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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Optimal numerical differentiation using three function evaluations

Authors: J. Marshall Ash and Roger L. Jones
Journal: Math. Comp. 37 (1981), 159-167
MSC: Primary 65D05; Secondary 39A05
MathSciNet review: 616368
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Abstract: Approximation of $f’ (x)$ by a difference quotient of the form \[ {h^{ - 1}}[{a_1}f(x + {b_1}h) + {a_2}f(x + {b_2}h) + {a_3}f(x + {b_3}h)]\] is found to be optimized for a wide class of real-valued functions by the surprisingly asymmetric choice of ${\mathbf {b}} = ({b_1},{b_2},{b_3}) = (1/\sqrt 3 - 1,1/\sqrt 3 ,1/\sqrt 3 + 1)$. The nearly optimal choice of ${\mathbf {b}} = ( - 2,3,6)$ is also discussed.

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Keywords: Numerical differentiation, discretization error, truncation error, generalized difference quotient
Article copyright: © Copyright 1981 American Mathematical Society