我一直在尝试将方程y = Ax ^ 2 + Bx + C拟合到以下数据集（data.txt）：
-9.39398e+09 1.52819e-19
-9.07008e+09 1.50337e-19
-8.74617e+09 1.44628e-19
-8.42227e+09 1.37837e-19
-8.09817e+09 1.31042e-19
-7.77427e+09 1.24624e-19
-7.45037e+09 1.18873e-19
-7.12646e+09 1.11213e-19
-6.80256e+09 1.00253e-19
-6.47865e+09 8.95713e-20
-6.15475e+09 7.92741e-20
-5.83066e+09 6.94736e-20
-5.50675e+09 6.02803e-20
-5.18285e+09 5.1667e-20
-4.85894e+09 4.37442e-20
-4.53504e+09 3.65056e-20
-4.21113e+09 3.00328e-20
-3.88723e+09 2.4281e-20
-3.56332e+09 1.93223e-20
-3.23923e+09 1.49467e-20
-2.91533e+09 1.13226e-20
-2.59142e+09 8.33933e-21
-2.26752e+09 5.93767e-21
-1.94361e+09 4.05992e-21
-1.61971e+09 2.64039e-21
-1.29581e+09 1.623e-21
-9.71713e+08 9.0523e-22
-6.47809e+08 4.05351e-22
-3.23904e+08 8.97219e-23
0 0
3.23904e+08 8.97219e-23
6.47809e+08 4.05351e-22
9.71713e+08 9.0523e-22
1.29581e+09 1.623e-21
1.61971e+09 2.64039e-21
1.94361e+09 4.05992e-21
2.26752e+09 5.93767e-21
2.59142e+09 8.33933e-21
2.91533e+09 1.13226e-20
3.23923e+09 1.49467e-20
3.56332e+09 1.93223e-20
3.88723e+09 2.4281e-20
4.21113e+09 3.00328e-20
4.53504e+09 3.65056e-20
4.85894e+09 4.37442e-20
5.18285e+09 5.1667e-20
5.50675e+09 6.02803e-20
5.83066e+09 6.94736e-20
6.15475e+09 7.92741e-20
6.47865e+09 8.95713e-20
6.80256e+09 1.00253e-19
7.12646e+09 1.11213e-19
7.45037e+09 1.18873e-19
7.77427e+09 1.24624e-19
8.09817e+09 1.31042e-19
8.42227e+09 1.37837e-19
8.74617e+09 1.44628e-19
9.07008e+09 1.50337e-19
9.39398e+09 1.52819e-19
在gnuplot中，我键入了命令：
fit a*x**2 + b*x + c 'data.txt' via a, b, c
吐出的结果是：
Final set of parameters Asymptotic Standard Error
======================= ==========================
a = -1.73185e-20 +/- 2.658e-11 (1.535e+11%)
b = 1 +/- 0.1325 (13.25%)
c = 1 +/- 1.076e+09 (1.076e+11%)
correlation matrix of the fit parameters:
a b c
a 1.000
b -0.000 1.000
c -0.739 0.000 1.000
当我使用a，b，c的值进行绘图时，数据与拟合线之间没有拟合。
所以我再次给出命令
fit a*x**2 + b*x + c 'data.txt' via a, b, c
然后，gnuplot吐出以下输出：
Final set of parameters Asymptotic Standard Error
======================= ==========================
a = -1.73185e-20 +/- 3.278e-21 (18.93%)
b = -2.24969e-22 +/- 1.645e-11 (7.313e+12%)
c = 1 +/- 0.1327 (13.27%)
correlation matrix of the fit parameters:
a b c
a 1.000
b 0.000 1.000
c -0.739 -0.001 1.000
a，b，c的值已更改，但这也不能令人满意地拟合数据。所以我再次给出命令：
fit a*x**2 + b*x + c 'data.txt' via a, b, c
这次吐出了以下结果：
Final set of parameters Asymptotic Standard Error
======================= ==========================
a = 1.96019e-39 +/- 5.98e-33 (3.051e+08%)
b = -2.24969e-22 +/- 2.98e-23 (13.25%)
c = -1.11942e-21 +/- 2.421e-13 (2.162e+10%)
correlation matrix of the fit parameters:
a b c
a 1.000
b 0.000 1.000
c -0.739 -0.000 1.000
这也不能令人满意。
然后再次重复执行fit命令并获得以下信息：
After 4 iterations the fit converged.
final sum of squares of residuals : 2.4063e-39
abs. change during last iteration : -2.64182e-48
Hmmmm.... Sum of squared residuals is zero. Can't compute errors.
Final set of parameters
=======================
a = 1.96019e-39
b = 2.01689e-41
c = -1.11942e-21
现在，a，b，c的值非常适合数据。
我的问题是：
为什么a，b，c的值不能很好地适合第一次，第二次和第三次尝试的数据？
我可以使用最后计算出的a，b，c值吗？
在上一次尝试的输出中，我应该担心以下消息：“ Hmmmm...。残差平方和为零。无法计算错误。” ？
我认为您的值的数值范围太宽，我猜想由于截断错误，某些计算（例如残差）是错误的。
如果您将第一列乘以10 ^ -9，第二列乘以10 ^ 18来规范化数据
-9.39398 0.152819
-9.07008 0.150337
-8.74617 0.144628
-8.42227 0.137837
-8.09817 0.131042
-7.77427 0.124624
-7.45037 0.118873
-7.12646 0.111213
-6.80256 0.100253
-6.47865 0.0895713
-6.15475 0.0792741
-5.83066 0.0694736
-5.50675 0.0602803
-5.18285 0.051667
-4.85894 0.0437442
-4.53504 0.0365056
-4.21113 0.0300328
-3.88723 0.024281
-3.56332 0.0193223
-3.23923 0.0149467
-2.91533 0.0113226
-2.59142 0.00833933
-2.26752 0.00593767
-1.94361 0.00405992
-1.61971 0.00264039
-1.295810 0.001623
-0.971713 0.00090523
-0.647809 0.000405351
-0.323904 0.0000897219
0 0
0.323904 0.0000897219
0.647809 0.000405351
0.971713 0.00090523
1.295810 0.001623
1.61971 0.00264039
1.94361 0.00405992
2.26752 0.00593767
2.59142 0.00833933
2.91533 0.0113226
3.23923 0.0149467
3.56332 0.0193223
3.88723 0.024281
4.21113 0.0300328
4.53504 0.0365056
4.85894 0.0437442
5.18285 0.051667
5.50675 0.0602803
5.83066 0.0694736
6.15475 0.0792741
6.47865 0.0895713
6.80256 0.100253
7.12646 0.111213
7.45037 0.118873
7.77427 0.124624
8.09817 0.131042
8.42227 0.137837
8.74617 0.144628
9.07008 0.150337
9.39398 0.152819
然后
gnuplot> fit a*x**2 + b*x + c 'data.txt' via a, b, c
iter chisq delta/lim lambda a b c
0 2.4049966612e-03 0.00e+00 4.62e-02 1.960970e-03 6.941376e-15 -1.162140e-03
1 2.4049966612e-03 -1.79e-09 4.62e-03 1.960970e-03 6.941376e-15 -1.162140e-03
iter chisq delta/lim lambda a b c
After 1 iterations the fit converged.
final sum of squares of residuals : 0.002405
rel. change during last iteration : -1.78522e-14
degrees of freedom (FIT_NDF) : 56
rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 0.00655335
variance of residuals (reduced chisquare) = WSSR/ndf : 4.29464e-05
Final set of parameters Asymptotic Standard Error
======================= ==========================
a = 0.00196097 +/- 3.136e-05 (1.599%)
b = 6.94138e-15 +/- 0.0001416 (2.04e+12%)
c = -0.00116214 +/- 0.00128 (110.1%)
correlation matrix of the fit parameters:
a b c
a 1.000
b 0.001 1.000
c -0.745 -0.001 1.000
稳定
如果A，B，C是归一化数据中的拟合参数，则可以通过以下方法获取初始的a，b，c参数（与原始数据相对应的参数）：
a = A / (10^18 * 10^18) = A * 10^-36
b = B / (10^9 * 10^18) = B * 10^-27
c = C / 10^18 = C * 10^-18
说明/更新2：
令（xi，yi），i = 1,2，.. n为您的初始数据，您要拟合：
y ≈ a x^2 + b x + c
但是，对于您的计算机，最好使用：
Y ≈ A X^2 + B X + C
其中X = 10 ^ -9 x，Y = 10 ^ 18 y
要从A，B，c中找到a，b，c，只需进行识别即可：
Y ≈ A X^2 + B X + C => 10^18 y ≈ A 10^-18 x^2 + B 10^-9 x + C
=> y ≈ A 10^-36 x^2 + B 10^-27 x + C 10^-18
to be compared to y ≈ a x^2 + b x + c
因此，您有：
a = A * 10^-36
b = B * 10^-27
c = C * 10^-18
现在可以吗？
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