A peculiar behavior of NSolve[]

Question

NSolve[Power[x + 1, x + 1] - Power[x, x + 2] == 0, x, Reals]

returns:

{{x -> 3.14104}}

Great.

However,

NSolve[Power[x + 1.5, x + 1.5] - Power[x, x + 3] == 0, x, Reals]

runs forever. :(

Now,

NSolve[Power[x + 2, x + 2] - Power[x, x + 4] == 0, x, Reals]

returns

{{x -> 3.47175}}

Again great.

I expected "2.5" case would run forever, just like "1.5" case. But:

NSolve[Power[x + 2.5, x + 2.5] - Power[x, x + 5] == 0, x, Reals]

Solve::ratnz: Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result. >>

and outputs the answer:

{{x -> 3.61691}}

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Since the solution is in real numbers, from numerical math point of view, all cases above are equal. Why such behavior of NSolve[] then?

Answer 1

In the NSolve rather than just limiting x to Reals, limit it to a finite interval of interest, say 2 < x < 6

f[x_, a_] = Power[x + a, x + a] - Power[x, x + 2 a];

data = Table[
   {a, x /. NSolve[{f[x, a] == 0, 2 < x < 6}, x][[1]]},
   {a, 1/10, 10, 1/10}];

ListLinePlot[data, AxesLabel -> {"a", "root"}]

Answer 2

tl;dr Don't expect NSolve to give sensible results for equations with no analytic solution.

If you are interested in a "numerical math point of view", use FindRoot or a related function to solve numerically. NSolve tries to find a transformation to an equation it knows exact solutions for, then converts those solutions to machine-precision numbers when returning them.

For example, take the equation x - Sin[x] == 1:

NSolve[x - Sin[x] == 1, x]
(* NSolve::nsmet: This system cannot be solved with the methods available to NSolve. >> *)

FindRoot[x - Sin[x] == 1, {x, 1}]
(* {x -> 1.93456} *)

Your equations look different to NSolve. Note that it first converts approximate to exact numbers (hence the message "solving a corresponding exact system"), so that your troublesome equation is being treated as:

SetPrecision[Power[x + 1.5, x + 1.5] - Power[x, x + 3] == 0, Infinity]

$$ \left(x+\frac{3}{2}\right)^{x+\frac{3}{2}}=x^{x+3} $$

I don't know why NSolve/Solve get hung up on this particular equation, it probably makes a bad guess early on and misses the transformation it used on the other ones.

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Just to demonstrate that Mathematica is handling your equations differently, we can look at the output of Solve instead of NSolve:

Solve[Power[x + 1, x + 1] - Power[x, x + 2] == 0, x, Reals]

$$ \text{Root}\left[\left\{(\text{$\#$1}+1) \left(\frac{\text{$\#$1}+1}{\text{$\#$1}}\right)^{\text{$\#$1}}-\text{$\#$1}^2\&,3.141\ldots\right\}\right] $$

Solve[SetPrecision[Power[x + 2.5, x + 2.5] - Power[x, x + 5] == 0, 
  Infinity], x, Reals]

$$ \text{Root}\left[\left\{\frac{25 \sqrt{2 \text{$\#$1}+5} e^{\text{$\#$1} \log \left(\text{$\#$1}+\frac{5}{2}\right)-\text{$\#$1} \log (\text{$\#$1})}}{\text{$\#$1}^5}+\frac{20 \sqrt{2 \text{$\#$1}+5} e^{\text{$\#$1} \log \left(\text{$\#$1}+\frac{5}{2}\right)-\text{$\#$1} \log (\text{$\#$1})}}{\text{$\#$1}^4}+\frac{4 \sqrt{2 \text{$\#$1}+5} e^{\text{$\#$1} \log \left(\text{$\#$1}+\frac{5}{2}\right)-\text{$\#$1} \log (\text{$\#$1})}}{\text{$\#$1}^3}-4 \sqrt{2}\&,3.616\ldots\right\}\right] $$

Very different results...