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The Newton formula for solving (1) is
J(X(i)) Delta X(i)=-f(X(i)) (2)
where J(X(i)) is the Jacobian or circuit matrix. It can be shown
that the sensitivity equations used to compute the sensitivities of
one response variable, X(j), with respect to a set (or vector), p, of
parameters are: (see original).
where the superscript, T, indicates transpose and Epsilon (j) is the
unit vector (see original).
B is a matrix that can be uniquely specified.
In an ICD (Interactive Circuit Design) program the pre-ordered matrix has the
structure of Fig. 1A and is stored sparsely in Fig. 1B, where Regions I, II and III
are stored sparsely (i.e., only the non zero elements), column, row, and column
wise, respectively, and Region IV is stored densely in the one-dimensional array
of Fig. 1B.
Thus in order to solve (3), the Jacobian matrix of (2) must be transposed
using only the array of Fig. 1B. Method
The ICD program uses the modified nodal method of formulating the circuit
equations. Therefore, almost all non zero elements above the diagonal have
mirror images (with respect to the diagonal) below the diagonal. Structural
symmetry is offset by unilateral or (unidirectional) coupling such as those caused
by bipolar transistors. In ordering th...