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# Method for the Transportation of a Pre-Ordered Sparsely Stored Matrix for Computing Analytic Sensitivities

IP.com Disclosure Number: IPCOM000049573D
Original Publication Date: 1982-Jun-01
Included in the Prior Art Database: 2005-Feb-09
Document File: 2 page(s) / 36K

IBM

## Related People

Ho, CW: AUTHOR [+2]

## Abstract

Introduction. Computer-Aided Design (CAD) programs solve the nonlinear algebraic equation: f(x,p)=0 f, x Epsilon R/n/, p Epsilon R/m/

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Method for the Transportation of a Pre-Ordered Sparsely Stored Matrix for Computing Analytic Sensitivities

Introduction

Computer-Aided Design (CAD) programs solve the nonlinear algebraic equation: f(x,p)=0 (1)

f, x Epsilon R/n/, p Epsilon R/m/

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...