By Sachin S Sapatnekar
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Since U is upper triangular‚ one may start from the last equation to find then proceed to the second last equation‚ and so on until the first equation‚ to find the in backward order‚ and this explains why the procedure is referred to as backward substitution. 61)‚ we solve Ly = b using forward substitution to obtain the elements of y as Next‚ applying backward substitution we find the elements of x from Ux = y as Pivoting During the process of LU factorization‚ it is very possible that a diagonal element with a value of zero may be encountered.
The formulæ for the roots of cubic and quartic equations are provided in Appendix A. For this reason‚ and because higher-order AWE approximations are often unstable‚ it is common to select Next‚ given these poles‚ the approximant is represented as a sum of partial fractions‚ and the corresponding residues are calculated. The final step of inverse Laplace transforming this sum yields the response as a sum of exponentials. A typical approach to finding an reduced order system is to start with a desired order and to compute the corresponding approximant.
However‚ in the case of an RC tree‚ admits a closed form‚ and the Elmore delay at node can be written down by inspection as follows. 6. The node is the input node. The procedure described above may be used to compute the Elmore delay to various nodes in the circuit. For example‚ using the upstream resistance formula‚ we can compute The downstream capacitance formula may also be used to write these terms as The reader is invited to verify that the expressions are indeed identical. Several techniques [LM84‚ Wya85‚ CS89‚ CK89] have been suggested for a fast computational approximation to the Elmore delay of a general RC mesh.