最速下降法(Steepest descent)与共轭梯度法(Conjugate gradient)

忘是亡心i 2022-12-28 12:38 242阅读 0赞

# 目标问题： #

# ![\\text\{min\}~ f = x\_1^2+10x\_2^2.][text_min_ f _ x_1_2_10x_2_2.] #

### 初始化参数：![x\_0=\[5, 5\]^\\text\{T\},~k=0.][x_0_5_ 5_text_T_k_0.] ###

![\\nabla f = \\binom\{2x\_1\}\{20x\_2\}][nabla f _ _binom_2x_1_20x_2]

### 在第一步迭代时(k=0)，共轭梯度法等同于最速下降法： ###

![x\_1 = x\_0-\\alpha\_0\\nabla f\_0=\\begin\{bmatrix\} 5-10\\alpha\_0\\\\ 5-100\\alpha\_0 \\end\{bmatrix\}][x_1 _ x_0-_alpha_0_nabla f_0_begin_bmatrix_ 5-10_alpha_0_ 5-100_alpha_0 _end_bmatrix]

### 采用牛顿迭代法搜寻k=0时适当的步长因子： ###

![f(\\alpha)=(5-10\\alpha)^2+10(5-100\\alpha)^2.][f_alpha_5-10_alpha_2_10_5-100_alpha_2.]

![\\alpha\_\{m=0\}=1, m=0][alpha_m_0_1_ m_0]

![\\dot f(\\alpha)=2(5-10\\alpha)(-10)+20(5-100\\alpha)(-100)][dot f_alpha_2_5-10_alpha_-10_20_5-100_alpha_-100]

![=-100+200\\alpha-10000+20000\\alpha = -10100+200200\\alpha][-100_200_alpha-10000_20000_alpha _ -10100_200200_alpha]

![\\ddot f(\\alpha)=200200][ddot f_alpha_200200]

![gif.latex][]![\\alpha\_1 = \\alpha\_0 - \\frac\{\\dot f(\\alpha\_0)\}\{\\ddot f(\\alpha\_0)\}=1-\\frac\{190100\}\{200200\}=0.0505.][alpha_1 _ _alpha_0 - _frac_dot f_alpha_0_ddot f_alpha_0_1-_frac_190100_200200_0.0505.]

![|\\alpha\_1 - \\alpha\_0|>10^\{-3\},~\\text\{not convergence\}.][alpha_1 - _alpha_0_10_-3_text_not convergence_.]

![m = m+1=1.][m _ m_1_1.]

![\\alpha\_2 = \\alpha\_1- \\frac\{dot f(\\alpha)\}\{\\ddot f(\\alpha\_1)\}=0.0505-\\frac\{10.1\}\{200200\}\\approx 0.0505.][alpha_2 _ _alpha_1- _frac_dot f_alpha_ddot f_alpha_1_0.0505-_frac_10.1_200200_approx 0.0505.]

![|\\alpha\_2 - \\alpha\_1|>10^\{-3\},~\\text\{convergence\}.][alpha_2 - _alpha_1_10_-3_text_convergence_.]

### 当k=1时，将![\\alpha\_2][alpha_2]带入原迭代方程，并采用最速下降法迭代： ###

![x\_1 = x\_0-\\alpha\_0\\nabla f\_0=\\begin\{bmatrix\} 5-10\\alpha\_0\\\\ 5-100\\alpha\_0 \\end\{bmatrix\}=\\begin\{bmatrix\} 5-10\\times 0.0505\\\\ 5-100\\times 0.0505\\end\{bmatrix\}=\\begin\{bmatrix\} 4.495\\\\ -0.05 \\end\{bmatrix\}][x_1 _ x_0-_alpha_0_nabla f_0_begin_bmatrix_ 5-10_alpha_0_ 5-100_alpha_0 _end_bmatrix_begin_bmatrix_ 5-10_times 0.0505_ 5-100_times 0.0505_end_bmatrix_begin_bmatrix_ 4.495_ -0.05 _end_bmatrix]

![||x\_1-x\_0||=||\\begin\{bmatrix\} 4.495-5\\\\ -0.05-5 \\end\{bmatrix\}||=||\\begin\{bmatrix\} -0.505\\\\ -5.05 \\end\{bmatrix\}||=5.0752>10^\{-3\}.~(\\text\{not convergence\})][x_1-x_0_begin_bmatrix_ 4.495-5_ -0.05-5 _end_bmatrix_begin_bmatrix_ -0.505_ -5.05 _end_bmatrix_5.0752_10_-3_._text_not convergence]

![k = k+1=1.][k _ k_1_1.]

![x\_2 = x\_1-\\alpha\_1\\nabla f\_1=\\begin\{bmatrix\} 4.495-8.99\\alpha\_1\\\\ -0.05+\\alpha\_1 \\end\{bmatrix\}][x_2 _ x_1-_alpha_1_nabla f_1_begin_bmatrix_ 4.495-8.99_alpha_1_ -0.05_alpha_1 _end_bmatrix]

### 再次采用牛顿迭代法搜寻k=1时适当的步长因子： ###

![f(\\alpha)=(4.495-8.99\\alpha)^2+10(-0.05-\\alpha)^2.][f_alpha_4.495-8.99_alpha_2_10_-0.05-_alpha_2.]

![\\alpha\_\{m=0\}=1, m=0][alpha_m_0_1_ m_0]

![\\dot f(\\alpha)=2(4.495-8.99\\alpha)(-8.99)+20(-0.05+\\alpha)=181.64\\alpha-81.82][dot f_alpha_2_4.495-8.99_alpha_-8.99_20_-0.05_alpha_181.64_alpha-81.82]

![\\ddot f(\\alpha)=181.64.][ddot f_alpha_181.64.]

![gif.latex][]![\\alpha\_1 = \\alpha\_0 - \\frac\{\\dot f(\\alpha\_0)\}\{\\ddot f(\\alpha\_0)\}=1-\\frac\{99.82\}\{181.64\}=0.4505.][alpha_1 _ _alpha_0 - _frac_dot f_alpha_0_ddot f_alpha_0_1-_frac_99.82_181.64_0.4505.]

![|\\alpha\_1 - \\alpha\_0|>10^\{-3\},~\\text\{not convergence\}.][alpha_1 - _alpha_0_10_-3_text_not convergence_.]

![m = m+1=1.][m _ m_1_1.]

![\\alpha\_2 = \\alpha\_1- \\frac\{dot f(\\alpha)\}\{\\ddot f(\\alpha\_1)\}=0.4505-\\frac\{0.0088\}\{181.64\}\\approx 0.4505.][alpha_2 _ _alpha_1- _frac_dot f_alpha_ddot f_alpha_1_0.4505-_frac_0.0088_181.64_approx 0.4505.]

![|\\alpha\_2 - \\alpha\_1|>10^\{-3\},~\\text\{convergence\}.][alpha_2 - _alpha_1_10_-3_text_convergence_.]

### 将![\\alpha\_2][alpha_2]带入原迭代方程，并采用最速下降法迭代： ###

![=\\begin\{bmatrix\} 4.495-8.99\\times 0.4505\\\\ -0.05+ 0.4505\\end\{bmatrix\}=\\begin\{bmatrix\} 4.445\\\\ -0.4005 \\end\{bmatrix\}][begin_bmatrix_ 4.495-8.99_times 0.4505_ -0.05_ 0.4505_end_bmatrix_begin_bmatrix_ 4.445_ -0.4005 _end_bmatrix]

。。。。。。利用最速下降法的迭代方法，重复多次的迭代。

--------------------

### 当k=1时采用共轭梯度法迭代： ###

![x\_2 = x\_1 + \\alpha\_1 d\_1][x_2 _ x_1 _ _alpha_1 d_1]

![\\nabla f\_1\\cdot \\nabla f\_1 = 81.82][nabla f_1_cdot _nabla f_1 _ 81.82]

![\\nabla f\_0\\cdot \\nabla f\_0 = 10100][nabla f_0_cdot _nabla f_0 _ 10100]

![d\_1 = -\\nabla f\_1+\\frac\{\\nabla f\_1\\cdot \\nabla f\_1\}\{\\nabla f\_0\\cdot \\nabla f\_0\}\\cdot d\_0=\\begin\{bmatrix\} -8.99+\\frac\{81.82\}\{10100\}(-10)\\\\ 1+\\frac\{81.82\}\{10100\}(-100) \\end\{bmatrix\}=\\begin\{bmatrix\} -9.0710\\\\ 0.1899 \\end\{bmatrix\}][d_1 _ -_nabla f_1_frac_nabla f_1_cdot _nabla f_1_nabla f_0_cdot _nabla f_0_cdot d_0_begin_bmatrix_ -8.99_frac_81.82_10100_-10_ 1_frac_81.82_10100_-100_ _end_bmatrix_begin_bmatrix_ -9.0710_ 0.1899 _end_bmatrix]

![x\_2 = x\_1+\\alpha\_1 d\_1=\\begin\{bmatrix\} 4.495-9.071\\alpha\_1\\\\ -0.05+0.1899\\alpha\_1 \\end\{bmatrix\}][x_2 _ x_1_alpha_1 d_1_begin_bmatrix_ 4.495-9.071_alpha_1_ -0.05_0.1899_alpha_1 _end_bmatrix]

### 采用牛顿迭代法搜寻k=1时适当的步长因子： ###

![f(\\alpha)=(4.495-9.071\\alpha)^2+10(-0.05-0.1899\\alpha)^2.][f_alpha_4.495-9.071_alpha_2_10_-0.05-0.1899_alpha_2.]

![\\alpha\_\{m=0\}=1, m=0][alpha_m_0_1_ m_0]

![\\dot f(\\alpha)=2(4.495-9.071\\alpha)(-9.071)+20(-0.05+0.1899\\alpha)(0.1899)][dot f_alpha_2_4.495-9.071_alpha_-9.071_20_-0.05_0.1899_alpha_0.1899]

![=181.64\\alpha-81.82][181.64_alpha-81.82]

![\\ddot f(\\alpha)=165.2873][ddot f_alpha_165.2873]

![gif.latex][]![\\alpha\_1 = \\alpha\_0 - \\frac\{\\dot f(\\alpha\_0)\}\{\\ddot f(\\alpha\_0)\}=1-\\frac\{83.5491\}\{165.2873\}=0.4945.][alpha_1 _ _alpha_0 - _frac_dot f_alpha_0_ddot f_alpha_0_1-_frac_83.5491_165.2873_0.4945.]

![|\\alpha\_1 - \\alpha\_0|>10^\{-3\},~\\text\{not convergence\}.][alpha_1 - _alpha_0_10_-3_text_not convergence_.]

![m = m+1=1.][m _ m_1_1.]

![\\alpha\_2 = \\alpha\_1- \\frac\{dot f(\\alpha)\}\{\\ddot f(\\alpha\_1)\}=0.4945-\\frac\{0.0036\}\{165.2873\}\\approx 0.4945.][alpha_2 _ _alpha_1- _frac_dot f_alpha_ddot f_alpha_1_0.4945-_frac_0.0036_165.2873_approx 0.4945.]

![|\\alpha\_2 - \\alpha\_1|>10^\{-3\},~\\text\{convergence\}.][alpha_2 - _alpha_1_10_-3_text_convergence_.]

### 将![\\alpha\_2][alpha_2]带入原迭代方程，并采用共轭梯度法迭代： ###

![x\_2 = x\_1-\\alpha\_1 d\_1=\\begin\{bmatrix\} 4.495-9.071\\alpha\_1\\\\ -0.05+0.1899\\alpha\_1 \\end\{bmatrix\}][x_2 _ x_1-_alpha_1 d_1_begin_bmatrix_ 4.495-9.071_alpha_1_ -0.05_0.1899_alpha_1 _end_bmatrix]

![=\\begin\{bmatrix\} 4.495-9.071\\times 0.4945\\\\ -0.05+ 0.1899\\times 0.4945\\end\{bmatrix\}=\\begin\{bmatrix\} 0.0094\\\\ -0.0439 \\end\{bmatrix\}][begin_bmatrix_ 4.495-9.071_times 0.4945_ -0.05_ 0.1899_times 0.4945_end_bmatrix_begin_bmatrix_ 0.0094_ -0.0439 _end_bmatrix]

![||x\_2-x\_1||~\\text\{close to\} ~0][x_2-x_1_text_close to_ _0]

。。。。。。利用共轭梯度法的迭代方法，重复多次的迭代。

[text_min_ f _ x_1_2_10x_2_2.]: https://private.codecogs.com/gif.latex?%5Ctext%7Bmin%7D~%20f%20%3D%20x_1%5E2&plus;10x_2%5E2.
[x_0_5_ 5_text_T_k_0.]: https://private.codecogs.com/gif.latex?x_0%3D%5B5%2C%205%5D%5E%5Ctext%7BT%7D%2C~k%3D0.
[nabla f _ _binom_2x_1_20x_2]: https://private.codecogs.com/gif.latex?%5Cnabla%20f%20%3D%20%5Cbinom%7B2x_1%7D%7B20x_2%7D
[x_1 _ x_0-_alpha_0_nabla f_0_begin_bmatrix_ 5-10_alpha_0_ 5-100_alpha_0 _end_bmatrix]: https://private.codecogs.com/gif.latex?x_1%20%3D%20x_0-%5Calpha_0%5Cnabla%20f_0%3D%5Cbegin%7Bbmatrix%7D%205-10%5Calpha_0%5C%5C%205-100%5Calpha_0%20%5Cend%7Bbmatrix%7D
[f_alpha_5-10_alpha_2_10_5-100_alpha_2.]: https://private.codecogs.com/gif.latex?f%28%5Calpha%29%3D%285-10%5Calpha%29%5E2&plus;10%285-100%5Calpha%29%5E2.
[alpha_m_0_1_ m_0]: https://private.codecogs.com/gif.latex?%5Calpha_%7Bm%3D0%7D%3D1%2C%20m%3D0
[dot f_alpha_2_5-10_alpha_-10_20_5-100_alpha_-100]: https://private.codecogs.com/gif.latex?%5Cinline%20%5Cdot%20f%28%5Calpha%29%3D2%285-10%5Calpha%29%28-10%29&plus;20%285-100%5Calpha%29%28-100%29
[-100_200_alpha-10000_20000_alpha _ -10100_200200_alpha]: https://private.codecogs.com/gif.latex?%5Cinline%20%3D-100&plus;200%5Calpha-10000&plus;20000%5Calpha%20%3D%20-10100&plus;200200%5Calpha
[ddot f_alpha_200200]: https://private.codecogs.com/gif.latex?%5Cinline%20%5Cddot%20f%28%5Calpha%29%3D200200
[gif.latex]: https://private.codecogs.com/gif.latex?
[alpha_1 _ _alpha_0 - _frac_dot f_alpha_0_ddot f_alpha_0_1-_frac_190100_200200_0.0505.]: https://private.codecogs.com/gif.latex?%5Calpha_1%20%3D%20%5Calpha_0%20-%20%5Cfrac%7B%5Cdot%20f%28%5Calpha_0%29%7D%7B%5Cddot%20f%28%5Calpha_0%29%7D%3D1-%5Cfrac%7B190100%7D%7B200200%7D%3D0.0505.
[alpha_1 - _alpha_0_10_-3_text_not convergence_.]: https://private.codecogs.com/gif.latex?%7C%5Calpha_1%20-%20%5Calpha_0%7C%3E10%5E%7B-3%7D%2C~%5Ctext%7Bnot%20convergence%7D.
[m _ m_1_1.]: https://private.codecogs.com/gif.latex?m%20%3D%20m&plus;1%3D1.
[alpha_2 _ _alpha_1- _frac_dot f_alpha_ddot f_alpha_1_0.0505-_frac_10.1_200200_approx 0.0505.]: https://private.codecogs.com/gif.latex?%5Calpha_2%20%3D%20%5Calpha_1-%20%5Cfrac%7Bdot%20f%28%5Calpha%29%7D%7B%5Cddot%20f%28%5Calpha_1%29%7D%3D0.0505-%5Cfrac%7B10.1%7D%7B200200%7D%5Capprox%200.0505.
[alpha_2 - _alpha_1_10_-3_text_convergence_.]: https://private.codecogs.com/gif.latex?%7C%5Calpha_2%20-%20%5Calpha_1%7C%3E10%5E%7B-3%7D%2C~%5Ctext%7Bconvergence%7D.
[alpha_2]: https://private.codecogs.com/gif.latex?%5Cdpi%7B150%7D%20%5Calpha_2
[x_1 _ x_0-_alpha_0_nabla f_0_begin_bmatrix_ 5-10_alpha_0_ 5-100_alpha_0 _end_bmatrix_begin_bmatrix_ 5-10_times 0.0505_ 5-100_times 0.0505_end_bmatrix_begin_bmatrix_ 4.495_ -0.05 _end_bmatrix]: https://private.codecogs.com/gif.latex?x_1%20%3D%20x_0-%5Calpha_0%5Cnabla%20f_0%3D%5Cbegin%7Bbmatrix%7D%205-10%5Calpha_0%5C%5C%205-100%5Calpha_0%20%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D%205-10%5Ctimes%200.0505%5C%5C%205-100%5Ctimes%200.0505%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D%204.495%5C%5C%20-0.05%20%5Cend%7Bbmatrix%7D
[x_1-x_0_begin_bmatrix_ 4.495-5_ -0.05-5 _end_bmatrix_begin_bmatrix_ -0.505_ -5.05 _end_bmatrix_5.0752_10_-3_._text_not convergence]: https://private.codecogs.com/gif.latex?%7C%7Cx_1-x_0%7C%7C%3D%7C%7C%5Cbegin%7Bbmatrix%7D%204.495-5%5C%5C%20-0.05-5%20%5Cend%7Bbmatrix%7D%7C%7C%3D%7C%7C%5Cbegin%7Bbmatrix%7D%20-0.505%5C%5C%20-5.05%20%5Cend%7Bbmatrix%7D%7C%7C%3D5.0752%3E10%5E%7B-3%7D.~%28%5Ctext%7Bnot%20convergence%7D%29
[k _ k_1_1.]: https://private.codecogs.com/gif.latex?k%20%3D%20k&plus;1%3D1.
[x_2 _ x_1-_alpha_1_nabla f_1_begin_bmatrix_ 4.495-8.99_alpha_1_ -0.05_alpha_1 _end_bmatrix]: https://private.codecogs.com/gif.latex?x_2%20%3D%20x_1-%5Calpha_1%5Cnabla%20f_1%3D%5Cbegin%7Bbmatrix%7D%204.495-8.99%5Calpha_1%5C%5C%20-0.05&plus;%5Calpha_1%20%5Cend%7Bbmatrix%7D
[f_alpha_4.495-8.99_alpha_2_10_-0.05-_alpha_2.]: https://private.codecogs.com/gif.latex?f%28%5Calpha%29%3D%284.495-8.99%5Calpha%29%5E2&plus;10%28-0.05-%5Calpha%29%5E2.
[dot f_alpha_2_4.495-8.99_alpha_-8.99_20_-0.05_alpha_181.64_alpha-81.82]: https://private.codecogs.com/gif.latex?%5Cdot%20f%28%5Calpha%29%3D2%284.495-8.99%5Calpha%29%28-8.99%29&plus;20%28-0.05&plus;%5Calpha%29%3D181.64%5Calpha-81.82
[ddot f_alpha_181.64.]: https://private.codecogs.com/gif.latex?%5Cddot%20f%28%5Calpha%29%3D181.64.
[alpha_1 _ _alpha_0 - _frac_dot f_alpha_0_ddot f_alpha_0_1-_frac_99.82_181.64_0.4505.]: https://private.codecogs.com/gif.latex?%5Calpha_1%20%3D%20%5Calpha_0%20-%20%5Cfrac%7B%5Cdot%20f%28%5Calpha_0%29%7D%7B%5Cddot%20f%28%5Calpha_0%29%7D%3D1-%5Cfrac%7B99.82%7D%7B181.64%7D%3D0.4505.
[alpha_2 _ _alpha_1- _frac_dot f_alpha_ddot f_alpha_1_0.4505-_frac_0.0088_181.64_approx 0.4505.]: https://private.codecogs.com/gif.latex?%5Calpha_2%20%3D%20%5Calpha_1-%20%5Cfrac%7Bdot%20f%28%5Calpha%29%7D%7B%5Cddot%20f%28%5Calpha_1%29%7D%3D0.4505-%5Cfrac%7B0.0088%7D%7B181.64%7D%5Capprox%200.4505.
[begin_bmatrix_ 4.495-8.99_times 0.4505_ -0.05_ 0.4505_end_bmatrix_begin_bmatrix_ 4.445_ -0.4005 _end_bmatrix]: https://private.codecogs.com/gif.latex?%3D%5Cbegin%7Bbmatrix%7D%204.495-8.99%5Ctimes%200.4505%5C%5C%20-0.05&plus;%200.4505%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D%204.445%5C%5C%20-0.4005%20%5Cend%7Bbmatrix%7D
[x_2 _ x_1 _ _alpha_1 d_1]: https://private.codecogs.com/gif.latex?x_2%20%3D%20x_1%20&plus;%20%5Calpha_1%20d_1
[nabla f_1_cdot _nabla f_1 _ 81.82]: https://private.codecogs.com/gif.latex?%5Cnabla%20f_1%5Ccdot%20%5Cnabla%20f_1%20%3D%2081.82
[nabla f_0_cdot _nabla f_0 _ 10100]: https://private.codecogs.com/gif.latex?%5Cnabla%20f_0%5Ccdot%20%5Cnabla%20f_0%20%3D%2010100
[d_1 _ -_nabla f_1_frac_nabla f_1_cdot _nabla f_1_nabla f_0_cdot _nabla f_0_cdot d_0_begin_bmatrix_ -8.99_frac_81.82_10100_-10_ 1_frac_81.82_10100_-100_ _end_bmatrix_begin_bmatrix_ -9.0710_ 0.1899 _end_bmatrix]: https://private.codecogs.com/gif.latex?d_1%20%3D%20-%5Cnabla%20f_1&plus;%5Cfrac%7B%5Cnabla%20f_1%5Ccdot%20%5Cnabla%20f_1%7D%7B%5Cnabla%20f_0%5Ccdot%20%5Cnabla%20f_0%7D%5Ccdot%20d_0%3D%5Cbegin%7Bbmatrix%7D%20-8.99&plus;%5Cfrac%7B81.82%7D%7B10100%7D%28-10%29%5C%5C%201&plus;%5Cfrac%7B81.82%7D%7B10100%7D%28-100%29%20%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D%20-9.0710%5C%5C%200.1899%20%5Cend%7Bbmatrix%7D
[x_2 _ x_1_alpha_1 d_1_begin_bmatrix_ 4.495-9.071_alpha_1_ -0.05_0.1899_alpha_1 _end_bmatrix]: https://private.codecogs.com/gif.latex?x_2%20%3D%20x_1&plus;%5Calpha_1%20d_1%3D%5Cbegin%7Bbmatrix%7D%204.495-9.071%5Calpha_1%5C%5C%20-0.05&plus;0.1899%5Calpha_1%20%5Cend%7Bbmatrix%7D
[f_alpha_4.495-9.071_alpha_2_10_-0.05-0.1899_alpha_2.]: https://private.codecogs.com/gif.latex?f%28%5Calpha%29%3D%284.495-9.071%5Calpha%29%5E2&plus;10%28-0.05-0.1899%5Calpha%29%5E2.
[dot f_alpha_2_4.495-9.071_alpha_-9.071_20_-0.05_0.1899_alpha_0.1899]: https://private.codecogs.com/gif.latex?%5Cdot%20f%28%5Calpha%29%3D2%284.495-9.071%5Calpha%29%28-9.071%29&plus;20%28-0.05&plus;0.1899%5Calpha%29%280.1899%29
[181.64_alpha-81.82]: https://private.codecogs.com/gif.latex?%3D181.64%5Calpha-81.82
[ddot f_alpha_165.2873]: https://private.codecogs.com/gif.latex?%5Cddot%20f%28%5Calpha%29%3D165.2873
[alpha_1 _ _alpha_0 - _frac_dot f_alpha_0_ddot f_alpha_0_1-_frac_83.5491_165.2873_0.4945.]: https://private.codecogs.com/gif.latex?%5Calpha_1%20%3D%20%5Calpha_0%20-%20%5Cfrac%7B%5Cdot%20f%28%5Calpha_0%29%7D%7B%5Cddot%20f%28%5Calpha_0%29%7D%3D1-%5Cfrac%7B83.5491%7D%7B165.2873%7D%3D0.4945.
[alpha_2 _ _alpha_1- _frac_dot f_alpha_ddot f_alpha_1_0.4945-_frac_0.0036_165.2873_approx 0.4945.]: https://private.codecogs.com/gif.latex?%5Calpha_2%20%3D%20%5Calpha_1-%20%5Cfrac%7Bdot%20f%28%5Calpha%29%7D%7B%5Cddot%20f%28%5Calpha_1%29%7D%3D0.4945-%5Cfrac%7B0.0036%7D%7B165.2873%7D%5Capprox%200.4945.
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