Secant Method
المؤلف:
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T.
المصدر:
"Secant Method, False Position Method, and Ridders Method." §9.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press
الجزء والصفحة:
pp. 347-352
14-12-2021
1895
Secant Method
A root-finding algorithm which assumes a function to be approximately linear in the region of interest. Each improvement is taken as the point where the approximating line crosses the axis. The secant method retains only the most recent estimate, so the root does not necessarily remain bracketed. The secant method is implemented in the Wolfram Language as the undocumented option Method -> Secant in FindRoot[eqn, ![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/SecantMethod/Inline1.gif" style="height:15px; width:5px" />x, x0, x1![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/SecantMethod/Inline2.gif" style="height:15px; width:5px" />].
When the algorithm does converge, its order of convergence is
 |
(1)
|
where 
is a constant and 
is the golden ratio.
 |
(2)
|
 |
(3)
|
 |
(4)
|
so
 |
(5)
|
The secant method can be implemented in the Wolfram Language as
SecantMethodList[f_, {x_, x0_, x1_}, n_] :=
NestList[Last[#] - {0, (Function[x, f][Last[#]]*
Subtract @@ #)/Subtract @@
Function[x, f] /@ #}&, {x0, x1}, n]
REFERENCES:
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Secant Method, False Position Method, and Ridders' Method." §9.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 347-352, 1992.
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