ABSTRACT
In many branches of engineering and science it is desirable to be able to math ematically determine the state of a system based on a set of physical relation ships. These physical relationships may be determined from characteristics such as circuit topology, mass, weight, or force to name a few. For example, the injected currents, network topology, and branch impedances govern the voltages at each node of a circuit. In many cases, the relationship between the known, or input, quantities and the unknown, or output, states is a linear relationship. Therefore, a linear system may be generically modeled as
Ax — b (2 .1)
where b is the n x 1 vector of known quantities, x is the n x 1 unknown state vector, and A is the n x n matrix that relates x to b. For the time being, it will be assumed that the matrix A is invertible, or non-singular; thus, each vector b will yield a unique corresponding vector x. Thus the matrix A r 1 exists and
x* = A~xb (2 .2 )
is the unique solution to equation (2 .1). The natural approach to solving equation (2.1) is to directly calculate the
inverse of A and multiply it by the vector b. One method to calculate A ^ 1 is to use Cramer’s rule :
= — l — (Aij)T for i — 1 , . . . , n, j — 1 , . . . , n (2.3)
where A^1(i , j ) is the i j th entry of A ^ 1 and Ay is the cofactor of each entry a,ij of A. This method requires the calculation of (n + 1) determinants which results in 2(n + 1)! multiplications to find A-1 ! For large values of n, the calculation requirement grows too rapidly for computational tractability; thus, alternative approaches have been developed.
