MMP.9:4.2 - Derive the discarded contribution before approximating it

Try to express the discarded variables through their own evolution, initial conditions and the retained history. In a discrete law, iterate the discarded update and substitute each earlier term until the dependence has a usable form. In a differential law, solve or integrate the discarded equation under the supplied retained history, then substitute the result into the retained equation.

For constant compatible matrices and supplied initial values, consider:

x_dot=A*x+B*y, y_dot=C*x+D*y.

Solving the second equation while treating x(s) as its input gives:

y(t)=exp(D*t)*y0 + integral_0^t exp(D*(t-s))*C*x(s) ds.

Substitution gives the retained law:

x_dot(t)=A*x(t)+B*exp(D*t)*y0 + integral_0^t B*exp(D*(t-s))*C*x(s) ds.

The first added term carries the discarded initial condition. The integral carries the past influence of x through y. With a forcing term in the discarded equation, its propagated contribution also appears in the integral. These terms identify what a proposed memory approximation would replace. Unknown y0 remains an initial uncertainty; it cannot be set to zero solely because y is being removed.

Look for a less costly way to compute the derived expression. Equal decay modes can be combined; a sum of exponentials can be updated through a few auxiliary variables. For example, v(t)=integral_0^t exp(-lambda*(t-s))*x(s) ds satisfies v_dot=x-lambda*v, v(0)=0. This replaces storage of the whole history with an evolving value. Carry a nonzero initial contribution separately or incorporate its matching initial condition. Count the required auxiliary values and update work before claiming a computational saving.

For nonlinear discarded dynamics the same elimination question remains, but the response to retained history may require solving a nonlinear problem. Use the derived dependence to select an approximation, retained variable or bound; an implicit formula that still requires the original computation has not yet supplied a cheaper model.