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An algorithm for computing polar Kirchhoff networks of high dimension
U.S.S.R. Comput.Maths.Math.Phys.,Vol.29,No.6,p.217,1989
0041-5553/89 $i0.00+0.00 ~ 1 9 9 1 Pergamon Press plc
Printed in Great Britain
AN ALGORITHM FOR COMPUTINGPOLARKIRCHHOFF NETWORKSOF HIGH DIMENSION* N.P. PILIKOV
The subject of this paper is the so-called flow-distribution problem, which arises when computing steady regimes of hydraulic, electrical etc. circuits. We introduce the concept of a polar Kirchhoff network, which is a mathematical model of physical systems of varying nature, i.e., we consider a weighted graph G=(V,E), where V is the vertex set and E the set of arcs of the graph. ( 'The full text of the paper is deposited at VINITI, 2158-B89, 104, 1989). To each arc (v,,v~)~E of G there correspond certain positive real numbers c~.$,x~.j,y~.j, where c~.~ is the weight of the arc, x~.j the flow in the arc, y~.~ the potential difference between the vertices v, and vl, and also the relations y,.~=c,.~x](x,.~), where l(z,.~) is a function, continuous in the half-interval 0
I(b)'
l-'(b) '
l(ab)=i(a)/(b),
l-'(ab) =l-'(a)l-'(b),
where a, b are certain positive real numbers and either ~ as xl.1~0, |im[(x,j)= 0 as x,.$-~~,
{
or
lira l(zi,D =
{
0
as as
Xi.j~O, x~.r ~.
Kirchhoff's laws.for currents and voltages hold in the graph G = ( ~ E). To each vertex ~V of G there corresponds a number ~, called the outer flow, such that if q(>0 the outer flow is assumed to flow into the vertex and if ~(<0 it is assumed to emanate from the vertex. It is required to determine the flows xi.~>0 in all arcs (v,.v~)~V of G. The quest for an efficient method to solve this problem has a hundred-year-old history, and computers have been used to that end for the past 40 odd years. It is now accepted that the most efficient of the possible methods of solution (extremal, algebraic, combinations of extremal and algebraic methods) are various modifications of the Newton-Raphson method. Data published in recent years have shown that the Newton-Raphson method furnishes a reliable solution of the problem for graphs with up to 500-1000 arcs. It .should be noted that when this method is used, convergence to the solution cannot be guaranteed, and in practice the method frequently fails to converge. AU the same time certain practical problems, e.g., in the computer-aided design of very large integrated circuits, have dimensions reaching several tens of thousands of arcs or more, and these problems require some guarantee of convergence for the method of solution. This paper proposed an essentially new numerical method (algorithm), with whose help it is always possible to solve the flow-distribution problem for several tens of thousands of arcs. In addition, certain theoretical results are obtained which enable one to decompose the original graph G and thereby increase the dimensions of the problems that can be solved by one or two orders of magnitude. In conclusion I would like to express by sincere gratitude to V.V. Morozov for frequent discussions, lasting many hours, on the work reported here over the past three years, and also for his attentive interest in this paper and abundant advice, which contributed much to its improvement.
Translated by D.L.
~Zh.~yoh{8Z.M~t.n~t.F~z.,29,12,1908,1989
0041-5553/89 $i0.00+0.00 ~ 1 9 9 1 Pergamon Press plc
Printed in Great Britain
AN ALGORITHM FOR COMPUTINGPOLARKIRCHHOFF NETWORKSOF HIGH DIMENSION* N.P. PILIKOV
The subject of this paper is the so-called flow-distribution problem, which arises when computing steady regimes of hydraulic, electrical etc. circuits. We introduce the concept of a polar Kirchhoff network, which is a mathematical model of physical systems of varying nature, i.e., we consider a weighted graph G=(V,E), where V is the vertex set and E the set of arcs of the graph. ( 'The full text of the paper is deposited at VINITI, 2158-B89, 104, 1989). To each arc (v,,v~)~E of G there correspond certain positive real numbers c~.$,x~.j,y~.j, where c~.~ is the weight of the arc, x~.j the flow in the arc, y~.~ the potential difference between the vertices v, and vl, and also the relations y,.~=c,.~x](x,.~), where l(z,.~) is a function, continuous in the half-interval 0
I(b)'
l-'(b) '
l(ab)=i(a)/(b),
l-'(ab) =l-'(a)l-'(b),
where a, b are certain positive real numbers and either ~ as xl.1~0, |im[(x,j)= 0 as x,.$-~~,
{
or
lira l(zi,D =
{
0
as as
Xi.j~O, x~.r ~.
Kirchhoff's laws.for currents and voltages hold in the graph G = ( ~ E). To each vertex ~V of G there corresponds a number ~, called the outer flow, such that if q(>0 the outer flow is assumed to flow into the vertex and if ~(<0 it is assumed to emanate from the vertex. It is required to determine the flows xi.~>0 in all arcs (v,.v~)~V of G. The quest for an efficient method to solve this problem has a hundred-year-old history, and computers have been used to that end for the past 40 odd years. It is now accepted that the most efficient of the possible methods of solution (extremal, algebraic, combinations of extremal and algebraic methods) are various modifications of the Newton-Raphson method. Data published in recent years have shown that the Newton-Raphson method furnishes a reliable solution of the problem for graphs with up to 500-1000 arcs. It .should be noted that when this method is used, convergence to the solution cannot be guaranteed, and in practice the method frequently fails to converge. AU the same time certain practical problems, e.g., in the computer-aided design of very large integrated circuits, have dimensions reaching several tens of thousands of arcs or more, and these problems require some guarantee of convergence for the method of solution. This paper proposed an essentially new numerical method (algorithm), with whose help it is always possible to solve the flow-distribution problem for several tens of thousands of arcs. In addition, certain theoretical results are obtained which enable one to decompose the original graph G and thereby increase the dimensions of the problems that can be solved by one or two orders of magnitude. In conclusion I would like to express by sincere gratitude to V.V. Morozov for frequent discussions, lasting many hours, on the work reported here over the past three years, and also for his attentive interest in this paper and abundant advice, which contributed much to its improvement.
Translated by D.L.
~Zh.~yoh{8Z.M~t.n~t.F~z.,29,12,1908,1989