1. Tutorial

This is the pypol tutorial. If you haven’t installed it yet, see below.

1.1. First steps with pypol

1.1.1. Download and install

First of all, you need to get the files. To do this, go to the github repository or to PyPI and download the right file.

If you downloaded the source package you need to unpack it:

$ tar -xzfv pypol_-0.4.tar.gz

Now run the setup.py script:

$ python setup.py install

Sometimes (particularly on Unix systems) you have to allow setup.py to run with root security privileges:

$ python setup.py install
running install
install_dir /usr/local/lib/python2.7/dist-packages/
Checking .pth file support in /usr/local/lib/python2.7/dist-packages/
error: can't create or remove files in install directory

The following error occurred while trying to add or remove files in the
installation directory:

    [Errno 13] Permission denied: '/usr/local/lib/python2.7/dist-packages/test-easy-install-4687.pth'

The installation directory you specified (via --install-dir, --prefix, or
the distutils default setting) was:

    /usr/local/lib/python2.7/dist-packages/

Perhaps your account does not have write access to this directory?  If the
installation directory is a system-owned directory, you may need to sign in
as the administrator or "root" account.  If you do not have administrative
access to this machine, you may wish to choose a different installation
directory, preferably one that is listed in your PYTHONPATH environment
variable.

For information on other options, you may wish to consult the
documentation at:

  http://packages.python.org/distribute/easy_install.html

Please make the appropriate changes for your system and try again.

So try:

$ sudo python setup.py install
[sudo] password for user:
running install
install_dir /usr/local/lib/python2.7/dist-packages/
Checking .pth file support in /usr/local/lib/python2.7/dist-packages/
/usr/bin/python -E -c pass
TEST PASSED: /usr/local/lib/python2.7/dist-packages/ appears to support .pth files
running bdist_egg
running egg_info
writing pypol_.egg-info/PKG-INFO
writing top-level names to pypol_.egg-info/top_level.txt
writing dependency_links to pypol_.egg-info/dependency_links.txt
reading manifest file 'pypol_.egg-info/SOURCES.txt'
reading manifest template 'MANIFEST.in'
warning: no files found matching 'README.textile'
writing manifest file 'pypol_.egg-info/SOURCES.txt'
installing library code to build/bdist.linux-i686/egg
running install_lib
running build_py
creating build
creating build/lib.linux-i686-2.7
creating build/lib.linux-i686-2.7/pypol
copying pypol/__init__.py -> build/lib.linux-i686-2.7/pypol
copying pypol/core.py -> build/lib.linux-i686-2.7/pypol
copying pypol/series.py -> build/lib.linux-i686-2.7/pypol
copying pypol/funcs.py -> build/lib.linux-i686-2.7/pypol
copying pypol/roots.py -> build/lib.linux-i686-2.7/pypol
creating build/bdist.linux-i686
creating build/bdist.linux-i686/egg
creating build/bdist.linux-i686/egg/pypol
copying build/lib.linux-i686-2.7/pypol/__init__.py -> build/bdist.linux-i686/egg/pypol
copying build/lib.linux-i686-2.7/pypol/core.py -> build/bdist.linux-i686/egg/pypol
copying build/lib.linux-i686-2.7/pypol/series.py -> build/bdist.linux-i686/egg/pypol
copying build/lib.linux-i686-2.7/pypol/funcs.py -> build/bdist.linux-i686/egg/pypol
copying build/lib.linux-i686-2.7/pypol/roots.py -> build/bdist.linux-i686/egg/pypol
byte-compiling build/bdist.linux-i686/egg/pypol/__init__.py to __init__.pyc
byte-compiling build/bdist.linux-i686/egg/pypol/core.py to core.pyc
byte-compiling build/bdist.linux-i686/egg/pypol/series.py to series.pyc
byte-compiling build/bdist.linux-i686/egg/pypol/funcs.py to funcs.pyc
byte-compiling build/bdist.linux-i686/egg/pypol/roots.py to roots.pyc
creating build/bdist.linux-i686/egg/EGG-INFO
copying pypol_.egg-info/PKG-INFO -> build/bdist.linux-i686/egg/EGG-INFO
copying pypol_.egg-info/SOURCES.txt -> build/bdist.linux-i686/egg/EGG-INFO
copying pypol_.egg-info/dependency_links.txt -> build/bdist.linux-i686/egg/EGG-INFO
copying pypol_.egg-info/top_level.txt -> build/bdist.linux-i686/egg/EGG-INFO
zip_safe flag not set; analyzing archive contents...
creating 'dist/pypol_-0.4-py2.7.egg' and adding 'build/bdist.linux-i686/egg' to it
removing 'build/bdist.linux-i686/egg' (and everything under it)
Processing pypol_-0.4-py2.7.egg
creating /usr/local/lib/python2.7/dist-packages/pypol_-0.4-py2.7.egg
Extracting pypol_-0.4-py2.7.egg to /usr/local/lib/python2.7/dist-packages
Adding pypol- 0.4 to easy-install.pth file

Installed /usr/local/lib/python2.7/dist-packages/pypol_-0.4-py2.7.egg
Processing dependencies for pypol-==0.4
Finished processing dependencies for pypol-==0.4

1.1.2. Running the tests

If you want to run pypol’s tests you only have to run the python files in the tests directory:

$ python test_pypol.py
================================= test session starts =================================
platform linux2 -- Python 2.6.6 -- pytest-2.0.0
collected 40 items

test_pypol.py .................s......................

======================== 39 passed, 1 skipped in 0.31 seconds =========================
$ python test_funcs.py -v
================================= test session starts =================================
platform linux2 -- Python 2.6.6 -- pytest-2.0.0 -- /usr/bin/python
collected 12 items

test_funcs.py:34: TestFuncs.testDivisible PASSED
test_funcs.py:38: TestFuncs.testRandomPoly PASSED
test_funcs.py:53: TestFuncs.testPolyder PASSED
test_funcs.py:59: TestFuncs.testPolyint PASSED
test_funcs.py:68: TestFuncs.testPolyint_ PASSED
test_funcs.py:78: TestFuncs.testInterpolation PASSED
test_funcs.py:81: TestFuncs.testBinCoeff PASSED
test_funcs.py:90: TestFuncs.testHarmonic PASSED
test_funcs.py:104: TestFuncs.testGeneralizedHarmonic PASSED
test_funcs.py:115: TestFuncs.testStirling PASSED
test_funcs.py:124: TestFuncs.testStirling2 PASSED
test_funcs.py:132: TestFuncs.testBellNumbers PASSED

============================== 12 passed in 0.80 seconds ==============================

or run them with setup.py:

$ cd pypol_-0.4
$ sudo python setup.py test
running test
install_dir /usr/local/lib/python2.6/dist-packages/
Searching for pytest
Reading http://pypi.python.org/simple/pytest/
Reading http://pytest.org
Best match: pytest 2.0.0
Processing pytest-2.0.0-py2.6.egg
pytest 2.0.0 is already the active version in easy-install.pth
Installing py.test script to /usr/local/bin
Installing py.test-2.6 script to /usr/local/bin

Using /usr/local/lib/python2.6/dist-packages/pytest-2.0.0-py2.6.egg
Processing dependencies for pytest
Finished processing dependencies for pytest
================================= test session starts =================================
platform linux2 -- Python 2.6.6 -- pytest-2.0.0 -- /usr/bin/python
collected 86 items

tests/test_funcs.py:34: TestFuncs.testDivisible PASSED
tests/test_funcs.py:38: TestFuncs.testRandomPoly PASSED
tests/test_funcs.py:53: TestFuncs.testPolyder PASSED
tests/test_funcs.py:59: TestFuncs.testPolyint PASSED
tests/test_funcs.py:68: TestFuncs.testPolyint_ PASSED
tests/test_funcs.py:78: TestFuncs.testInterpolation PASSED
tests/test_funcs.py:81: TestFuncs.testBinCoeff PASSED
tests/test_funcs.py:90: TestFuncs.testHarmonic PASSED
tests/test_funcs.py:104: TestFuncs.testGeneralizedHarmonic PASSED
tests/test_funcs.py:115: TestFuncs.testStirling PASSED
... cut ...
tests/test_series.py:156: TestSeries.testTouchard PASSED
tests/test_series.py:164: TestSeries.testBell PASSED
tests/test_series.py:174: TestSeries.testGegenbauer PASSED
tests/test_series.py:180: TestSeries.testLaguerre PASSED
tests/test_series.py:186: TestSeries.testGeneralizedLaguerre PASSED
tests/test_series.py:192: TestSeries.testBernoulli PASSED
tests/test_series.py:202: TestSeries.testBernoulliNumbers PASSED
tests/test_series.py:217: TestSeries.testEuler PASSED
tests/test_series.py:225: TestSeries.testEulerNumbers PASSED
tests/test_series.py:235: TestSeries.testGenocchi PASSED

======================== 85 passed, 1 skipped in 3.38 seconds =========================

1.1.3. Building this documentation

To build the documentation, change directory and go to pypol/doc:

$ cd pypol_-0.4/doc

• On Windows

  Run make.bat <target>

• On Unix/Mac OS X

  Run make <target>

Where <target> is one of:

html	to make standalone HTML files
singlehtml	to make a single large HTML file
dirhtml	to make HTML files named index.html in directories
pickle	to make pickle files
json	to make JSON files
htmlhelp	to make HTML files and a HTML help project
qthelp	to make HTML files and a qthelp project
devhelp	to make HTML files and a Devhelp project
epub	to make an epub
latex	to make LaTeX files, you can set PAPER=a4 or PAPER=letter
latexpdf	to make LaTeX files and run them through pdflatex
text	to make text files
man	to make manual pages
changes	to make an overview of all changed/added/deprecated items
linkcheck	to check all external links for integrity
doctest	to run all doctests embedded in the documentation (if enabled)

1.2. Cookbook

Here is pypol cookbook. All examples assume:

>>> from pypol import *

1.2.1. Creating a polynomial

Use pypol.poly1d(), pypol.poly1d_2(), pypol.polynomial(), or pypol.monomial() to create a polynomial:

>>> p = poly1d([1, 2, -3, 4])
>>> p
+ x^3 + 2x^2 - 3x + 4
>>> q = poly1d_2([[3, 9], [-5, 6]])
>>> q
+ 3x^9 - 5x^6
>>> r = polynomial('.3x^4 - 2x^5 + 4x')
>>> r
- 2x^5 + 3/10x^4 + 4x
>>> m = monomial(-3)
>>> m
- 3
>>> m.monomials
((-3, {}),)
>>> m2 = monomial(-3, x=1, y=3, z=2)
>>> m2
- 3xy^3z^2
>>> m2.monomials
((-3, {'y': 3, 'x': 1, 'z': 2}),)
>>> len(m2)
1

1.2.2. The pypol.Polynomial class

>>> p = poly1d([1, 2, -3, 4])
>>> type(p)
<class 'pypol.core.Polynomial'>
>>> p.monomials
((1, {'x': 3}), (2, {'x': 2}), (-3, {'x': 1}), (4, {}))
>>> p.coefficients
[1, 2, -3, 4]
>>> p.letters
('x',)
>>> p.append(-2)
>>> p
+ x^3 + 2x^2 - 3x + 2
>>> p.append('4xy')
>>> p
+ x^3 + 2x^2 + 4xy - 3x + 2
>>> p.letters
('x', 'y')
>>> del p[1]
>>> p
+ x^3 + 4xy - 3x + 2
>>> p.gcd
+ 1
>>> p.lcm
+ 12x^3y
>>> p.degree
3

See also

pypol.Polynomial class reference.

1.2.3. Operations

>>> p / q
Traceback (most recent call last):
  File "<pyshell#20>", line 1, in <module>
    p / q
  File "/core.py", line 1436, in __divmod__
    raise ValueError('The polynomials are not divisible')
ValueError: The polynomials are not divisible
>>> q / p
+ 3x^6 - 6x^5 + 21x^4 - 77x^3 + 241x^2 - 797x + 2625
>>> divmod(q, p)
(+ 3x^6 - 6x^5 + 21x^4 - 77x^3 + 241x^2 - 797x + 2625,
 - 8605x^2 + 11063x - 10500)
>>> quot, rem = divmod(q, p)
>>> quot, rem
(+ 3x^6 - 6x^5 + 21x^4 - 77x^3 + 241x^2 - 797x + 2625,
 - 8605x^2 + 11063x - 10500)
>>> quot * p + rem
+ 3x^9 - 5x^6
>>> quot * p + rem == q
True
>>> j = poly1d([-3, 2, 1])
>>> j
- 3x^2 + 2x + 1
>>> j * -3
+ 9x^2 - 6x - 3
>>> j * '-x^3'
+ 3x^5 - 2x^4 - x^3
>>> j * ((1, {'y': 3}), (-2, {}))
- 3x^2y^3 + 2xy^3 + y^3 + 6x^2 - 4x - 2
>>> j
- 3x^2 + 2x + 1
>>> k = poly1d([1, 2])
>>> k
+ x + 2
>>> j + k
- 3x^2 + 3x + 3
>>> j - k
- 3x^2 + x - 1
>>> j + -k == j - k
True

See also

Operators

1.2.4. Differentiation and integration

Currently, there are three functions: pypol.funcs.polyder() (to find the derivative), pypol.funcs.polyint() (to find the indefinite integral) and pypol.funcs.polyint_() (to find the definite integral):

>>> p = poly1d([1, 3, -3, -1])
>>> p
+ x^3 + 3x^2 - 3x - 1
>>> funcs.polyder(p)
+ 3x^2 + 6x - 3
>>> funcs.polyder(p, 2)
+ 6x + 6
>>> funcs.polyder(p, 2) == funcs.polyder(funcs.polyder(p))
True
>>> funcs.polyder(p, 3)
+ 6
>>> funcs.polyder(p, 4)

>>> funcs.polyder(p, 5)

>>> funcs.polyint(p)
+ 1/4x^4 + x^3 - 3/2x^2 - x
>>> funcs.polyint(p, 2)
+ 1/20x^5 + 1/4x^4 - 1/2x^3 - 1/2x^2
>>> funcs.polyint(p, 2) == funcs.polyint(funcs.polyint(p))
True
>>> funcs.polyint(p, 2, [3, 1]) ## Integration costants
+ 1/20x^5 + 1/4x^4 - 1/2x^3 - 1/2x^2 + 3x + 1
>>> funcs.polyint(p, 3, [3, 1, -4, 3, 2]) ## Integration costants, polyint will use only the first three (m = 3)
+ 1/120x^6 + 1/20x^5 - 1/8x^4 - 1/6x^3 + 3/2x^2 + x - 4
>>> funcs.polyint_(p, 10, -2) ## Definite integral
3348.0
>>> funcs.polyint_(p, -10, -2) ## Definite integral
1368.0
>>> funcs.polyint_(p, -10, -10) ## If the limits are equal the result will be 0
0.0

1.2.5. Series

The pypol.series module defines some functions, like:

>>> from pypol.series import *
>>> fibonacci(2)
+ x
>>> fibonacci(21)
+ x^20 + 19x^18 + 153x^16 + 680x^14 + 1820x^12 + 3003x^10 + 3003x^8 + 1716x^6 + 495x^4 + 55x^2 + 1
>>> hermite_prob(12)
+ x^12 - 66x^10 + 1485x^8 - 13860x^6 + 51975x^4 - 62370x^2 + 10395
>>> hermite_phys(12)
+ 4096x^12 - 135168x^10 + 1520640x^8 - 7096320x^6 + 13305600x^4 - 7983360x^2 + 665280
>>> chebyshev_t(19)
+ 262144x^19 - 1245184x^17 + 2490368x^15 - 2723840x^13 + 1770496x^11 - 695552x^9 + 160512x^7 - 20064x^5 + 1140x^3 - 19x
>>> chebyshev_u(17) ## Chebyshev polynomials of the second kind
+ 131072x^17 - 524288x^15 + 860160x^13 - 745472x^11 + 366080x^9 - 101376x^7 + 14784x^5 - 960x^3 + 18x
>>> abel(12)
+ x^12 - 132ax^11 + 7920a^2x^10 - 285120a^3x^9 + 6842880a^4x^8 - 114960384a^5x^7 + 1379524608a^6x^6 - 11824496640a^7x^5 + 70946979840a^8x^4 - 283787919360a^9x^3 + 681091006464a^10x^2 - 743008370688a^11x
>>> abel(9, 'k')
+ x^9 - 72kx^8 + 2268k^2x^7 - 40824k^3x^6 + 459270k^4x^5 - 3306744k^5x^4 + 14880348k^6x^3 - 38263752k^7x^2 + 43046721k^8x
>>> laguerre_g(2)
+ 1/2a^2 + 3/2a - ax + 1 + 1/2x^2 - 2x
>>> laguerre_g(2, 't')
+ 1/2t^2 + 3/2t - tx + 1 + 1/2x^2 - 2x
>>> bernoulli(2)
+ x^2 - x + 1/6
>>> bern_num(2)
Fraction(1, 6)
>>> euler(3)
+ x^3 - 3/2x^2 + 1/4

1.2.6. Root-finding

The pypol.roots module implements some root-finding algorithms:

>>> from pypol.roots import *
>>> k = poly1d([3, -4, -1, 4])
>>> k
+ 3x^3 - 4x^2 - x + 4
>>> newton(k, 100)
-0.859475828371609
>>> newton(k, -10)
-0.859475828371609
>>> k(newton(k, -10))
0.0
>>> newton(k, complex(100, 1))
(1.0964045808524712-0.5909569632973221j)
>>> k(newton(k, complex(100, 1)))
-1.1102230246251565e-16j
>>> newton(k, complex(100, -1))
(1.0964045808524712+0.5909569632973221j)
>>> k(newton(k, complex(100, -1)))
1.1102230246251565e-16j
>>> halley(k, 100)
-0.859475828371609
>>> householder(k, 100)
-0.859475828371609
>>> halley(k, 1j)
(1.0964045808524712-0.5909569632973221j)
>>> householder(k, complex(.4, .9))
(1.0964045808524712+0.5909569632973221j)
>>> schroeder(k, 100)
-0.859475828371609
>>> schroeder(k, 100j)
(1.0964045808524712-0.5909569632973221j)
>>> schroeder(k, -100j)
(1.0964045808524712+0.5909569632973221j)
>>> cubic(k) ## All in one
(-0.8594758283716091, (1.0964045808524712+0.590956963297322j), (1.0964045808524712-0.590956963297322j))