Implement a symbolic version of the arg function

[TimothyClemans]

It would be nice if there were a symbolic arg function, just like the symbolic sin, cos, etc., functions. Then the following would happen:

sage: f = arg(x); f
arg(x)
sage: f.subs(x=1+I)
arg(1+I)

Now we have

sage: arg(1+I)
0.785398163397
sage: type(arg(1+I))
<type 'sage.rings.real_double.RealDoubleElement'>

I.e., the arg in Sage is currently the numerical person's arg, not the symbolic person's. It just casts to CDF.

"The function should return the argument of a complex function." - Ronan Paixão

Apply

• attachment:trac_4498-symbolic_arg.cleanup.patch
• attachment:trac_4498-arg_evalf.patch

[TimothyClemans]

arg(x)
///

Traceback (most recent call last):
 File "<stdin>", line 1, in <module>
 File "/home/ronan/.sage/sage_notebook/worksheets/admin/0/code/127.py",
line 6, in <module>
   exec compile(ur'arg(x)' + '\n', '', 'single')
 File
"/home/ronan/progs/sage/local/lib/python2.5/site-packages/ZODB3-3.7.0-py2.5-linux-i686.egg/", line 1, in <module>

 File
"/home/ronan/progs/sage/local/lib/python2.5/site-packages/sage/misc/functional.py", line 67, in arg
   except AttributeError: return CDF(x).arg()
 File "complex_double.pyx", line 286, in
sage.rings.complex_double.ComplexDoubleField_class.__call__
(sage/rings/complex_double.c:3324)
 File
"/home/ronan/progs/sage/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 1454, in _complex_double_
   raise TypeError
TypeError

[mabshoff]

Please post a complete session. As is the above is not very clear.

Cheers,

Michael

[ronanpaixao]

Is this really an enchancement? As it is, just using arg(x) already raises an error and everything that needs arg() must be implemented numerically.

[was]

Is this really an enchancement? As it is, just using arg(x) already raises an error and everything that needs arg() must be implemented numerically.

Yes, this is an enhancement since it is implementing new functionality. It would be a bug fix if there were a bug in the existing arg function, where it produced invalid results on supported input.

-- William

[burcin]

This was also reported in #6220. I will close that as duplicate.

[ktkohl]

symbolic arg function

[kcrisman]

There are a few small formatting issues, and it would be good to add an example showing that arg(sqrt(2)+i) remaining symbolic (as opposed to arctan(1/sqrt(2))) still evaluates correctly numerically. Otherwise looks fine. Currently running tests, as arg is likely used in a lot of places in Sage...

[kcrisman]

File "/Users/karl-dietercrisman/Downloads/sage-4.8.alpha5/devel/sage-main/sage/symbolic/random_tests.py", line 16:
    sage: [f for (one,f,arity) in _mk_full_functions()]
Expected:
    [Ei, abs, arccos, arccosh, arccot, arccoth, arccsc, arccsch,
    arcsec, arcsech, arcsin, arcsinh, arctan, arctan2, arctanh,
    binomial, ceil, conjugate, cos, cosh, cot, coth, csc, csch,
    dickman_rho, dilog, dirac_delta, elliptic_e, elliptic_ec,
    elliptic_eu, elliptic_f, elliptic_kc, elliptic_pi, erf, exp,
    factorial, floor, heaviside, imag_part, integrate,
    kronecker_delta, log, polylog, real_part, sec, sech, sgn, sin,
    sinh, tan, tanh, unit_step, zeta, zetaderiv]
Got:
    [Ei, abs, arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, arcsech, arcsin, arcsinh, arctan, arctan2, arctanh, arg, binomial, ceil, conjugate, cos, cosh, cot, coth, csc, csch, dickman_rho, dilog, dirac_delta, elliptic_e, elliptic_ec, elliptic_eu, elliptic_f, elliptic_kc, elliptic_pi, erf, exp, factorial, floor, heaviside, imag_part, integrate, kronecker_delta, log, polylog, real_part, sec, sech, sgn, sin, sinh, tan, tanh, unit_step, zeta, zetaderiv]
**********************************************************************
File "/Users/karl-dietercrisman/Downloads/sage-4.8.alpha5/devel/sage-main/sage/symbolic/random_tests.py", line 238:
    sage: random_expr(5, verbose=True)
Expected:
    About to apply dirac_delta to [1]
    About to apply arccsch to [0]
    About to apply <built-in function add> to [0, arccsch(0)]
    arccsch(0)
Got:
    About to apply dirac_delta to [1]
    About to apply arcsec to [0]
    About to apply <built-in function add> to [0, arcsec(0)]
    arcsec(0)

[kcrisman]

I think the Maxima translation may not be correct.

 -- Function: carg (<z>)
     Returns the complex argument of <z>.  The complex argument is an
     angle `theta' in `(-%pi, %pi]' such that `r exp (theta %i) = <z>'
     where `r' is the magnitude of <z>.

     `carg' is a computational function, not a simplifying function.

     See also `abs' (complex magnitude), `polarform', `rectform',
     `realpart', and `imagpart'.

     Examples:

          (%i1) carg (1);
          (%o1)                           0
          (%i2) carg (1 + %i);
                                         %pi
          (%o2)                          ---
                                          4
          (%i3) carg (exp (%i));
          (%o3)                           1
          (%i4) carg (exp (%pi * %i));
          (%o4)                          %pi
          (%i5) carg (exp (3/2 * %pi * %i));
                                          %pi
          (%o5)                         - ---
                                           2
          (%i6) carg (17 * exp (2 * %i));
          (%o6)                           2


(%o3)                                true

See also Barton Willis'  parg, though that only works if having loaded to_poly_solve.

[kcrisman]

Otherwise, all tests pass!

[kcrisman]

Do this to check the fix works.

sage: maxima(arg(x))
atan2(0,x)
sage: maxima(arg(2+i))
atan(1/2)
sage: maxima(arg(sqrt(2)+i))
atan(1/sqrt(2))
sage: arg(2+i)
arctan(1/2)
sage: arg(sqrt(2)+i)
arg(sqrt(2) + I)

It also seems to help with the sqrt(2) issue, in a manner of speaking. One could tell someone to do

sage: arg(sqrt(2)+i).simplify()
arctan(1/2*sqrt(2))

[ktkohl]

revision of arg function--apply instead of the previous patch

[ktkohl]

symbolic arg--fixed whitespace issues--use this one instead of others

[burcin]

I uploaded a new patch with minor modifications to Karen's. In particular, it

• removes the duplicate commands that appear in the EXAMPLES and TESTS blocks for Function_arg.
• import CC directly instead of calling ComplexField in _evalf_().

I give a positive review to Karen's patch. If someone can take a quick look at my changes to check if I didn't mess anything up, this can be merged.

[kcrisman]

It doesn't appear that you messed anything up, except ...

sage: arg(3.0)
---------------------------------------------------------------------------
<snip>
   1426             return x.arg()
   1427         except AttributeError:
-> 1428             from sage.rings.complex_field import CC
   1429             x = CC(x)
   1430             return x.arg()

ImportError: cannot import name CC

So apparently that won't work. Otherwise the changes are fine.

[burcin]

Apparently I didn't run tests, latex output was also broken.

attachment:trac_4498-arg_evalf.patch , to be applied after attachment:trac_4498-symbolic_arg.cleanup.patch , implements a new _evalf_() function which keeps the precision of the input. This one needs a real review. :)

[kcrisman]

This makes a lot more sense. Running tests...

Why parent_d and not parent? Just wondering in case there is a convention I should be aware of.

[burcin]

Using parent as the name of the keyword argument masks the imported parent() function, which I used in the function body.

[kcrisman]

I wondered; that makes sense.

All looks well. Just a question - do you want to include any of the following as tests?

sage: arg(long(1000))
0
sage: arg(1j)
1.57079632679490
sage: arg(1J)
1.57079632679490
sage: arg(complex(0,1))
1.57079632679490
sage: arg(complex(1,0))
0.000000000000000
sage: arg(int(10))
0

It's not a big deal to me either way, I just wanted to test them.

[jdemeyer]

This patch conflicts with #9130. Either this one or #9130 should be rebased.