# Giacpy

Name: giacpy A Cython frontend to the c++ library giac. (Computer Algebra System) Frederic Han frederic.han@imj-prg.fr 2012 Frederic Han GPL v2 or above https://www.imj-prg.fr/~frederic.han/xcas/giacpy/

Access from python to the Computer Algebra System giac via libgiac

# Introduction

This is an interface to be able to use from Python the Giac features.

• Giac is a general purpose Computer algebra system by Bernard Parisse released under GPLv3.

• http://www-fourier.ujf-grenoble.fr/~parisse/giac.html
• It is build on C and C++ libraries: PARI, NTL (arithmetic), CoCoA (Groebner basis), GSL (numerics), GMP (big integers), MPFR (bigfloats)
• It provides (fast) algorithms for multivariate polynomial operations (product, GCD, factorisation) and
• symbolic computations: solver, simplifications, limits/series, integration, sommation...
• Linear Algebra with numerical or symbolic coefficients.
• giacpy is an interface to this library. It is built with cython. Graphic output is obtained with qcas by Loic Lecoq: http://git.tuxfamily.org/qcas/qcas.git

# Short Usage

Example:

```>>> import giacpy  # outputs various messages
>>> giacpy.ifactor(2**128+1)
59649589127497217*5704689200685129054721
>>> from giacpy import giac
>>> x,y,z=giac('x,y,z')
>>> f=(x+y+z+1)**15+1
>>> g=(f*(f+1)).normal()
>>> print g.nops()
>>> print g.factor().nops()
>>> f.diff()
```

Help:

```>>> help("giacpy")
>>> from giacpy import normal
>>> print(normal.__doc__) ; # to have help on some giac keyword
>>> solve.htmlhelp('fr') ; # (may be not avaible on your system) to have detailled help on some giac keyword
>>> htmlhelp()  ; # to have help the global help pages.

* Graphics 2D Output: (cf. help('giacpy') for examples)
If your version of giacpy has qt support, you can send graphics to qcas with the .qcas() method. For experimental interactive geometry see: help(qcas)
```

# Install

• To build the extension from sources (unix):

• You need the giac library, gmp and python headers. Ex: giac, libgmp-dev python-dev
• Then execute the command: python setup.py build_ext (or try the: make or make local)
• If you need some options see: python setup.py build_ext --help
• To install giacpy on unix (needs libgiac): python setup.py install
• For binaries of giacpy: http://webusers.imj-prg.fr/~frederic.han/xcas/giacpy/

• To run tests you can try: make test or run: python -m doctest giacpy.pyx -v (in the directory of giapy.so if it is not installed)

# Short Tutorial on the giac function

This function evaluate a python object with the giac library.

• It creates in python a Pygen element and evaluate it with giac:
```>>> from giacpy import giac,pi
>>> x,y=giac('x,y');type(x)
<type 'giacpy.Pygen'>
>>> (x+2*y).cos().texpand()
cos(x)*(2*cos(y)**2-1)-sin(x)*2*cos(y)*sin(y)
```

## Coercion, Pygen and internal giac variables:

• The most usefull objects will be the Python object of type Pygen.
```>>> from giacpy import *
>>> x,y,z=giac('x,y,z')
>>> f=sum([x[i] for i in range(5)])**15/(y+z);f.coeff(x,12)
(455*(x)**3+1365*(x)**2*x+1365*(x)**2*x+1365*(x)**2*x+1365*x*(x)**2+2730*x*x*x+2730*x*x*x+1365*x*(x)**2+2730*x*x*x+1365*x*(x)**2+455*(x)**3+1365*(x)**2*x+1365*(x)**2*x+1365*x*(x)**2+2730*x*x*x+1365*x*(x)**2+455*(x)**3+1365*(x)**2*x+1365*x*(x)**2+455*(x)**3)/(y+z)
```
• The Python object y of type Pygen is not an internal giac variable. (Most of the time you won't need to use internal giac variables).
```>>> type(y);giac('y:=1');y
<type 'giacpy.Pygen'>
1
y
```
• There are some natural coercion to Pygen elements:
```>>> pi>3.14 ; pi >3.15 ; giac(3)==3
True
False
True
```

## Lists of Pygen and Giac lists:

• Here l1 is a giac list and l2 a python list of Pygen type objects.
```>>> l1=giac(range(10)); l2=[1/(i**2+1) for i in l1]
>>> sum(l2)
33054527/16762850
```

So l1+l1 is done in giac and means a vector addition. But l2+l2 is done in Python so it is the list concatenation.

```>>> l1+l1
[0,2,4,6,8,10,12,14,16,18]
>>> l2+l2
[1, 1/2, 1/5, 1/10, 1/17, 1/26, 1/37, 1/50, 1/65, 1/82, 1, 1/2, 1/5, 1/10, 1/17, 1/26, 1/37, 1/50, 1/65, 1/82]
```
• Here V is not a Pygen element. We need to push it to giac to use a giac method like dim, or we need to use an imported function.
```>>> V=[ [x[i]**j for i in range(9)] for j in range(9)]
>>> giac(V).dim()
[9,9]
>>> det_minor(V).factor()
(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))*(x-(x))
```
• Modular objects with %
```>>> V=ranm(5,5) % 2;
>>> ker(V).rowdim()+V.rank()
5
>>> a=giac(7)%3;a;a%0;7%3
1 % 3
1
1
```

Do not confuse with the full python integers:

```>>> type(7%3);type(a)
<type 'int'>
<type 'giacpy.Pygen'>
```

## Syntaxes with reserved or unknown Python symbols:

• In general equations needs symbols such as = < > or that have another meaning in Python. So those objects must be quoted.
```>>> from giacpy import *
>>> x=giac('x')
>>> (1+2*sin(3*x)).solve(x)
list[-pi/3/6,7*pi/18]
```
```>>> solve('sin(3*x)>2*sin(x)',x)
Traceback (most recent call last):
...
RuntimeError: Unable to find numeric values solving equation. For trigonometric equations this may be solved using assumptions, e.g. assume(x>-pi && x<pi) Error: Bad Argument Value
```
• You can also add some hypothesis to a giac symbol:
```>>> assume('x>-pi && x<pi')
x
>>> solve('sin(3*x)>2*sin(x)',x)
list[((x>((-5*pi)/6)) and (x<((-pi)/6))),((x>0) and (x<(pi/6))),((x>(5*pi/6)) and (x<pi))]
```
• To remove those hypothesis use the giac function: purge
```>>> purge('x')
assume[[],[line[-pi,pi]],[-pi,pi]]
>>> solve('x>0')
list[x>0]
```
• Same problems with the ..
```>>> from giacpy import *
>>> x=giac('x')
>>> f=1/(5+cos(4*x));f.int(x)
1/2/(2*sqrt(6))*(atan(2*tan(4*x/2)/sqrt(6))+pi*floor(4*x/2/pi+1/2))
>>> fMax(f,'x=-0..pi').simplify()
pi/4,3*pi/4
>>> fMax.help()
"Returns the abscissa of the maximum of the expression.
Expr,[Var]
fMax(-x^2+2*x+1,x)
fMin"
>>> sum(1/(1+x**2),'x=0..infinity').simplify()
(pi*exp(pi)**2+pi+exp(pi)**2-1)/(2*exp(pi)**2-2)
```

# Changelog

• Version 0.2:
• Add a comparison function to Pygen. (with coersion)
• Add a basic definition for most giac functions.
• Version 0.2.1:
• Add __neg__ and __pos__ support for Pygen. (Ex: -pi)
• Change __repr__ to hide too long outputs.
• Make ** be the default printing for powers in giac.
• Version 0.2.2:
• Change Pygen() to Pygen('NULL'). (Ex: rand())
• Add direct acces to the python double value of a Pygen: a._double
• Add conversion to giac modulars via the operator %
• Add ctrl-c support during list initialisation and iteration
• Modification of __getitem__ to allow formal variables with indexes.
• Add htmlhelp method for Pygen objects.
• Improve the giac initialisation of Python long integers. (basic Horner method instead of strings)
• Improve help(giac) and doctests
• Add support for the slice notation with giac lists
• Version 0.2.3:
• Fix Pygen() None initialisation. Add crash test and improve speed in _wrap_gen
• Add a GiacSettings class with some frontends to the cas settings.
• Version 0.2.4:
• Update giac 1.1 keywords.
• Version 0.3:
• Add a qt output for 2d graphics via qcas.
• Fixes for giac 1.1
• Version 0.4:
• Fixes for Python 3 compatibility

• Qt/qcas can be disabled at compilation. (cf setup.py)

• 0.4.1:
• test if the html doc is present locally, otherwise open the web doc.
• 0.4.2:
• add digits and epsilon in GiacSetting.
• Fix for interruptions of giac operators.
• Put all the GiacKeywords in a new class: GiacFunction to enable docstrings from giac.
• 0.4.3:
• Update qcas to current version. (svg export added)
• New evaluation with threads to have better interruptions.
• 0.4.4:
• Add sqrt and complex flags in giac settings.
• Add support for multi indexes. Ex A[1,2].
• Version 0.5:
• 0.5.0:
• Put all the Qt/Graphics functions in an independant submodule