Numerical integration and plot failing

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Here’s a workaround for plotting:

list_plot( [[t, F(t)] for t in (1,1.1..2)], plotjoined=True )

with a live example. For integration you could likewise evaluate the function at discrete points, add all interior evaluations plus half of the values at the endpoints and multiply by the step size (i.e., a discrete integral).

A different approach that works even in a Python 2 based Sage:

sage: var("y,z");
sage: cauchy(z) = y.subs(solve(z*y^3 + y^2 - 2*z*y + 2, y)[0])
sage: F(z) = arctan2(imag(cauchy(z)),real(cauchy(z)))
sage: numerical_integral(F, 1, 2)
(0.6116545934235456, 6.790730127365591e-15)
sage: plot(F(z),(z,1,2))

The plot is really slow. It may take more than 20 seconds. See this SageMath cell.

NOTE : This has replaced, (not complemented) my original answer concluding that this was a (Python) bug. Apologies for the confusion...

As pointed out by Serge Lelièvre, this seems a limitation of Python 2 floats, which does not exist in Python 3, where one can do :

sage: def F(u): return arg((u-3)^(1/3))
sage: F(1)
arg((-2)^(1/3))
sage: F(1).n()
1.04719755119660
sage: F(2).n()
1.04719755119660
sage: plot(F,(1,2))
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: integrate(F(x),x,1,2)
1/3*pi

See this sage-devel thread for further illumination...

In the original example:

sage: cauchy(z)=y.substitute(solve(z*y^3 +y^2 - 2*z*y+2, y)[0])
sage: def G(u): return arg(cauchy(u))
sage: numerical_integral(lambda u:G(u), 1, 2)
(0.6116545934235456, 6.790730127365591e-15)

This works for me in (Python 3 based) Sage 8.8.beta7:

sage: y, z = var('y z')
sage: cauchy = y.substitute(solve(z*y^3 + y^2 - 2*z*y + 2, y)[0])
sage: def F(u):
....:     return arg(cauchy).substitute(z=u).n()
....:
sage: numerical_integral(F, 1, 2)
(0.6116545934235456, 6.790730127365591e-15)

Here are links to Jupyter notebooks on CoCalc showing

• failure to integrate F with Py2-based SageMath 8.7
• success integrating F with Py3-based SageMath 8.8.beta7

This also fails with Py2-based SageMath 8.8.beta7.

So the "easy" solution for you is: switch to Python3-based Sage!

For this, follow the instructions to install from source, paying attention to the discussion about Python 3.

Or if you want to turn your existing SageMath into a Python3-based one, you could open a terminal, cd to your Sage directory, and type

make distclean
make configure
./configure --with-python=3
make

which will work if you have all the prerequisites.