4.2 The Quasi Full-Newton Algorithm

In [39] a method was proposed which was termed full-Newton algorithm. However, this approach is very similar to the two-level method proposed in the same paper hence it is termed ``quasi'' full-Newton in this thesis. The basic idea will be demonstrated for the circuit shown in Fig. 4.1. The combined device and circuit equations read

f(x, V) = 0 (4.13)

I(x, V) + G ^. (V - V_S) = 0   . (4.14)

Applying Newton's method to (4.13) and (4.14) one gets

$$\mathbb {J} \Delta \mathbb {J} \Delta$$ = - f(x, V) (4.15)

$${\frac{\partial I}{\partial \mathbf{x}}} \Delta {\frac{\partial I}{\partial V}} \Delta \Delta$$ = - I(x, V) - G ^. (V - V_S)   . (4.16)

Rearranging (4.15) yields

$$\Delta \mathbb {J} \left(\vphantom{-\mathbf{f}(\mathbf{x},V) -\mathbb{J}_{V} \cdot \Delta V}\right. \mathbb {J} \Delta \left.\vphantom{-\mathbf{f}(\mathbf{x},V) -\mathbb{J}_{V} \cdot \Delta V}\right)$$ (4.17)

which can be rewritten as

$$\Delta \Delta \hat{x} \mathbb {J} \mathbb {J} \Delta$$ = (4.18)

$$\Delta \hat{x} \mathbb {J}$$ = (4.19)

Substituting (4.18) in (4.16) yields

$$\left(\vphantom{ -\frac{\partial I}{\partial \mathbf{x}}\cdot\mathbb{J}_{\mathbf{x}}^{-1}\cdot\mathbb{J}_{V}+\frac{\partial I}{\partial V} +G \cdot }\right. {\frac{\partial I}{\partial \mathbf{x}}} \mathbb {J} \mathbb {J} {\frac{\partial I}{\partial V}} \left.\vphantom{ -\frac{\partial I}{\partial \mathbf{x}}\cdot\mathbb{J}_{\mathbf{x}}^{-1}\cdot\mathbb{J}_{V}+\frac{\partial I}{\partial V} +G \cdot }\right) \Delta {\frac{\partial I}{\partial \mathbf{x}}} \Delta \hat{x}$$ (4.20)

This equation can be rewritten as

$$\left(\vphantom{G_{\mathit{eq}} + G}\right. \left.\vphantom{G_{\mathit{eq}} + G}\right) \Delta$$ (4.21)

with

$${\frac{\partial I}{\partial \mathbf{x}}} \mathbb {J} \mathbb {J} {\frac{\partial I}{\partial V}}$$ G_eq = (4.22)

$${\frac{\partial I}{\partial \mathbf{x}}} \Delta \hat{x}$$ I_T = (4.23)

Equation (4.21) is similar in form to that obtained by the two-level Newton algorithm. Hence, similar methods can be used to embed distributed devices into a circuit simulator and to provide a decoupling between both simulators even for this quasi full-Newton algorithm.