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In [10], Park et al. proposed a FMD method with initial mode selection.
Then Newton's method with initial point x 0 ∈ G considered in [16] can be written in a coordinate-free form as follows.
The dispersion relation is solved by a modified Newton Raphson method with initial values given by an asymptotic expansion or a winding integral method.
Following [17], we define Newton's method with initial point x 0 for f on a Lie group as follows: x n + 1 = x n ⋅ exp ( − d f x n − 1 f ( x n ) ) for each n = 0, 1, …. (3.1).
(28) In Fig. 3 we show a numerical simulation prescribed by the interface method, with initial data equivalent to that from the full space time simulation shown in Fig. 2.
The existence of stable subharmonic motions found is confirmed by solving the differential equation of motion numerically by means of a time-difference method, with initial conditions being supplied by the harmonic balance approximation.
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The system may be solved by numerical analysis methods with initial conditions, like the Runge-Kutta-Gear method.
Subject recruitment was by snowball methods with initial identification made by research contacts within Indigenous health.
A series of six open population estimates (Jolly-Seber method) varied in results as expected for this method [26] with initial and ending estimates in the series showing high variance.
As an example, for Lena image, the Full-search method performs 475,799,552 MSE calculations, while the proposed method, with 200 initial countries and 10 initial empires, performs 1,026,341 MSE computations; it means about 463 times less than the Full-search algorithm.
Then the sequence { x n } generated by Newton's method (3.1) with initial point x 0 is well defined and converges to a zero x ∗ of f.
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