NotImplementedError: Unable to find critical points for Abs(f(x))

[galenseilis]

It would be particularly useful to some of my projects if it were possible for me to find the maximum of the absolute value of a single variable function. Here is a reproducible example:

from sympy import *
x = Symbol('x')
maximum(Abs(x**3), x, Interval(0,3))

I've noted that max(|f|) = max(|min(f)|, |max(f)|) is a useful simplification that can be implemented explicitly.

from sympy import *
x = Symbol('x')
Max(Abs(minimum(x**3, x, Interval(0,3))), Abs(maximum(x**3, x, Interval(0,3))))

As a feature request, it would be beneficial if SymPy did this (or something better) automatically.

[oscargus]

If you define x to be real as x = Symbol('x', real=True) it returns the correct result.

Here, one could think that the code would replace your x with another x that has real assumptions (as is done in e.g. the integral code).

However, your observation should be relevant in addition to that.

[oscargus]

An example where your observation is useful is maximum(Abs(x**3*y), x, Interval(0,3)) with x and y both real.

Although

Max(Abs(minimum(x**3*y, x, Interval(0,3))), Abs(maximum(x**3*y, x, Interval(0,3)))))

gives Max(27*y, -Min(0, 27*y)) it is clearly better than NotImplementedError: relationship did not evaluate: y >= 0...

The function function_range computes both the maximum and minimum values, so something similar should be applied for the minimum value by checking the sign of the min and max without abs and figure our what the abs min is.

[oscarbenjamin]

If you define x to be real as x = Symbol('x', real=True) it returns the correct result.

Here the function is differentiated:

sympy/sympy/calculus/util.py

Line 193 in a63cb72

solution = solveset(f.diff(symbol), symbol, interval)

Depending on whether x has real assumptions different results are obtained:

In [2]: x = Symbol('x')

In [3]: Abs(x**3).diff(x)
Out[3]: 
⎛⎛    2              3   ⎞ ⎛    2    d                         d               2    d        ⎞   ⎛  3                2   ⎞ ⎛    2    d                      
⎜⎝3⋅re (x)⋅im(x) - im (x)⎠⋅⎜3⋅re (x)⋅──(im(x)) + 6⋅re(x)⋅im(x)⋅──(re(x)) - 3⋅im (x)⋅──(im(x))⎟ + ⎝re (x) - 3⋅re(x)⋅im (x)⎠⋅⎜3⋅re (x)⋅──(re(x)) - 6⋅re(x)⋅im(
⎝                          ⎝         dx                        dx                   dx       ⎠                             ⎝         dx                     
────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
                                                                                                    3                                                       
                                                                                                   x                                                        

   d               2    d        ⎞⎞     ⎛ 3⎞
x)⋅──(im(x)) - 3⋅im (x)⋅──(re(x))⎟⎟⋅sign⎝x ⎠
   dx                   dx       ⎠⎠         
────────────────────────────────────────────
                                            
                                            

In [4]: x = Symbol('x', real=True)

In [5]: Abs(x**3).diff(x)
Out[5]: 
 2                  
x ⋅sign(x) + 2⋅x⋅│x│

Without the real assumption we get:

In [6]: x = Symbol('x')

In [7]: solveset(Abs(x**3).diff(x), x, Reals)
Out[7]: 
      ⎧  │               ⎛ 2 d        ⎞⎫   ⎧  │              ⎛ 2 d        ⎞⎫
{0} ∪ ⎨x │ x ∊ (-∞, 0) ∧ ⎜x ⋅──(x) = 0⎟⎬ ∪ ⎨x │ x ∊ (0, ∞) ∧ ⎜x ⋅──(x) = 0⎟⎬
      ⎩  │               ⎝   dx       ⎠⎭   ⎩  │              ⎝   dx       ⎠⎭

In [8]: solveset(Abs(x**3).diff(x), x, Interval(0, 3))
Out[8]: 
      ⎧  │              ⎛ 2 d        ⎞⎫
{0} ∪ ⎨x │ x ∊ (0, 3] ∧ ⎜x ⋅──(x) = 0⎟⎬
      ⎩  │              ⎝   dx       ⎠⎭

In [9]: _.doit()
Out[9]: 
      ⎧  │              ⎛ 2    ⎞⎫
{0} ∪ ⎨x │ x ∊ (0, 3] ∧ ⎝x  = 0⎠⎬
      ⎩  │                      ⎭

I think it should be possible to resolve that ConditionSet further.

[pranjii]

Can I get this?

[oscargus]

@pranjii there is a solution in #21985 so probably you better spend your efforts on some other issue.