Derivative subroutines

Subroutine FUNDR. This subroutine calculates the required derivatives for REGRES by central difference, using EVAL to calculate the objective functions. 

Finite element library subroutines containing shape functions and their derivatives in terms of local coordinates. 

The only requirement is to modify the measure of integration similar to subroutine FLOW. Other terms remain unchanged (see Chapter 4 for derivation of the working equations of the scheme). 

POLYRATE can be used for computing reaction rates from either the output of electronic structure calculations or using an analytic potential energy surface. If an analytic potential energy surface is used, the user must create subroutines to evaluate the potential energy and its derivatives then relink the program. POLYRATE can be used for unimolecular gas-phase reactions, bimolecular gas-phase reactions, or the reaction of a gas-phase molecule or adsorbed molecule on a solid surface. 

Numerical Derivatives The results given above can be used to obtain numerical derivatives when solving problems on the computer, in particular for the Newton-Raphson method and homotopy methods. Suppose one has a program, subroutine, or other function evaluation device that will calculate/given x. One can estimate the value of the first derivative at Xq using... 

Call to ABF subroutine, which computes the derivative of the free energy using (4.33) and adds the adaptive biasing force. It takes as input the current acceleration al and returns the new acceleration with ABF added. /... 

SLOPER REM Subroutine to calculate the coefficient matrix, SLEQ 60SU8 EQUATIONS calculate the derivatives... 

Subroutine PLOTTER actually plots the relative values of the elements of an array called ploty. In this subroutine it is necessary only to set the values of ploty according to the labels specified in GRAFINIT. It is not necessary to plot all of the dependent variables, y, and it is possible to plot derived quantities like the concentrations of dissolved carbon species, by simply specifying what is wanted in GRAFINIT and PLOTTER. The cores of these two subroutines, GRINC and PLTC, do not need to be modified for different simulations. 

Chapter 8 presented the last of the computational approaches that I find widely useful in the numerical simulation of environmental properties. The routines of Chapter 8 can be applied to systems of several interacting species in a one-dimensional chain of identical reservoirs, whereas the routines of Chapter 7 are a somewhat more efficient approach to that chain of identical reservoirs that can be used when there is only one species to be considered. Chapter 7 also presented subroutines applicable to a generally useful but simple climate model, an energy balance climate model with seasonal change in temperature. Chapter 6 described the peculiar features of equations for changes in isotope ratios that arise because isotope ratios are ratios and not conserved quantities. Calculations of isotope ratios can be based directly on calculations of concentration, with essentially the same sources and sinks, provided that extra terms are included in the equations for rates of change of isotope ratios. These extra terms were derived in Chapter 6. 

Several numerical subroutine libraries contain safeguarded Newton codes using the ideas previously discussed. When first and second derivatives can be computed quickly and accurately, a good safeguarded Newton code is fast, reliable, and locates a local optimum very accurately. We discuss this NLP software in Section 8.9. 

Since there are many trays and most are described by Eqs. (5.29) through (5.32), it is logical to use dimensioned variables and to evaluate derivatives and integrate using FORTRAN DO loops. It also makes sense to use a SUBROUTINE or FUNCTION to find given x , because the same equation is used over and over again. 

To use the module M24 you should supply two subroutines. As in the previous methods the one starting at line 900 will evaluate the value F of the function. The second subroutine, starting at line 800, gives the current value of the derivative f (x) to the variable D. To start the iteration we need a single initial guess X. Once again we use the ideal gas volume as initial estimate. The lines different from the lines of the previous program are ... 

The subroutine between lines 4792 - 4806 provides divided difference approximation of the appropriate segment of the Jacobian matrix, stored in the array G(NY,NP). In some applications the efficiency of the minimization can be considerably increased replacing this general purpose routine by analytical derivatives for the particular model. In that case, however, Y(NY) should be also updated here. 

The module M52 is used to solve the error-in-variables estimation problem. The main program contains the starting estimates of the unknown parameters A = = RT in line 230. The subroutine starting at line 700 computes the current values of the two functional relatione. The partial derivatives with respect to the observed variables are computed in lines 600-622. 

Interference Corrections. In certain cases, the y-rays of one element isotope are close to the y-ray energy of an isotope used to determine another element. In these cases, it was necessary to correct for this interference. Subroutines for interference corrections were added to SPECTRA to eliminate performing such corrections manually. The correction factors used were normally derived from nuclear data information. But we did check these corrections by irradiating mixtures of these elements in various concentrations. The interference corrections used are shown in Table VI. The correction for 28Al on 27Mg was determined empirically because it was specific to the irradiation location in the PBR. 

Technically, COSMO-RS meets all requirements for a thermodynamic model in a process simulation. It is able to evaluate the activity coefficients of the components at a given mixture composition vector, x, and temperature, T. As shown in Appendix C of [Cl 7], even the analytic derivatives of the activity coefficients with respect to temperature and composition, which Eire required in many process simulation programs for most efficient process optimization, can be evaluated within the COSMO-RS framework. Within the COSMOt/ierra program these analytic derivatives Eire available at negligible additionEd expense. COSMOt/ierra can Eilso be csdled as a subroutine, Euid hence a simulator program can request the activity coefficients and derivatives whenever it needs such input. 

The only difference in the Jacobian for a fractionator with reaction is the inclusion of the partial derivatives of the new reaction functions Rj,i and QRj with respect to the unknown variables. In the implementation of this model of a fractionator with reaction occurring in a stage, we require a user written subroutine to supply Rj,i and QRj. The user has the option of also coding 9Rj,i / 3Tj, 3Rj,i / 3Lj, . .. or having these... 

With this change to the Jacobian generator and a user supplied routine to evaluate the extents of reaction and at the user s option the partial derivatives 3r/3x, 3r/3l, and 3R/3T along with the ECES generated subroutines, this form of process can be successfully modeled. This type of model has been successfully used to simulate proprietary processes developed by our clients. 

The above has been programmed as a general subroutine U DERIV (see Appendix C), which returns both the wanted derivative (first or second) as well as the coefficients that produced it. 

The above treatment includes the current approximation on an unequal grid, and the subroutine U DERIV can compute it. It is, however, a little unwieldy, and a simpler interface to it is also mentioned in the same Appendix, function GU, which only requires the three arguments (C, x, n). Similarly, the function CU computes Co from a given concentration profile and a known current,... 

As for the choice between direct discretisation on an arbitrarily spaced grid or the formulae for the semi-transformed or the transformed diffusion equation, the present author now inclines towards the first of these. Formulae for the derivatives on arbitrarily spaced points are given in Chap. 3 and Appendix A, and the general subroutine U DERIV is referred to in Appendix C. 

Here, the algorithm described in Chap. 3, Sect. 3.8, implemented in the very general subroutine U DERIV referred to in Appendix C can provide both the derivatives on an arbitrarily spaced set of points (.r, u). However, the reader may wish to restrict the expressions to those involving only up to four points (for which there are some good arguments, see Chap. 8, Sect. 8.4). This can be coupled with current approximations using up to four points. For this number of points, the expressions are not unreasonably long, and a few useful ones are therefore presented here. 

In some programs such as CVRUCAT or the subroutine U DERIV, matrix inversion is needed. This is best done by using LU decomposition, as described in Press et al. (1986). The subroutine MATINV does this. It assumes a square matrix, of the exact size given, so the best way to call it is by using a section of that size, for example... 

For arbitrarily spaced intervals, we require procedures for first and second derivatives, and some other subroutines. 

This subroutine is a general routine for computing the first or second derivative on a number n of points, referred to the ith one among that number, on an arbitrarily spaced grid of points. The derivatives are computed as a linear sum of terms, and the coefficients in that sum are also passed back, for use, for example, in the discretisation of boundary conditions or the spatial second derivative. The number of points is in principle unrestricted, but the routine will fail for values n > 12, where the accuracy abruptly drops. A value, in any case, exceeding about 8, is perhaps impractical. This routine can be used instead of the algebraic expressions shown in Chap. 7, or if n values greater than 4 are required. 

---