Geodesic
The error of Euler's method for floating-point arithmetic computations
Sibirskij žurnal industrialʹnoj matematiki, Tome 14 (2011) no. 3, pp. 37-49

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We present an algorithm that enables us to find the optimal number of steps in Euler's method, in the sense of computational precision, while solving a Cauchy problem for a system of linear differential equations with constant coefficients. We include numerical examples of applications of this method for evaluating a solution to the Cauchy problem at a point and constructing solutions to systems of nonlinear ordinary differential equations.
Keywords: Euler's method, Cauchy problem, system of ordinary differential equations, floating-point arithmetic, computational error. 
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     author = {E. A. Kalinina and O. N. Samarina},
     title = {The error of {Euler's} method for floating-point arithmetic computations},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {37--49},
     publisher = {mathdoc},
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     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2011_14_3_a4/}
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E. A. Kalinina; O. N. Samarina. The error of Euler's method for floating-point arithmetic computations. Sibirskij žurnal industrialʹnoj matematiki, Tome 14 (2011) no. 3, pp. 37-49. http://geodesic.mathdoc.fr/item/SJIM_2011_14_3_a4/