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Gauss, Jacobi, Seidel, Richardson, Krylov: the Invention of Iterative Methods

576 views · Published 28 May 2013 · 52:14 · Indexed 6 October 2026

Channel: Institut Henri Poincaré · 2013 · Science & Technology

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10 ans du Groupe Calcul : Histoire du Calcul

Par Martin Gander (Université de Genève) 
Abstract: The invention of iterative methods for systems of linear equations now
spans almost two centuries. These methods were invented for the same
reasons as they are used today, namely to reduce computational cost.
Gauss states in a letter to his friend in 1823: "you will in the
future hardly eliminate directly, at least not when you have more than
two unknowns". The basic methods of Gauss and Jacobi are however not
used any more today, except as special components of more
sophisticated methods. Richardson's paper from 1910 was very
influential in this sense, and is a model of a modern numerical
analysis paper: modelling, discretization, approximate solution of the
discrete problem, and a real application. It contains also a
fundamentally new idea for an iterative method which later became the
Chebyshev semi-iterative method. It was however the work of Stiefel,
Hestenes and Lanczos in the early 1950 which sparked the success story
of Krylov methods. But why are these methods called Krylov methods ?
In order to get an understanding, we will have to take a closer look
at the publication of Krylov from 1931, where Krylov explains several
numerical methods for solving systems of second order ordinary
differential equations, and in particular a method of Leverrier, which
contains after some manipulations what we now know as being a Krylov
space.

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