Joseph F. Grcar

6059 Castlebrook Drive
Castro Valley, CA   94552
Phone: 510-581-1353


Professional Experience

The History of "Gaussian" Elimination

Gaussian elimination is universally known as "the" method for solving simultaneous linear equations.  As Leonhard Euler wrote in 1765, it is the most natural way of proceeding ("der natürlichste Weg").  Because Gaussian elimination solves linear problems directly, it is an important technique in computational science and engineering, through which it makes continuing, albeit indirect, contributions to advancing knowledge and to human welfare.

What is natural depends on the context, so the algorithm has changed many times with the problems to be solved and with competing technology.  The sole development in ancient times was in China.  An independent origin in modern Europe has had three phases.  First came the schoolbook lesson beginning with Isaac Newton, whose contribution has only recently been recognized.  Next were methods for professional hand computers which began with Carl Friedrich Gauss.  Lastly was the interpretation in matrix algebra by several authors including John von Neumann and Alan Turing.  Gaussian elimination is living mathematics that has successfully mutated for hundreds of years to meet changing social needs.  Perhaps the only certainty about future algorithms is their name.

This talk has been given to undergraduate and graduate students in computer science and mathematics seminars and colloquia at several institutions: Stanford Univ. - Univ. of California at Berkeley - Purdue Univ. - Univ. of San Francisco - California State Univ. East Bay

Related publication by speaker
J. F. Grcar, "How Ordinary Elimination Became Gaussian Elimination," Historia Mathematica, to appear (available online).  DOI:/10.1016/

John von Neumann and the Origins of Modern Scientific Computing

Scientific computing has existed as both a mathematical speciality and as an occupation for over two hundred years.  The first professional, human computers were employed by astronomical observatories to express observations in celestial coordinates and to calculate ephermerides.  Computers entered government cartographic bureaus and then agricultural agencies after Carl Friedrich Gauss introduced least squares methods and then modern statistics and econometrics were developed.  A variety of mechanical aids were invented to help the human computers including manual calculators and differential analyzers.

The invention of modern computers (those digital, electronic, and programmable) in the mid 1940s represented a paradigm shift in what could be achieved through calculation.  From his wartime military duties John von Neumann acquired what he described as an "obscene" interest in mechanized calculations.  No one was better situated than he to understand the advances that could be realized but also the whole range of technical obstacles that had to be overcome.  As a necessary prerequisite to that work von Neumann created computer science through a series of influential reports that described the design and use of the computers being built first at the Univ. of Pennsylvania and later at the Institute of Advanced Study.

This talk has been given to undergraduate and graduate students in computer science and mathematics seminars and colloquia at several institutions: Univ. of Southern California - Univ. of California at Davis - Humboldt State Univ. - Univ. of Illinois at Urbana - Purdue Univ. - Univ. of Colorado at Boulder - Department of Energy Office of Advanced Scientific Computing Research Principal Investigator Annual Meeting after-dinner speech - Eidgenössische Technische Hochschule - Univ. of San Francisco - Stanford Univ. - Santa Clara Univ. - San Jose State Univ.- Stanford Univ. (twice) - Lawrence Berkeley National Laboratory - Univ. of California at Berkeley

Related publication by speaker
J. F. Grcar, "John von Neumann's Analysis of Gaussian Elimination and the Origins of Modern Numerical Analysis," SIAM Review, in press.

Review of a presentation

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