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Landmarks in the History of Iterative Methods
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DOI：10.1137/24M1680428.png)
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Abstract
En 中文
One of the ways to help make computer science respectable is to show that it is deeply rooted in history ...(Donald E. Knuth, Comm. ACM, 15 (1972), p. 671). A great many of the respectable modern numerical methods proceed iteratively, and we give an overview of them in the final section 11. Teaching and learning science from a historical perspective also leads to a ``respectable deeper understanding. The first problems requiring iterative processes were square-root calculations in Babylon, Greece, and India. More complicated problems such as sine tables in the Arabic, Indian, and medieval calculations, including Kepler's Problem, were performed with fixed point iterations. With Newton, Raphson, and Simpson we enter the ``respectable realm of methods based on derivatives. Mourraille and Cayley contribute geometric insights in both R and C, while Fourier, Cauchy, and Kantorovich provide rigorous error estimations. Surprisingly, even linear problems became interesting for very large dimensions, beginning with the work of Gauss, Seidel, Young, Richardson, and Krylov to domain decomposition and multigrid methods. We explain all of these methods and illustrate them using the Montreal test problem.
Keywords:
iterative methods
fixed point iterations
Newton's method
Krylov methods
domain decomposition
multigrid
Journal
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6.1
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888
Citations:
1.2W
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