Symbolic min/max

[pelegm]

The Python builtins max and min are not user-friendly because they give totally wrong results on symbolic input:

sage: max(1/(1-x), 1/x)
-1/(x - 1)

There exist functions max_symbolic and min_symbolic but people generally don't know those functions. They expect max and min to "just work".

One way this puzzles users is when plotting eg the following:

sage: plot(max(1/(1-x), 1/x), (x, 0, 1), ymin=0, ymax=10))

... for which a workaround is to plot using a lambda function:

sage: plot(lambda x: max(1/(1-x), 1/x), (0, 1), ymin=0, ymax=10)

Depends on #24838

CC: @mathzeta @slel

Component: symbolics

Keywords: min, max, days79

Issue created by migration from https://trac.sagemath.org/ticket/21945

[mathzeta]

comment:2

The functions max_symbolic and min_symbolic exist. They can not make strong deductions, but at least they do know how to convert to sympy's Max and Min.

[pelegm]

comment:3

Replying to @mathzeta:

The functions max_symbolic and min_symbolic exist. They can not make strong deductions, but at least they do know how to convert to sympy's Max and Min.

Thanks for the info. The names are not intuitive, but perhaps we can live with it. Here's a short list of what these creatures should be able to do:

• Assume x and y are positive. Then min_symbolic(x, y) and max_symbolic(x, y) should return True.
• Similarly, for x and y which are both negative, nonnegative and nonpositive.
• Similarly, if x is assumed to be negative and y is assumed to be positive, then min and max should behave respectively.
• min and max of real symbols should be real. Similarly for algebraic, rational, integer...
• Should min_symbolic(x, x) simplify to x?

[mathzeta]

comment:4

Replying to @pelegm:

• Should min_symbolic(x, x) simplify to x?

It currently does. Regarding your other points, unfortunately, it seems those are not available. See also #17700 and #19162.
A "non-trivial" thing that does work is that

var("x,y,z")
assume(x>z)
assume(z>y)
print max_symbolic(x,y,z).simplify()
print min_symbolic(x,y,z).simplify()

prints x and y.

[jdemeyer]

Description changed:

--- 
+++ 
@@ -1,16 +1,8 @@
-It would be good to have a symbolic min/max operators, that keep (finite, say) list of arguments, each may be an expression.
+The Python builtins `max` and `min` are not user-friendly because they give totally wrong results on symbolic input:
 
-This is how it works in SymPy:
+```
+sage: max(1/(1-x), 1/x)
+-1/(x - 1)
+```
 
-```python
->>> x = Symbol('x', positive=True)
->>> y = Symbol('y', positive=True)
->>> z = Symbol('z', negative=True)
->>> Min(x, y).is_positive
-True
->>> Max(x, z).is_positive
-True
->>> Min(x, z).is_nonnegative
-False
-```
-and so on.  Note that they use `Min` and `Max` instead of the built-in `min` and `max`.
+There exist functions `max_symbolic` and `min_symbolic` but people generally don't know those functions. They expect `max` and `min` to "just work".

[slel]

Description changed:

--- 
+++ 
@@ -6,3 +6,15 @@
 ```
 
 There exist functions `max_symbolic` and `min_symbolic` but people generally don't know those functions. They expect `max` and `min` to "just work".
+
+Note that one way this puzzles users is when plotting eg the following:
+
+```
+sage: plot(max(1/(1-x), 1/x), (x, 0,1))
+```
+
+For this, a workaround is to plot using a lambda function:
+
+```
+sage: plot(lambda x: max(1/(1-x), 1/x), (0,1))
+```

[slel]

Description changed:

--- 
+++ 
@@ -7,14 +7,14 @@
 
 There exist functions `max_symbolic` and `min_symbolic` but people generally don't know those functions. They expect `max` and `min` to "just work".
 
-Note that one way this puzzles users is when plotting eg the following:
+One way this puzzles users is when plotting eg the following:
 
 ```
-sage: plot(max(1/(1-x), 1/x), (x, 0,1))
+sage: plot(max(1/(1-x), 1/x), (x, 0, 1), ymin=0, ymax=10))
 ```
 
-For this, a workaround is to plot using a lambda function:
+... for which a workaround is to plot using a lambda function:
 
 ```
-sage: plot(lambda x: max(1/(1-x), 1/x), (0,1))
+sage: plot(lambda x: max(1/(1-x), 1/x), (0, 1), ymin=0, ymax=10)
 ```

[rwst]

comment:8

Replying to @pelegm:

• Assume x and y are positive. Then min_symbolic(x, y) and max_symbolic(x, y) should return True.
• Similarly, for x and y which are both negative, nonnegative and nonpositive.
• Similarly, if x is assumed to be negative and y is assumed to be positive, then min and max should behave respectively.
• min and max of real symbols should be real. Similarly for algebraic, rational, integer...

Actually this is quite achievable. Pynac already has such logic for several functions. However, it would help if min/max_symbolic would be made GinacFunctions (like most elementary functions). Note also #18630, #24368, #24352, #22162 and pynac/pynac#293