Mathematica：FindRoot错误

Question

FindRoot[
 27215. - 7.27596*10^-12 x + 52300. x^2 - 9977.4 Log[1. - 1. x] == 0
 , 
 {x, 0.000001}
]

收敛到解决方案{x -> -0.0918521}但是如何在解决方案之前让Mathematica避免以下错误消息：

FindRoot::nlnum: The function value {Indeterminate} is not a list of numbers with dimensions {1} at {x} = {1.}. >>

我正在使用FindRoot来解决一些非常混乱的表达式。我有时也收到以下错误，虽然Mathematica仍然会得到答案，但我想知道是否有办法避免它：

FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances. >> 

Answer 1

您获得的解决方案并非实际解决方案。该消息表明出现了问题，FindRoot返回x的最后一个值。这是FindRoot的“更多信息”下的最后一项：

• 如果FindRoot未成功找到您在MaxIterations步骤中指定的准确度的解决方案，则会返回找到的解决方案的最新近似值。然后，您可以再次应用FindRoot，并将此近似值作为起点。

例如，在这种情况下，也没有解决方案：

FindRoot[x^2 + 1 == 0, {x, 1}]

您将收到FindRoot::jsing警告， Mathematica 返回{x -> 0.}（这是最近的近似值）。

类似的情况，但有一个Log函数：

FindRoot[1 + Log[1 + x]^2 == 0, {x, 2}]

提供与您所看到的相似的FindRoot::nlnum并返回{x -> 0.000269448}（在这种情况下，这是最近的近似值）。

这是一个相同功能的图，用于说明目的：

如果您想要包含复杂的根，请考虑FindRoot的文档的这一部分（在“更多信息”下）：

• 您始终可以通过向起始值添加0.I来告诉FindRoot搜索复杂的根。

因此，例如，您可以在一个复杂的根附近取一个起始值，如下所示：

FindRoot[x^2 + 1 == 0, {x, 1 + 1. I}]

哪些会聚（没有消息）到{x -> 8.46358*10^-23 + 1. I}（所以基本上I）。

或者在其他复杂根附近有一个起始值：

FindRoot[x^2 + 1 == 0, {x, 1 - 1. I}]

你基本上会得到-I（确切地说，你得到{x -> 8.46358*10^-23 - 1. I}）。

Answer 2

这个等式没有真正的解决方案。 Mathematica最终得到了函数最小值附近，并报告了这一点，因为这是算法收敛的地方。

Plot[27215. - 7.27596*10^-12 x + 52300. x^2 - 9977.4 Log[1. - 1. x],
 {x, -2, 0.09}, AxesOrigin -> {0, 0}]

Mathematica警告你：

In[30]:= x /. 
 Table[FindRoot[
   27215. - 7.27596*10^-12 x + 52300. x^2 - 9977.4 Log[1. - 1. x] == 
    0, {x, y}], {y, -0.01, 0.01, 0.0002}]



During evaluation of In[30]:= FindRoot::nlnum: The function value {Indeterminate} is not a list of numbers with dimensions {1} at {x} = {1.}. >>

During evaluation of In[30]:= FindRoot::nlnum: The function value {Indeterminate} is not a list of numbers with dimensions {1} at {x} = {1.}. >>

During evaluation of In[30]:= FindRoot::nlnum: The function value {Indeterminate} is not a list of numbers with dimensions {1} at {x} = {1.}. >>

During evaluation of In[30]:= General::stop: Further output of FindRoot::nlnum will be suppressed during this calculation. >>

During evaluation of In[30]:= FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances. >>

During evaluation of In[30]:= FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances. >>

During evaluation of In[30]:= FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances. >>

During evaluation of In[30]:= General::stop: Further output of FindRoot::lstol will be suppressed during this calculation. >>

Out[30]= {-0.0883278, -0.0913649, -0.0901617, -0.0877546, -0.0877383, \
-0.088508, -0.0937041, -0.0881606, -0.0912122, -0.0899562, \
-0.0876965, -0.0879619, -0.0877441, -0.101551, -0.0915088, \
-0.0880611, -0.0959972, -0.0930364, -0.0902243, -0.0877198, \
-0.0881157, -0.107205, -0.103746, -0.100439, -0.0972646, -0.094208, \
-0.0912554, -0.0878633, -0.089473, -0.0884659, -0.0876997, \
-0.0876936, -0.0879112, -0.104396, -0.100987, -0.0976638, -0.0879892, \
-0.087777, -0.0881334, -0.0880071, -0.0880255, -0.0880285, \
-0.0880345, -0.0911966, -0.0879797, -0.0890295, -0.087701, \
-0.0952537, -0.0941312, -0.0929994, -0.0918578, -0.0885677, \
-0.0895444, -0.0883719, -0.103914, -0.102701, -0.0885007, -0.0915083, \
-0.098988, -0.0963068, -0.0891533, -0.0907357, -0.0881215, \
-0.0893928, -0.108191, -0.104756, -0.101456, -0.0982737, -0.0951949, \
-0.0922072, -0.0892996, -0.0878794, -0.0877164, -0.0896659, \
-0.0886859, -0.0876952, -0.0909219, -0.0899049, -0.0888758, \
-0.0878343, -0.0952044, -0.0941281, -0.0887345, -0.0919322, \
-0.0886726, -0.0876955, -0.0877232, -0.0878879, -0.0877578, \
-0.101642, -0.0916633, -0.0991254, -0.0877255, -0.0936139, \
-0.0907846, -0.0877205, -0.0877454, -0.0881589, -0.0893507, \
-0.0878747, -0.0876961}