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Improved Newton-Raphson method
Applies an improved Newton-Raphson root-finding method to
a non-linear equation. |
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| Equation |
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Non-linear equation, F(x),
for which a root is sought. |
| y0 |
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Right hand side in F(x) = y0
(usually 0). |
| x0 |
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Initial guess for x, the root. |
| itmax |
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Maximum number of iterations. |
| tol |
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Tolerance in x. |
| Show values |
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Shows the intermediate calculations. |
Applies a modified, improved Newton-Raphson root-finding method
[Chun, 2006] to a given function (to be selected)
F(x) = y0. A value x* will be
sought iteratively, given an initial guess, x0.
If the method converges, it will be, for
f(x) = F − y0 = 0:
X
= x − f(x) ⁄ f ′(x)
− 2 f(ξ) ⁄ f ′(x) +
f(ξ) f ′(ξ) ⁄
f ′²(x)
= ξ − [2 − f ′(ξ) ⁄
f ′(x)] f(ξ) ⁄
f ′(x)
with ξ = x −
f(x) ⁄ f ′(x)
Above, Eq. 1 (Ex. 2 in the reference) is
x − 2 − exp(−x) = 0, with
x0 = 2 (x* = 2.12); and Eq. 2 (Ex. 4) is
exp(x) − 3 x² = 0,
with x0 = 0.5
(x* = 0.91).
An Excel solution…
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References:
• Chun, Changbum,
2006, "A new iterative method for solving nonlinear equations",
Applied Mathematics and Computation, 178, pp 415–422
(Chun.zip).
• Weisstein, Eric W.,
"Newton's
Method", from Mathworld—A Wolfram Web Resource.
http://mathworld.wolfram.com/NewtonsMethod.html . |
