Network sensitivity and noise analysis simplified

Formulas are easily derived for computing the sensitivity of all the response variables of a network with respect to variation of a single parameter, and the computations can be carried out very efficiently. The converse problem of computing the sensitivity of a single response variable with respect to variations of several parameters, though apparently more difficult, has recently been solved with equal efficiency by appealing to the concept of an "adjoint network." This concept, however, is shown here to be superfluous, since equivalent (and slightly simpler) formulas can be derived using well-known matrix manipulations alone. The problem of computing the signal/noise ratio of a single network response variable has also been solved by using the adjoint network, concept. But, here again, standard matrix manipulations suffice to yield the same results with less conceptual encumbrance. Thus the adjoint network approach, though still valid, proves to be unnecessary for solving these two problems.