#!/usr/bin/env python
# coding: utf-8

# # Table of Contents
# * [1. Demonstration of the ``numpy.polynomial`` package](#1.-Demonstration-of-the-numpy.polynomial-package)
# 	* [1.1 And especially a small hand-made pretty printing function for ``Polynomial`` objects](#1.1-And-especially-a-small-hand-made-pretty-printing-function-for-Polynomial-objects)
# 	* [1.2 First goal: pretty print in ASCII text](#1.2-First-goal:-pretty-print-in-ASCII-text)
# 	* [1.3 Second goal: pretty-print in $\LaTeX{}$ code](#1.3-Second-goal:-pretty-print-in-$\LaTeX{}$-code)
# 	* [1.4 A bonus for the end: add this pretty-printer as the default one in IPython:](#1.4-A-bonus-for-the-end:-add-this-pretty-printer-as-the-default-one-in-IPython:)
# 

# # 1. Demonstration of the ``numpy.polynomial`` package

# ## 1.1 And especially a small hand-made pretty printing function for ``Polynomial`` objects

# For both [Python 2](https://docs.python.org/2/) and [Python 3](https://docs.python.org/3/), [numpy](https://numpy.org) has a very nice module to work with [polynomials](https://en.wikipedia.org/wiki/Polynomial): [numpy.polynomial](https://docs.scipy.org/doc/numpy/reference/routines.polynomials.classes.html).
# 
# If you are not familiar with it, I **highly** recommend you to read [this tutorial](https://docs.scipy.org/doc/numpy/reference/routines.polynomials.classes.html#basics).
# 
# > - Note: this small tutorial was inspired by [this question asked on StackOverflow](https://stackoverflow.com/questions/28646336/pretty-printing-polynomials-in-ipython-notebook/37510036#37510036) (and by my answer).
# > - This [Jupyter Notebook](http://jupyter.org/) was written by [Lilian Besson](https://github.com/Naereen/).

# ----
# 
# Now, let assume you already know everything about Python, and the ``numpy.polynomial`` package.
# So you know that it should be imported like this:

# In[1]:


from numpy.polynomial import Polynomial as P


# And we can then define the monome $X$ as ``P([0, 1])``, defined by the list of its coefficients in the *canonical* basis $(X_i)_{i \in \mathbb{N}}$:

# In[2]:


X = P([0, 1])
print("We defined the monome X =", X)
X


# The main issue with this output (either ``poly([ 0.  1.])`` or ``Polynomial([ 0.,  1.], [-1,  1], [-1,  1])``) is its lack of *sexyness*: it gives the required information (coefficients, domain, window etc, see [here](https://docs.scipy.org/doc/numpy/reference/generated/numpy.polynomial.polynomial.Polynomial.html#numpy.polynomial.polynomial.Polynomial) for more details) but it is too far from the mathematical notation.
# 
# If we define $Q(X) = 1 + 17 X^3$, we would like Python (or IPython, or in this case, the Jupyter Notebook) to display this polynomial nicely, either as ASCII text: ``1 + 17 * X**3`` (valid Python code), or as a nice $\LaTeX{}$ code: ``1 + 17 X^3``.

# In[3]:


Q = 1 + 17 * X ** 3
print("Q(X) =", Q)
Q


# ----

# ## 1.2 First goal: pretty print in ASCII text

# Our first task will be to implement a small function, or an overload of the [numpy.polynomial.Polynomial](https://docs.scipy.org/doc/numpy/reference/generated/numpy.polynomial.polynomial.Polynomial.html) class to be able to pretty-print a ``Polynomial`` object nicely.
# 
# > Note: more details, along with a documentation, some examples and a fully [pylint](https://www.pylint.org/)-compatible code are available on this [Bitbucket snippet](https://bitbucket.org/snippets/): [bitbucket.org/snippets/lbesson/j6dpz](https://bitbucket.org/snippets/lbesson/j6dpz#file-nicedisplay_numpy_polynomial_Polynomial.py).

# The function ``prettyprintPolynomial`` defined below implements a natural strategy to print a polynomial: it prints the coefficients, as ``a_i * X**i``, in increasing order.
# 
# This function takes care of all the special cases:
# 
# - it removes a trailing ``.0`` if one coefficient is an integer,
# - it displays the coefficient between ``(`` and ``)`` if it is negative,
# - it only displays the non-zero coefficients,
# - if one coefficient is ``1``, no need to display it.

# In[4]:


def prettyprintPolynomial(p):
    """ Small function to print nicely the polynomial p as we write it in maths, in ASCII text."""
    coefs = p.coef  # List of coefficient, sorted by increasing degrees
    res = ""  # The resulting string
    for i, a in enumerate(coefs):
        if int(a) == a:  # Remove the trailing .0
            a = int(a)
        if i == 0:  # First coefficient, no need for X
            if a > 0:
                res += "{a} + ".format(a=a)
            elif a < 0:  # Negative a is printed like (a)
                res += "({a}) + ".format(a=a)
            # a = 0 is not displayed 
        elif i == 1:  # Second coefficient, only X and not X**i
            if a == 1:  # a = 1 does not need to be displayed
                res += "X + "
            elif a > 0:
                res += "{a} * X + ".format(a=a)
            elif a < 0:
                res += "({a}) * X + ".format(a=a)
        else:
            if a == 1:
                res += "X**{i} + ".format(i=i)
            elif a > 0:
                res += "{a} * X**{i} + ".format(a=a, i=i)
            elif a < 0:
                res += "({a}) * X**{i} + ".format(a=a, i=i)
    return res[:-3] if res else ""


# We can check its behavior on some small polynomials:

# In[5]:


print("X =", prettyprintPolynomial(X))
print("Q(X) =", prettyprintPolynomial(Q))


# In[6]:


Q3 = -1 - 2*X - 17*X**3
print("Q_3(X) =", prettyprintPolynomial(Q3))
print("- Q_3(X) =", prettyprintPolynomial(-Q3))


# We can create a more complicated polynomial $Q_4(X) = (1 + 2 X + 17 X^3) ^ {12}$ and check that it is also nicely printed:

# In[7]:


Q4 = (1 + 2*X + 17*X**3) ** 12
print("Q_4(X) =", prettyprintPolynomial(Q4))


# ----

# ## 1.3 Second goal: pretty-print in $\LaTeX{}$ code

# We will simply modify the function ``prettyprintPolynomial`` to use $\LaTeX{}$ code instead of ASCII text: the string has to be between ``$``, the multiplications are without symbols (e.g., $17 X$ for ``17 * X``), and the powers are with the ``^`` symbol instead of ``**`` :

# In[8]:


def Polynomial_to_LaTeX(p):
    """ Small function to print nicely the polynomial p as we write it in maths, in LaTeX code."""
    coefs = p.coef  # List of coefficient, sorted by increasing degrees
    res = ""  # The resulting string
    for i, a in enumerate(coefs):
        if int(a) == a:  # Remove the trailing .0
            a = int(a)
        if i == 0:  # First coefficient, no need for X
            if a > 0:
                res += "{a} + ".format(a=a)
            elif a < 0:  # Negative a is printed like (a)
                res += "({a}) + ".format(a=a)
            # a = 0 is not displayed 
        elif i == 1:  # Second coefficient, only X and not X**i
            if a == 1:  # a = 1 does not need to be displayed
                res += "X + "
            elif a > 0:
                res += "{a} \;X + ".format(a=a)
            elif a < 0:
                res += "({a}) \;X + ".format(a=a)
        else:
            if a == 1:
                # A special care needs to be addressed to put the exponent in {..} in LaTeX
                res += "X^{i} + ".format(i="{%d}" % i)
            elif a > 0:
                res += "{a} \;X^{i} + ".format(a=a, i="{%d}" % i)
            elif a < 0:
                res += "({a}) \;X^{i} + ".format(a=a, i="{%d}" % i)
    return "$" + res[:-3] + "$" if res else ""


# We can quickly try the same examples as before:

# In[9]:


print("X =", Polynomial_to_LaTeX(X))
print("Q(X) =", Polynomial_to_LaTeX(Q))


# But we want the $\LaTeX{}$ code to be pretty-printed by IPython, not just displayed like this.
# For this, the internal function ``Latex`` from the module ``IPython.display`` is required:

# In[10]:


from IPython.display import Latex


# In[11]:


print("X =")
Latex(Polynomial_to_LaTeX(X))


# In[12]:


print("Q(X) =")
Latex(Polynomial_to_LaTeX(Q))


# Allright! It starts to look like what we wanted!
# Let's try with a bigger polynomial, as we did before:

# In[13]:


Q4 = (1 + 2*X + 17*X**3) ** 12
print("Q_4(X) =")
Latex(Polynomial_to_LaTeX(Q4))


# Way nicer! Yay!

# ----

# ## 1.4 A bonus for the end: add this pretty-printer as the default one in IPython:

# This manipulation is showed in [IPython's documentation](https://nbviewer.jupyter.org/github/ipython/ipython/blob/3607712653c66d63e0d7f13f073bde8c0f209ba8/docs/examples/notebooks/display_protocol.ipynb).
# 
# But we can configure IPython to do this automatically for us as follows.
# We hook into the IPython display system and instruct it to use ``Polynomial_to_LaTeX`` for the ``latex`` mimetype, when encountering objects of the ``Polynomial`` type defined in the ``numpy.polynomial.polynomial`` module:

# In[14]:


ip = get_ipython()
latex_formatter = ip.display_formatter.formatters['text/latex']
latex_formatter.for_type_by_name('numpy.polynomial.polynomial',
                                 'Polynomial', Polynomial_to_LaTeX)


# Once our special printer has been loaded, all polynomials will be represented by their mathematical form instead (as $\LaTeX{}$ code displayed with [MathJax](https://www.mathjax.org/)):

# In[15]:


X


# One a last example:

# In[23]:


P = ((1 + X**2)**16)**16 % (1 - X**16)
print("str(P) =", str(P))
print("repr(P) =", repr(P))
P


# *That's all for today, folks!*