运用牛顿迭代法计算公式(exp(thetae-thetah)*tan(phi))/(sin(alpha-thetaa))=(tan(thetab)*cos(thetah))/(sin(alpha-thetae)*sin(thetaa));中的thetae,并使输出结果为角度值
时间: 2023-12-02 16:05:17 浏览: 55
牛顿迭代法是一种求解方程的数值方法。对于方程f(x)=0,牛顿迭代法的迭代公式为:x_{n+1} = x_n - f(x_n) / f'(x_n),其中x_n为第n次迭代的结果,x_{n+1}为下一次迭代的结果,f(x_n)为方程在x_n处的函数值,f'(x_n)为方程在x_n处的导数。
对于给定的公式(exp(thetae-thetah)*tan(phi))/(sin(alpha-thetaa))=(tan(thetab)*cos(thetah))/(sin(alpha-thetae)*sin(thetaa)),我们可以将其转化为f(thetae) = 0的形式,即:
f(thetae) = (exp(thetae-thetah)*tan(phi))/(sin(alpha-thetaa)) - (tan(thetab)*cos(thetah))/(sin(alpha-thetae)*sin(thetaa))
然后,我们可以使用牛顿迭代法求解方程f(thetae) = 0。具体步骤如下:
1. 选择初始值thetae_0,并设定迭代精度eps。
2. 迭代计算thetae_{n+1} = thetae_n - f(thetae_n) / f'(thetae_n),直到|thetae_{n+1} - thetae_n| < eps。
3. 输出结果thetae_{n+1}。
其中,f'(thetae)为f(thetae)的导数,在此处为:
f'(thetae) = -exp(thetae-thetah)*tan(phi)*cos(alpha-thetae)*sin(thetaa)/(sin(alpha-thetaa)*sin(thetae)^2) - (tan(thetab)*cos(thetah)*cos(alpha-thetae))/(sin(alpha-thetae)*sin(thetaa)^2)
为了输出结果为角度值,我们可以将计算出的thetae结果转化为角度制,即thetae = thetae / pi * 180。
下面是Python代码实现:
```python
import math
# 定义函数f(thetae)
def f(thetae):
exp_part = math.exp(thetae - thetah) * math.tan(phi)
sin_part = math.sin(alpha - thetaa)
tan_part = math.tan(thetab) * math.cos(thetah)
sin2_part = math.sin(alpha - thetae) * math.sin(thetaa)
return exp_part / sin_part - tan_part / sin2_part
# 定义函数f'(thetae)
def fp(thetae):
exp_part = math.exp(thetae - thetah) * math.tan(phi)
cos_part = math.cos(alpha - thetae)
sin_part = math.sin(thetaa)
tan_part = math.tan(thetab) * math.cos(thetah)
sin2_part = math.sin(alpha - thetae) * math.sin(thetaa)
cos2_part = math.cos(thetae) ** 2
return -exp_part * cos_part * sin_part / sin_part ** 2 - tan_part * cos_part / sin2_part ** 2
# 初始值和迭代精度
thetae_0 = 0.5
eps = 1e-6
# 迭代计算thetae
thetae_n = thetae_0
while True:
thetae_np1 = thetae_n - f(thetae_n) / fp(thetae_n)
if abs(thetae_np1 - thetae_n) < eps:
break
thetae_n = thetae_np1
# 输出结果
thetae_deg = thetae_np1 / math.pi * 180
print(thetae_deg)
```
其中,参数thetah、phi、alpha、thetaa和thetab需要根据具体问题给定。
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