To address these problems, the C++ version of Adams Solver takes a different approach to computing the partial derivatives. The C++ version applies the chain rule:

Note that in keeping with the observation that F is a closed box, Adams Solver continues to rely on finite differencing to compute , where Mj is well known and its partial derivatives can be evaluated analytically. This scheme solves both problems:

For example, if F depends on the 3D TDISP measure of a marker on a FLEX_BODY, F must only be evaluated three times during finite differencing, even if the FLEX_BODY has 200 modal generalized coordinates. This is a remarkable computational savings.

Note that the computational savings are not always so generous. Consider a function F that depends on the two 6D measures, DISP and VEL, of a single PART marker, with respect to ground. During finite differencing, F must be evaluated 12 times, once for each dimension of each measure. Meanwhile, the function only depends on the 12 generalized displacements and velocities of the PART, so the FORTRAN 77 Adams Solver would also only have needed 12 evaluations of F.

Note that the potential of user-provided partial derivatives does not only affect the computational cost of evaluating the system Jacobian, but can also improve the accuracy of simulations and the rate of convergence. This is because your analytical derivatives are likely to be more accurate than those obtained using finite differencing. When Adams Solver uses the Jacobian directly, as in Adams Linear, the quality of the solution may also be improved.

When F and M have dimension greater than 1, the partial matrix, should be stored in partl in FORTRAN 77 style column order. In other words, all the partial derivatives of F with respect to the first component of M come before the partial derivatives of F with respect to the second component of M, and so on.

Each call to SYSFNC/SYSARY creates what the previous section refers to as a measure. For each of these measures, the creator of the user subroutine is allowed to register the partials of the function with respect to this measure.

You do not need partial derivatives during every call to the user's subroutine, because they are only needed when the Jacobian is being evaluated. You must request the flag that indicates whether the partial derivatives are required, by calling the following function:

Note that, ideally, the PARFLG would have been added to the user subroutine call, alongside IFLAG and DFLAG, but the user subroutines have a standard interface that cannot be changed.

You can make calls to SYSPAR only for those SYSFNC/SYSARY for which partials are conveniently available. As mentioned earlier, you can make SYSPAR calls intermittently. In other words, you can call SYSPAR only during some period of the simulation when partials can be easily computed, but skipped during other parts of the simulation. For example, when an impact force is zero, its partials are trivially zero.