Jacobian matrix definition

Finally, and more profoundly, not all properties require explicit knowledge of the functional form of the rate equations. In particular, many network properties, such as control coefficients or the Jacobian matrix, only depend on the elasticities. As all rate equations discussed above yield, by definition, the assigned elasticities, a discussion which functional form is a better approximation is not necessary. In Section VIII we propose to use (variants of) the elasticities as bona fide parameters, without going the loop way via explicit auxiliary functions. 

Using matrix notation and recalling the definition of the Jacobian M° given in Eq. (130), the sum of partial derivatives straightforwardly translates into an additive relationship for the Jacobian matrix... 

In solving the underlying model problem, the Jacobian matrix is an iteration matrix used in a modified Newton iteration. Thus it usually doesn t need to be computed too accurately or updated frequently. The Jacobian s role in sensitivity analysis is quite different. Here it is a coefficient in the definition of the sensitivity equations, as is 3f/9a matrix. Thus accurate computation of the sensitivity coefficients depends on accurate evaluation of these coefficient matrices. In general, for chemically reacting flow problems, it is usually difficult and often impractical to derive and program analytic expressions for the derivative matrices. However, advances in automatic-differentiation software are proving valuable for this task [36]. 

J can now be defined by vector derivatives. First, however, it is partitioned into (m+ l)2 submatrices, each of size n by n. Subscripts are used to designate the submatrices. Thus, from the definition of the Jacobian matrix, we obtain... 

Proof. The quantity under the integral sign in the definition of A in (5.2) is the trace of the Jacobian matrix for the system (5.1) evaluated along the periodic orbit. Theorem 4.2 then applies. A periodic orbit for an autonomous system has one Floquet multiplier equal to 1. Since there are only two multipliers and one of them is 1, is the remaining one. The periodic orbit is asymptotically orbitally stable because, in view of Lemma 5.1, A<0. ... 

The rest point Eq always exists, and 2 exists with xj = 1 — A2 and p the root of (3.5) if 0 < A2 < 1, which is contained in our basic assumption (3.6). The existence of 1 is a bit more delicate. In keeping with the definitions in (3.3), define Aq = fli/(mi/(l)-l). Then 0< Aq < 1 corresponds to the survivability of the first population in a chemostat under maximal levels of the inhibitor. Easy computations show that 1 = (1 — Aq, 0,1) will exist if Aq > 0 and will have positive coordinates and be asymptotically stable in the X -p plane if 0 < Aq < 1. If 1 — Aq is negative then [ is neither meaningful nor accessible from the given initial conditions, since thex2-p plane is an invariant set. The stability of either 1 or 2 will depend on comparisons between the subscripted As. The local stability of each rest point depends on the eigenvalues of the linearization around those points. The Jacobian matrix for the linearization of (3.2) at i = 1,2, takes the form... 

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... 

Note that V L = 0 gives Eqs. (18.21), which are the definitions of the slack variables and need not be expressed in the KKT conditions. Note also that L = 2A.jZ, = 0, and, using Eqs. (18.21), Eqs. (18.26) result. These are the so-called complementary slackness equations. For constraint i, either the residual of the constraint is zero, g, = 0, or the Kuhn-Tucker multiplier is zero, X., = 0, or both are zero that is, when the constraint is inactive (gj > 0), the Kuhn-Tucker multiplier is zero, and when the Kuhn-Tucker multiplier is greater than zero, the constraint must be active (g, = 0). Stated differently, there is slackness in either the constraint or the Kuhn-Tucker multiplier. Finally, it is noted that V c x is the Jacobian matrix of the equality constraints, J x, and V g i is the Jacobian matrix of the inequality constraints, K[x). 

Now, since the Hessian is the second derivative matrix, it is real and symmetric, and therefore hermitian. Thus, all its eigenvalues are real, and it is positive definite if all its eigenvalues are positive. We find that minimization amounts to finding a solution to g(x)=0 in a region where the Hessian is positive definite. Convergence properties of iterative methods to solve this equation have earlier been studied in terms of the Jacobian. We now find that for this type of problems the Jacobian is in fact a Hessian matrix. 

In theorem 12 of P it was shown that the first term is that of a positive definite Gramian matrix. The second is clearly the general term of a positive semi-definite matrix we cannot assert that it is definite since one or more of the A Hi might be zero. But the sum of a positive definite and a positive semi-definite matrix is definite and so the Jacobian nowhere vanishes. 

The Jacobian of the system (7.10) is the Hessian of the function to minimize in the case of optimization it is a symmetric matrix and it is positive definite when the minimum is strong. This is no longer true in the case of generic systems (7.1). 

As far as multidimensional optimization problems are concerned, the matrix B can be a bad approximation of the Hessian (provided it is positive definite) yet still be able to guarantee a reduction in the merit function. Conversely, the matrix B involved in the solution of nonlinear systems should be a good estimate of the Jacobian. 

Finally, the Hessian matrix of the function (7.64) is symmetric (and positive definite, if the Jacobian) is nonsingular) and equal to the product j7j . 

If the Jacobian J is factorized QR and the matrix R is worked out in order to avoid any zeros in the main diagonal, then the matrix J J = R R is symmetric positive definite. 

With the symmetric, positive definite 6x6 mass matrix M, the 6 x 1-Jacobian if, the joint variables the sixdimensional joint reaction force / and the sixdimensional force g, which collects the external, applied forces and moments and the gyroscopic terms, the equations of motion are... 

It is clear from Eq. [60a] and the definition of a Jacobian that J[S(r) S(q)] is simply the determinant of the M X M matrix whose — element is exp(i r ). It is shown in Appendix B that this determinant is equal to (V/A Substituting this factor into Eq. [63] yields... 

Jacobian, there always exists some very small, positive value of the fractional step length that results in a decrease of the norm, and so it will be possible always to find some e [0, 1] that satisfies the descent criterion. Even if we do not use the exact Jacobian, but only an approximation of it, as long as the matrix exists and is positive-definite, it... 

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