systems analysis Systems analysis is "the process of studying a procedure or business to identify its goal and purposes and create systems and procedures that will efficiently achieve them". Another view sees system analysis as a problem-solving technique that b ...

, Diakoptics (Greek ''dia''–through + ''kopto''–cut, tear) or the "Method of Tearing" involves breaking a (usually physical)

problem Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...

down into subproblems which can be solved independently before being joined back together to obtain an ''exact'' solution to the whole problem. The term was introduced by

Gabriel Kron Gabriel Kron (1901 – 1968) was a Hungarian American electrical engineer who promoted the use of methods of linear algebra, multilinear algebra, and differential geometry in the field. His method of system decomposition and solution called ...

in a series "Diakoptics — The Piecewise Solution of Large-Scale Systems" published in

London, England London is the capital and largest city of England and the United Kingdom, with a population of just under 9 million. It stands on the River Thames in south-east England at the head of a estuary down to the North Sea, and has been a major s ...

by ''The Electrical Journal'' between June 7, 1957 and February 1959. The twenty-one installments were collected and published as a

book A book is a medium for recording information in the form of writing or images, typically composed of many pages (made of papyrus, parchment, vellum, or paper) bound together and protected by a cover. The technical term for this physical arr ...

of the same title in 1963. The term ''diakoptics'' was coined by Philip Stanley of the

Union College Union College is a private liberal arts college in Schenectady, New York. Founded in 1795, it was the first institution of higher learning chartered by the New York State Board of Regents, and second in the state of New York, after Columbia Co ...

Department of Philosophy.

Features

According to Kron, "Diakoptics, or the Method of Tearing, is a combined theory of a pair of storehouses of information, namely equations+graph, or matrices+graph, associated with a given physical or economic system.". What Kron was saying here is that in order to carry out the Method of Tearing, not only were the system equations needed, but also the topology of the system. Diakoptics was explained in terms of

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

by J. Paul Roth. Roth describes how

Kirchhoff's circuit laws Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhof ...

in an

electrical network An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, c ...

with a given

impedance matrix Impedance parameters or Z-parameters (the elements of an impedance matrix or Z-matrix) are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electr ...

admittance matrix In power engineering, nodal admittance matrix (or just admittance matrix) or Y Matrix or Ybus is an ''N x N'' matrix describing a linear power system with ''N'' buses. It represents the nodal admittance of the buses in a power system. In realist ...

can be solved for currents and voltages by using the

circuit topology The circuit topology of a folded linear polymer refers to the arrangement of its intra-molecular contacts. Examples of linear polymers with intra-molecular contacts are nucleic acids and proteins. Proteins fold via formation of contacts of variou ...

. Roth translates Kron’s "orthogonality conditions" into

exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...

s of homology or cohomology. Roth’s interpretation is confirmed by

Raoul Bott Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions whi ...

in reports in

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

. Roth says, "tearing consists essentially in deducing from the solution of one (easier to solve) network K^~ the solution of a network K having the same number of branches as K^~ and having the same isomorphism L between the groups of 1-chains and 1-cochains." Diakoptics can be seen applied for instance in the text ''Solution of Large Networks by Matrix Methods''. Diakoptics is peculiar as a decomposition method, in that it involves taking values on the "intersection layer" (the boundary between subsystems) into account. The method has been rediscovered by the parallel processing community under the name "

Domain Decomposition In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the soluti ...

". According to Keith Bowden, "Kron was undoubtedly searching for an ontology of engineering". Bowden also described "a multilevel hierarchical version of the method, in which the subsystems are recursively torn into subsubsystems". When

parallel computing Parallel computing is a type of computation in which many calculations or processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different fo ...

was provided by the

transputer The transputer is a series of pioneering microprocessors from the 1980s, intended for parallel computing. To support this, each transputer had its own integrated memory and serial communication links to exchange data with other transputers. T ...

, Keith Bowden described how diakoptics might be applied. It is an ongoing open question how the parallelism of Quantum Computing may be relevant.

Japan

In 1951 "a group of about twenty scientists and engineers was organized in Japan for the unifying study of basic problems in engineering sciences by means of geometry….reorganized in 1954 into a new organization called the Research Association of Applied Geometry (RAAG)." "Kron was...the first charter member from overseas. He remained an Honorary Member until he passed away. Meanwhile, the RAAG membership roll has been increased to two hundred and fifty, distributed throughout the world." Many articles on diakoptics, by Kron and others, were published in the ''Memoirs'' of RAAG. The journal ''Tensor'' (), published in

Sapporo, Japan ( ain, サッ・ポロ・ペッ, Satporopet, lit=Dry, Great River) is a city in Japan. It is the largest city north of Tokyo and the largest city on Hokkaido, the northernmost main island of the country. It ranks as the fifth most populous city ...

, placed Kron on its "honorary editorial board" in 1951. He contributed an article to the March 1955 issue.

United Kingdom

"Kron’s influence extends far beyond the U.S.A. The Tensor Society of Great Britain came into being to further the understanding and applications of tensor analysis." In 1950 it was founded by S. Austen Stigant as The Tensor Club of Great Britain, and began publishing ''Matrix and Tensor Quarterly'' (). According to Kron, it was S. Austen Stigant that first suggested Kron write his serial "Diakoptics" for the ''Electrical Journal''. In September 1961 the Club became a Society. In 1968 (volume 19) the ''Quarterly'' published a commemorative issue to note the death of Gabriel Kron. The ''Quarterly'' continued publishing until 1987. "C.H. Flurscheim and J.R. Mortlock of Associated Electrical Industries Ltd. ncouragedthe Power Systems Engineering Department of that company to investigate the applications of Diakoptics to the solution of practical load-flow problems and some difficult mechanical vibration problems; investigations which yielded results of considerable value."Alfred Brameller & D. W. Mortifee (1973) "The influence of Gabriel Kron in the United Kingdom", pp 143 to 151 in ''Gabriel Kron and Systems Theory''

References

{{Reflist * A. Brameller, M.N. John & M.R. Scott (1969) ''Practical Diakoptics for Electrical Networks'',

Chapman & Hall Chapman & Hall is an imprint owned by CRC Press, originally founded as a British publishing house in London in the first half of the 19th century by Edward Chapman and William Hall. Chapman & Hall were publishers for Charles Dickens (from 1840 ...

. * H.H. Happ (1973) "The Development of Diakoptics", pp 83 to 120 in ''Gabriel Kron and Systems Theory''. * Kron G. (1963) ''Diakoptics: The Piecewise Solution of Large Scale Systems'', MacDonald Publishing. Problem solving Systems analysis

Features

Japan

United Kingdom

See also

References