal procedure for solving the system. The system of equations (13.37) is nonlinear, and the Newton method is used to solve it. At each index i we have three 4x4 blocks in the Jacobian matrix L the left-hand block for the elements at index / — 1 M,, the middle block for index /, and Q the right-hand block for index t -I-1, that symbol chosen here in order to avoid clashes with the concentration symbol R. They produce a tridiagonal block system. For this example, three-point BDF was used, started with one BI step. 

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