Flow-Graph Solutions of Linear Algebraic Equations

A weighted, oriented topological structure, denoted by
G
and called a flow graph, is associated with a set of
m
equations in
n
variables, denoted by
KX = 0
, such that
K
is a connection matrix and
X
a vertex weight matrix of the associated graph. This same set of equations can be written as
A_{-v:}^{-} C(A+)'X = 0
where
A_{-}^{-v:}
and
A^{+}
are negative and positive incidence matrices and where
C
and
X
are respectively branch and vertex weight matrices of the graph. By familiar algebraic procedures, an expression for the weight
x_p
, of a nonreference vertex of
G
is obtained as a linear combination of the weights of the reference vertices (vertices with zero negative order) and can be written as
x_p = \Sigma_{j=1}^{s} \zeta p{\dot}r_{j}x_{r_j}
. To these algebraic results there correspond topological expressions in terms of subgraphs of
G
for the coefficients,
\zeta P{\dot}r_j
. A similar correspondence is obtained between the topological operation of deleting a vertex from the flow graph and the algebraic operation of eliminating a variable from the set of equations. These results are derived from the algebraic equations written in terms of the incidence and weight matrices of the graph. They are similar to those given for the familiar Signal-Flow-Graph, although they are more convient to use, since the topological properties of the flow graph depend only upon the algebraic properties of the set of equations. A flow graph can be drawn directly from an electric network diagram, and the flow-graph properties, used to obtain a solution of the network equations. Examples of this for two types of feedback networks are shown.