| OCR Text |
Show Besides total differentials, dG and dH, Equation (4.4) requires values of partial derivatives Fx, Fy, and Fxy. Since the values of partial derivatives depend on inputs to the operation (see Table 4.4), they are not constant in general. To avoid the calculation of partial derivatives for each input, the following two stage algorithm has been developed in which partial derivatives can be treated as constants. Two computation stages in the algorithm are initialization and evaluation. Let E be an arbitrary Boolean expression. During the initialization stage, the value of expression E is calculated with all independent input variables set to 0. The result is the initial value, denoted as £(0). Next, partial derivatives are calculated for all operations in expression E. Partial derivatives of an operation are constant values obtained from Table 4.4 and from inputs to the operation. The values of operation inputs have been calculated during the evaluation of E (0). The initialization stage thus determines the initial value -E'(O) and the corresponding partial derivatives for all operations in expression E. During the evaluation stage, the total differential d,E(v) for an input vector v is calculated by using the total differentials of operations. The calculation starts with 101 Figure 4.15. The Relation between Total Differentials |
