Trac #13620: initialization of padic polynomial from empty dict
Trac 12947: add doctest
Trac 12947: add tolerance to doctest
Merge branch 'develop' into ticket/12947-numerical_integral

Conflicts:
	src/sage/interfaces/maxima_lib.py
Merge branch 'develop' into ticket/13620
Trac #13620: initialization of padic polynomial from empty dict
Changed version number.
Update SPKG.txt and checksums.ini.
Changed checksum.
trac #16147: fix .abs() for elt of embedded number field
Added patch file to lrcalc.
Trac #16560: Upgrade lrcalc to 1.1.7

Using the upstream based off of here:
http://www.math.rutgers.edu/~asbuch/lrcalc/ but use the one in the
attachments as the root directory is `lrcalc-sage-1.1.7`.

This should (help) fix #14625.

URL: http://trac.sagemath.org/16560
Reported by: tscrim
Ticket author(s): Travis Scrimshaw
Reviewer(s): François Bissey
Trac #16147: abs for elements of number fields provided with a complex embedding ignores it

See [http://ask.sagemath.org/question/3967/qqextension-with-embedding-
incorrect-modulus]
{{{
sage: x = polygen(ZZ)
sage: f = x^3 - x - 1
sage: beta = f.complex_roots()[0]; beta
1.32471795724475
sage: K.<b> = NumberField(f, embedding=beta)
sage: b.abs()
0.868836961832709
sage: [r.abs() for r in f.complex_roots()]
[1.32471795724475, 0.868836961832709, 0.868836961832709]
}}}

URL: http://trac.sagemath.org/16147
Reported by: fwclarke
Ticket author(s): Vincent Delecroix
Reviewer(s): Francis Clarke
Trac #13620: initialization of padic polynomial fails for empty dictionary

Polynomials over the padics can be initialized from dictionaries.
{{{
sage: R.<t> = Qp(3)[]
sage: f = R.one()
sage: R(f.dict())
}}}

However, this fails for empty dictionaries:
{{{
sage: f = R.zero()
sage: R(f.dict())
ValueError: max() arg is an empty sequence
}}}

The attached patch fixes this.

-----

Apply

1. [[attachment:trac_13620.patch]]

to the sage repository.

URL: http://trac.sagemath.org/13620
Reported by: saraedum
Ticket author(s): Julian Rueth
Reviewer(s): Martin Albrecht
Trac #12947: Bug in integrating x*cos(x^3)

From this [https://groups.google.com/forum/?fromgroups#!topic/sage-
support/-Dzaion3VrA sage-support thread]:
{{{
sage: numerical_integral(x*cos(x^3),0,.5)
(0.1247560409610376, 1.3850702913602309e-15)
sage: integrate(x*cos(x^3),(x,0,1/2)).n()
-0.0677842754623305
}}}
Given
{{{
sage: integrate(x*cos(x^3),x)
1/12*((-I*sqrt(3) - 1)*gamma(2/3, -I*x^3) + (I*sqrt(3) - 1)*gamma(2/3,
I*x^3))*(x^3)^(1/3)/x
sage: integrate(x*cos(x^3),(x,0,1/2))
-1/3*gamma(2/3) + 1/6*gamma(2/3, -1/8*I) + 1/6*gamma(2/3, 1/8*I)
}}}
this is probably something where Maxima is returning too many imaginary
things and something isn't cancelling right either there or in Sage
proper?  We've had trouble with either side of this with incomplete
gamma functions before.

URL: http://trac.sagemath.org/12947
Reported by: kcrisman
Ticket author(s): Peter Bruin
Reviewer(s): Ralf Stephan