英语翻译Example 4.2.Consider the following nonlinear equation:This equation is a special case of the Kepler equation with e = 1 and M = 0 [21].For this equation G(x) is set to ekx.To determinethe ability of the proposed method for finding differe

英语翻译Example 4.2.Consider the following nonlinear equation:This equation is a special case of the Kepler equation with e = 1 and M = 0 [21].For this equation G(x) is set to ekx.To determinethe ability of the proposed method for finding differe
英语翻译
Example 4.2.Consider the following nonlinear equation:
This equation is a special case of the Kepler equation with e = 1 and M = 0 [21].For this equation G(x) is set to ekx.To determine
the ability of the proposed method for finding different roots,specially complex ones,initial values for z are set to
a + bi,that a and b are real numbers and vary from \220 to 20 with steps of 1.Therefore the algorithm runs for
412 = 1681 different initial values.The stopping criteria for each run is set to.The algorithm found 13 different
roots (one real and twelve complex roots).Fig.2 shows the roots found by the algorithm.The roots numbered from 1
to 13.Fig.3 shows the initial value regions of convergence of the 13 different roots.
From Fig.3 one can also find harmony in converging initial values to different roots.Therefore,if the initial value domain
is extended in the real axis direction,the algorithm may find other complex roots of (18).It is interesting to note that with
just real initial values the algorithm can find the all the roots in the related domain.That is,initial value domain of,
leads to 13 different roots that numbered before.
Note that we examine the special case of the Kepler equation in this example.One can vary e and M and find all of the
related roots in the domain of initial values.
6.Numerical examples
We now present some examples to illustrate the efficiency of the newly developed method.This section compares the
proposed method with fsolve command in MATLAB,the methods of Sánchez (w42and w63
Þ [22],the method of Ujevic [23],
the method of Jesheng et al [24] and Newton’s method (NM).All computations were done using MATLAB.The following stopping
criterion is used for computer programs:
,was used.
Here,the goal is to compare the power of methods for finding roots for a large amount of initial values.In the following
examples NR indicates the Number of Roots that a method has found in the prescribed initial value range,NF indicates Number
of Fails,Ave IT indicates average number of iterations for that method to find any root and finally,Ave F indicates average
number of function evaluations for a method to find a root.(Note that if a method fails to find any root that step gets a 30-
iteration penalty).
Example 6.1.Consider the following polynomial equation:
Obviously,this equation has 100 different roots (2 real and 98 complex roots).Initial values in this example are set to
±0.025 + bi,where b is a real number and varies from \20.999 to 1 with steps of 0.001.Therefore the algorithm runs for
2 \4 2000 ¼ 4000 different initial values.Table 2 shows the comparison between different methods.
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英语翻译Example 4.2.Consider the following nonlinear equation:This equation is a special case of the Kepler equation with e = 1 and M = 0 [21].For this equation G(x) is set to ekx.To determinethe ability of the proposed method for finding differe
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