About Gauss-Jordan elimination
Some clay tablets from the
Euphrates and Tigris valley indicate the earliest
cases, where systems of linear equations have appeared 4000 years ago.
The
Gauss-Jordan algorithm appeared first in the
Nine Chapters on the Mathematical Art, which was authored around 300 BC in China.
Due to a tradition of anonymity in that primary text, we might never know the details of
that development. But Chapter 8 in that text definitely already uses an early elimination technique.
Flash forward for almost 2000 years to Europe:
Isaac Newton wrote some notes in 1670, which
described elimination the first time in Europe. Also
Gottfried Leibniz developed a method for solving systems
of linear equations. But Leibniz used determinants.
Carl Friedrich Gauss in 1809 in
``Disquisitio de Elementis ellipticis Palladis" devised an algorithm to solve
least-squares problems. Gauss himself considered the method as ``commonly known" and used
them for astronomical purposes (he predicted earlier the path of the minor planet Ceres successfully).
Gauss developed later also finer methods like the
Gauss-Seidel algorithm.
This happened around 1823. His more sophisticated method was used for his work on the triangulation
of Hanover. The name
Gauss-Jordan elimination for transforming a matrix into reduced echelon form also
includes the name Jordan because a similar version was described by
Wilhelm Jordan in 1888 in his book
``Handbuch der Vermessungskunde" independently of
B.-I. Clasen who published
something similar in the same year. The geodesist Jordan also used the method
to minimze the squared error in observations. The story of elimination is also
related to the history of matrices. Formally, matrices were introduced in 1855 by Arthur Cayley
but according to Morris Kline the subject of matrices ``was well developed before it
was created". The method of elimination was also implemented in the first stored-program
digital computer, the
Manchester Baby
which in 1948 was historically
the first to run a computer program on an electronic stored-program computer.
This machine was further developed in the form of the
Manchester Mark 1,
which is not to be confused with the
Harvard Mark I
here at Harvard, which began computations already earlier, namely in 1944. The
Mark I (here an other website about it) did not
use Gaussian elimination yet. Also the Harvard Mark II from 1947 inverted matrices differently: (I-A)
-1 can
be written as 1+A+A
2 + ...
This idea was used for
input-output models of
the economist
Wassily Leontief. The series approach is a third and completely different way to solve
systems of linear equations. It is different from the determinant methods like Leibniz or
Cramer used and also different from the Gaussian elimination. It works
well if the matrix entries of A are small.
Here is some Literature: