update ref rstfile

tests/ipynbref/IntroNumPy.orig.rst

@@ -1,2077 +1,2077 b''
 An Introduction to the Scientific Python Ecosystem
 ==================================================
 
 While the Python language is an excellent tool for general-purpose
 programming, with a highly readable syntax, rich and powerful data types
 (strings, lists, sets, dictionaries, arbitrary length integers, etc) and
 a very comprehensive standard library, it was not designed specifically
 for mathematical and scientific computing. Neither the language nor its
 standard library have facilities for the efficient representation of
 multidimensional datasets, tools for linear algebra and general matrix
 manipulations (an essential building block of virtually all technical
 computing), nor any data visualization facilities.
 
 In particular, Python lists are very flexible containers that can be
 nested arbitrarily deep and which can hold any Python object in them,
 but they are poorly suited to represent efficiently common mathematical
 constructs like vectors and matrices. In contrast, much of our modern
 heritage of scientific computing has been built on top of libraries
 written in the Fortran language, which has native support for vectors
 and matrices as well as a library of mathematical functions that can
 efficiently operate on entire arrays at once.
 
 Scientific Python: a collaboration of projects built by scientists
 ------------------------------------------------------------------
 
 The scientific community has developed a set of related Python libraries
 that provide powerful array facilities, linear algebra, numerical
 algorithms, data visualization and more. In this appendix, we will
 briefly outline the tools most frequently used for this purpose, that
 make "Scientific Python" something far more powerful than the Python
 language alone.
 
 For reasons of space, we can only describe in some detail the central
 Numpy library, but below we provide links to the websites of each
 project where you can read their documentation in more detail.
 
 First, let's look at an overview of the basic tools that most scientists
 use in daily research with Python. The core of this ecosystem is
 composed of:
 
 -  Numpy: the basic library that most others depend on, it provides a
    powerful array type that can represent multidmensional datasets of
    many different kinds and that supports arithmetic operations. Numpy
    also provides a library of common mathematical functions, basic
    linear algebra, random number generation and Fast Fourier Transforms.
    Numpy can be found at `numpy.scipy.org <http://numpy.scipy.org>`_
 
 -  Scipy: a large collection of numerical algorithms that operate on
    numpy arrays and provide facilities for many common tasks in
    scientific computing, including dense and sparse linear algebra
    support, optimization, special functions, statistics, n-dimensional
    image processing, signal processing and more. Scipy can be found at
    `scipy.org <http://scipy.org>`_.
 
 -  Matplotlib: a data visualization library with a strong focus on
    producing high-quality output, it supports a variety of common
    scientific plot types in two and three dimensions, with precise
    control over the final output and format for publication-quality
    results. Matplotlib can also be controlled interactively allowing
    graphical manipulation of your data (zooming, panning, etc) and can
    be used with most modern user interface toolkits. It can be found at
    `matplotlib.sf.net <http://matplotlib.sf.net>`_.
 
 -  IPython: while not strictly scientific in nature, IPython is the
    interactive environment in which many scientists spend their time.
    IPython provides a powerful Python shell that integrates tightly with
    Matplotlib and with easy access to the files and operating system,
    and which can execute in a terminal or in a graphical Qt console.
    IPython also has a web-based notebook interface that can combine code
    with text, mathematical expressions, figures and multimedia. It can
    be found at `ipython.org <http://ipython.org>`_.
 
 While each of these tools can be installed separately, in our opinion
 the most convenient way today of accessing them (especially on Windows
 and Mac computers) is to install the `Free Edition of the Enthought
 Python Distribution <http://www.enthought.com/products/epd_free.php>`_
 which contain all the above. Other free alternatives on Windows (but not
 on Macs) are `Python(x,y) <http://code.google.com/p/pythonxy>`_ and
 `Christoph Gohlke's packages
 page <http://www.lfd.uci.edu/~gohlke/pythonlibs>`_.
 
 These four 'core' libraries are in practice complemented by a number of
 other tools for more specialized work. We will briefly list here the
 ones that we think are the most commonly needed:
 
 -  Sympy: a symbolic manipulation tool that turns a Python session into
    a computer algebra system. It integrates with the IPython notebook,
    rendering results in properly typeset mathematical notation.
    `sympy.org <http://sympy.org>`_.
 
 -  Mayavi: sophisticated 3d data visualization;
    `code.enthought.com/projects/mayavi <http://code.enthought.com/projects/mayavi>`_.
 
 -  Cython: a bridge language between Python and C, useful both to
    optimize performance bottlenecks in Python and to access C libraries
    directly; `cython.org <http://cython.org>`_.
 
 -  Pandas: high-performance data structures and data analysis tools,
    with powerful data alignment and structural manipulation
    capabilities; `pandas.pydata.org <http://pandas.pydata.org>`_.
 
 -  Statsmodels: statistical data exploration and model estimation;
    `statsmodels.sourceforge.net <http://statsmodels.sourceforge.net>`_.
 
 -  Scikit-learn: general purpose machine learning algorithms with a
    common interface; `scikit-learn.org <http://scikit-learn.org>`_.
 
 -  Scikits-image: image processing toolbox;
    `scikits-image.org <http://scikits-image.org>`_.
 
 -  NetworkX: analysis of complex networks (in the graph theoretical
    sense); `networkx.lanl.gov <http://networkx.lanl.gov>`_.
 
 -  PyTables: management of hierarchical datasets using the
    industry-standard HDF5 format;
    `www.pytables.org <http://www.pytables.org>`_.
 
 Beyond these, for any specific problem you should look on the internet
 first, before starting to write code from scratch. There's a good chance
 that someone, somewhere, has written an open source library that you can
 use for part or all of your problem.
 
 A note about the examples below
 -------------------------------
 
 In all subsequent examples, you will see blocks of input code, followed
 by the results of the code if the code generated output. This output may
 include text, graphics and other result objects. These blocks of input
 can be pasted into your interactive IPython session or notebook for you
 to execute. In the print version of this document, a thin vertical bar
 on the left of the blocks of input and output shows which blocks go
 together.
 
 If you are reading this text as an actual IPython notebook, you can
 press ``Shift-Enter`` or use the 'play' button on the toolbar
 (right-pointing triangle) to execute each block of code, known as a
 'cell' in IPython:
 
 In[71]:
 
 .. code:: python
 
     # This is a block of code, below you'll see its output
     print "Welcome to the world of scientific computing with Python!"
 
 .. parsed-literal::
 
     Welcome to the world of scientific computing with Python!
 
 
 Motivation: the trapezoidal rule
 ================================
 
 In subsequent sections we'll provide a basic introduction to the nuts
 and bolts of the basic scientific python tools; but we'll first motivate
 it with a brief example that illustrates what you can do in a few lines
 with these tools. For this, we will use the simple problem of
 approximating a definite integral with the trapezoid rule:
 
 .. math::
 
 
    \int_{a}^{b} f(x)\, dx \approx \frac{1}{2} \sum_{k=1}^{N} \left( x_{k} - x_{k-1} \right) \left( f(x_{k}) + f(x_{k-1}) \right).
 
 Our task will be to compute this formula for a function such as:
 
 .. math::
 
 
    f(x) = (x-3)(x-5)(x-7)+85
 
 integrated between :math:`a=1` and :math:`b=9`.
 
 First, we define the function and sample it evenly between 0 and 10 at
 200 points:
 
 In[1]:
 
 .. code:: python
 
     def f(x):
         return (x-3)*(x-5)*(x-7)+85
     
     import numpy as np
     x = np.linspace(0, 10, 200)
     y = f(x)
 
 We select :math:`a` and :math:`b`, our integration limits, and we take
 only a few points in that region to illustrate the error behavior of the
 trapezoid approximation:
 
 In[2]:
 
 .. code:: python
 
     a, b = 1, 9
     xint = x[logical_and(x>=a, x<=b)][::30]
     yint = y[logical_and(x>=a, x<=b)][::30]
 
 Let's plot both the function and the area below it in the trapezoid
 approximation:
 
 In[3]:
 
 .. code:: python
 
     import matplotlib.pyplot as plt
     plt.plot(x, y, lw=2)
     plt.axis([0, 10, 0, 140])
     plt.fill_between(xint, 0, yint, facecolor='gray', alpha=0.4)
     plt.text(0.5 * (a + b), 30,r"$\int_a^b f(x)dx$", horizontalalignment='center', fontsize=20);
 
-.. image:: tests/ipynbref/IntroNumPy.orig_files/IntroNumPy.orig_fig_00.svg
+.. image:: tests/ipynbref/IntroNumPy_orig_files/IntroNumPy_orig_fig_00.svg
 
 Compute the integral both at high accuracy and with the trapezoid
 approximation
 
 In[4]:
 
 .. code:: python
 
     from scipy.integrate import quad, trapz
     integral, error = quad(f, 1, 9)
     trap_integral = trapz(yint, xint)
     print "The integral is: %g +/- %.1e" % (integral, error)
     print "The trapezoid approximation with", len(xint), "points is:", trap_integral
     print "The absolute error is:", abs(integral - trap_integral)
 
 .. parsed-literal::
 
     The integral is: 680 +/- 7.5e-12
     The trapezoid approximation with 6 points is: 621.286411141
     The absolute error is: 58.7135888589
 
 
 This simple example showed us how, combining the numpy, scipy and
 matplotlib libraries we can provide an illustration of a standard method
 in elementary calculus with just a few lines of code. We will now
 discuss with more detail the basic usage of these tools.
 
 NumPy arrays: the right data structure for scientific computing
 ===============================================================
 
 Basics of Numpy arrays
 ----------------------
 
 We now turn our attention to the Numpy library, which forms the base
 layer for the entire 'scipy ecosystem'. Once you have installed numpy,
 you can import it as
 
 In[5]:
 
 .. code:: python
 
     import numpy
 
 though in this book we will use the common shorthand
 
 In[6]:
 
 .. code:: python
 
     import numpy as np
 
 As mentioned above, the main object provided by numpy is a powerful
 array. We'll start by exploring how the numpy array differs from Python
 lists. We start by creating a simple list and an array with the same
 contents of the list:
 
 In[7]:
 
 .. code:: python
 
     lst = [10, 20, 30, 40]
     arr = np.array([10, 20, 30, 40])
 
 Elements of a one-dimensional array are accessed with the same syntax as
 a list:
 
 In[8]:
 
 .. code:: python
 
     lst[0]
 
 Out[8]:
 
 .. parsed-literal::
 
     10
 
 In[9]:
 
 .. code:: python
 
     arr[0]
 
 Out[9]:
 
 .. parsed-literal::
 
     10
 
 In[10]:
 
 .. code:: python
 
     arr[-1]
 
 Out[10]:
 
 .. parsed-literal::
 
     40
 
 In[11]:
 
 .. code:: python
 
     arr[2:]
 
 Out[11]:
 
 .. parsed-literal::
 
     array([30, 40])
 
 The first difference to note between lists and arrays is that arrays are
 *homogeneous*; i.e. all elements of an array must be of the same type.
 In contrast, lists can contain elements of arbitrary type. For example,
 we can change the last element in our list above to be a string:
 
 In[12]:
 
 .. code:: python
 
     lst[-1] = 'a string inside a list'
     lst
 
 Out[12]:
 
 .. parsed-literal::
 
     [10, 20, 30, 'a string inside a list']
 
 but the same can not be done with an array, as we get an error message:
 
 In[13]:
 
 .. code:: python
 
     arr[-1] = 'a string inside an array'
 
 ::
 
     ---------------------------------------------------------------------------
     ValueError                                Traceback (most recent call last)
     /home/fperez/teach/book-math-labtool/<ipython-input-13-29c0bfa5fa8a> in <module>()
     ----> 1 arr[-1] = 'a string inside an array'
     
     ValueError: invalid literal for long() with base 10: 'a string inside an array'
 
 The information about the type of an array is contained in its *dtype*
 attribute:
 
 In[14]:
 
 .. code:: python
 
     arr.dtype
 
 Out[14]:
 
 .. parsed-literal::
 
     dtype('int32')
 
 Once an array has been created, its dtype is fixed and it can only store
 elements of the same type. For this example where the dtype is integer,
 if we store a floating point number it will be automatically converted
 into an integer:
 
 In[15]:
 
 .. code:: python
 
     arr[-1] = 1.234
     arr
 
 Out[15]:
 
 .. parsed-literal::
 
     array([10, 20, 30,  1])
 
 Above we created an array from an existing list; now let us now see
 other ways in which we can create arrays, which we'll illustrate next. A
 common need is to have an array initialized with a constant value, and
 very often this value is 0 or 1 (suitable as starting value for additive
 and multiplicative loops respectively); ``zeros`` creates arrays of all
 zeros, with any desired dtype:
 
 In[16]:
 
 .. code:: python
 
     np.zeros(5, float)
 
 Out[16]:
 
 .. parsed-literal::
 
     array([ 0.,  0.,  0.,  0.,  0.])
 
 In[17]:
 
 .. code:: python
 
     np.zeros(3, int)
 
 Out[17]:
 
 .. parsed-literal::
 
     array([0, 0, 0])
 
 In[18]:
 
 .. code:: python
 
     np.zeros(3, complex)
 
 Out[18]:
 
 .. parsed-literal::
 
     array([ 0.+0.j,  0.+0.j,  0.+0.j])
 
 and similarly for ``ones``:
 
 In[19]:
 
 .. code:: python
 
     print '5 ones:', np.ones(5)
 
 .. parsed-literal::
 
     5 ones: [ 1.  1.  1.  1.  1.]
 
 
 If we want an array initialized with an arbitrary value, we can create
 an empty array and then use the fill method to put the value we want
 into the array:
 
 In[20]:
 
 .. code:: python
 
     a = empty(4)
     a.fill(5.5)
     a
 
 Out[20]:
 
 .. parsed-literal::
 
     array([ 5.5,  5.5,  5.5,  5.5])
 
 Numpy also offers the ``arange`` function, which works like the builtin
 ``range`` but returns an array instead of a list:
 
 In[21]:
 
 .. code:: python
 
     np.arange(5)
 
 Out[21]:
 
 .. parsed-literal::
 
     array([0, 1, 2, 3, 4])
 
 and the ``linspace`` and ``logspace`` functions to create linearly and
 logarithmically-spaced grids respectively, with a fixed number of points
 and including both ends of the specified interval:
 
 In[22]:
 
 .. code:: python
 
     print "A linear grid between 0 and 1:", np.linspace(0, 1, 5)
     print "A logarithmic grid between 10**1 and 10**4: ", np.logspace(1, 4, 4)
 
 .. parsed-literal::
 
     A linear grid between 0 and 1: [ 0.    0.25  0.5   0.75  1.  ]
     A logarithmic grid between 10**1 and 10**4:  [    10.    100.   1000.  10000.]
 
 
 Finally, it is often useful to create arrays with random numbers that
 follow a specific distribution. The ``np.random`` module contains a
 number of functions that can be used to this effect, for example this
 will produce an array of 5 random samples taken from a standard normal
 distribution (0 mean and variance 1):
 
 In[23]:
 
 .. code:: python
 
     np.random.randn(5)
 
 Out[23]:
 
 .. parsed-literal::
 
     array([-0.08633343, -0.67375434,  1.00589536,  0.87081651,  1.65597822])
 
 whereas this will also give 5 samples, but from a normal distribution
 with a mean of 10 and a variance of 3:
 
 In[24]:
 
 .. code:: python
 
     norm10 = np.random.normal(10, 3, 5)
     norm10
 
 Out[24]:
 
 .. parsed-literal::
 
     array([  8.94879575,   5.53038269,   8.24847281,  12.14944165,  11.56209294])
 
 Indexing with other arrays
 --------------------------
 
 Above we saw how to index arrays with single numbers and slices, just
 like Python lists. But arrays allow for a more sophisticated kind of
 indexing which is very powerful: you can index an array with another
 array, and in particular with an array of boolean values. This is
 particluarly useful to extract information from an array that matches a
 certain condition.
 
 Consider for example that in the array ``norm10`` we want to replace all
 values above 9 with the value 0. We can do so by first finding the
 *mask* that indicates where this condition is true or false:
 
 In[25]:
 
 .. code:: python
 
     mask = norm10 > 9
     mask
 
 Out[25]:
 
 .. parsed-literal::
 
     array([False, False, False,  True,  True], dtype=bool)
 
 Now that we have this mask, we can use it to either read those values or
 to reset them to 0:
 
 In[26]:
 
 .. code:: python
 
     print 'Values above 9:', norm10[mask]
 
 .. parsed-literal::
 
     Values above 9: [ 12.14944165  11.56209294]
 
 
 In[27]:
 
 .. code:: python
 
     print 'Resetting all values above 9 to 0...'
     norm10[mask] = 0
     print norm10
 
 .. parsed-literal::
 
     Resetting all values above 9 to 0...
     [ 8.94879575  5.53038269  8.24847281  0.          0.        ]
 
 
 Arrays with more than one dimension
 -----------------------------------
 
 Up until now all our examples have used one-dimensional arrays. But
 Numpy can create arrays of aribtrary dimensions, and all the methods
 illustrated in the previous section work with more than one dimension.
 For example, a list of lists can be used to initialize a two dimensional
 array:
 
 In[28]:
 
 .. code:: python
 
     lst2 = [[1, 2], [3, 4]]
     arr2 = np.array([[1, 2], [3, 4]])
     arr2
 
 Out[28]:
 
 .. parsed-literal::
 
     array([[1, 2],
            [3, 4]])
 
 With two-dimensional arrays we start seeing the power of numpy: while a
 nested list can be indexed using repeatedly the ``[ ]`` operator,
 multidimensional arrays support a much more natural indexing syntax with
 a single ``[ ]`` and a set of indices separated by commas:
 
 In[29]:
 
 .. code:: python
 
     print lst2[0][1]
     print arr2[0,1]
 
 .. parsed-literal::
 
     2
     2
 
 
 Most of the array creation functions listed above can be used with more
 than one dimension, for example:
 
 In[30]:
 
 .. code:: python
 
     np.zeros((2,3))
 
 Out[30]:
 
 .. parsed-literal::
 
     array([[ 0.,  0.,  0.],
            [ 0.,  0.,  0.]])
 
 In[31]:
 
 .. code:: python
 
     np.random.normal(10, 3, (2, 4))
 
 Out[31]:
 
 .. parsed-literal::
 
     array([[ 11.26788826,   4.29619866,  11.09346496,   9.73861307],
            [ 10.54025996,   9.5146268 ,  10.80367214,  13.62204505]])
 
 In fact, the shape of an array can be changed at any time, as long as
 the total number of elements is unchanged. For example, if we want a 2x4
 array with numbers increasing from 0, the easiest way to create it is:
 
 In[32]:
 
 .. code:: python
 
     arr = np.arange(8).reshape(2,4)
     print arr
 
 .. parsed-literal::
 
     [[0 1 2 3]
      [4 5 6 7]]
 
 
 With multidimensional arrays, you can also use slices, and you can mix
 and match slices and single indices in the different dimensions (using
 the same array as above):
 
 In[33]:
 
 .. code:: python
 
     print 'Slicing in the second row:', arr[1, 2:4]
     print 'All rows, third column   :', arr[:, 2]
 
 .. parsed-literal::
 
     Slicing in the second row: [6 7]
     All rows, third column   : [2 6]
 
 
 If you only provide one index, then you will get an array with one less
 dimension containing that row:
 
 In[34]:
 
 .. code:: python
 
     print 'First row:  ', arr[0]
     print 'Second row: ', arr[1]
 
 .. parsed-literal::
 
     First row:   [0 1 2 3]
     Second row:  [4 5 6 7]
 
 
 Now that we have seen how to create arrays with more than one dimension,
 it's a good idea to look at some of the most useful properties and
 methods that arrays have. The following provide basic information about
 the size, shape and data in the array:
 
 In[35]:
 
 .. code:: python
 
     print 'Data type                :', arr.dtype
     print 'Total number of elements :', arr.size
     print 'Number of dimensions     :', arr.ndim
     print 'Shape (dimensionality)   :', arr.shape
     print 'Memory used (in bytes)   :', arr.nbytes
 
 .. parsed-literal::
 
     Data type                : int32
     Total number of elements : 8
     Number of dimensions     : 2
     Shape (dimensionality)   : (2, 4)
     Memory used (in bytes)   : 32
 
 
 Arrays also have many useful methods, some especially useful ones are:
 
 In[36]:
 
 .. code:: python
 
     print 'Minimum and maximum             :', arr.min(), arr.max()
     print 'Sum and product of all elements :', arr.sum(), arr.prod()
     print 'Mean and standard deviation     :', arr.mean(), arr.std()
 
 .. parsed-literal::
 
     Minimum and maximum             : 0 7
     Sum and product of all elements : 28 0
     Mean and standard deviation     : 3.5 2.29128784748
 
 
 For these methods, the above operations area all computed on all the
 elements of the array. But for a multidimensional array, it's possible
 to do the computation along a single dimension, by passing the ``axis``
 parameter; for example:
 
 In[37]:
 
 .. code:: python
 
     print 'For the following array:\n', arr
     print 'The sum of elements along the rows is    :', arr.sum(axis=1)
     print 'The sum of elements along the columns is :', arr.sum(axis=0)
 
 .. parsed-literal::
 
     For the following array:
     [[0 1 2 3]
      [4 5 6 7]]
     The sum of elements along the rows is    : [ 6 22]
     The sum of elements along the columns is : [ 4  6  8 10]
 
 
 As you can see in this example, the value of the ``axis`` parameter is
 the dimension which will be *consumed* once the operation has been
 carried out. This is why to sum along the rows we use ``axis=0``.
 
 This can be easily illustrated with an example that has more dimensions;
 we create an array with 4 dimensions and shape ``(3,4,5,6)`` and sum
 along the axis number 2 (i.e. the *third* axis, since in Python all
 counts are 0-based). That consumes the dimension whose length was 5,
 leaving us with a new array that has shape ``(3,4,6)``:
 
 In[38]:
 
 .. code:: python
 
     np.zeros((3,4,5,6)).sum(2).shape
 
 Out[38]:
 
 .. parsed-literal::
 
     (3, 4, 6)
 
 Another widely used property of arrays is the ``.T`` attribute, which
 allows you to access the transpose of the array:
 
 In[39]:
 
 .. code:: python
 
     print 'Array:\n', arr
     print 'Transpose:\n', arr.T
 
 .. parsed-literal::
 
     Array:
     [[0 1 2 3]
      [4 5 6 7]]
     Transpose:
     [[0 4]
      [1 5]
      [2 6]
      [3 7]]
 
 
 We don't have time here to look at all the methods and properties of
 arrays, here's a complete list. Simply try exploring some of these
 IPython to learn more, or read their description in the full Numpy
 documentation:
 
 ::
 
     arr.T             arr.copy          arr.getfield      arr.put           arr.squeeze
     arr.all           arr.ctypes        arr.imag          arr.ravel         arr.std
     arr.any           arr.cumprod       arr.item          arr.real          arr.strides
     arr.argmax        arr.cumsum        arr.itemset       arr.repeat        arr.sum
     arr.argmin        arr.data          arr.itemsize      arr.reshape       arr.swapaxes
     arr.argsort       arr.diagonal      arr.max           arr.resize        arr.take
     arr.astype        arr.dot           arr.mean          arr.round         arr.tofile
     arr.base          arr.dtype         arr.min           arr.searchsorted  arr.tolist
     arr.byteswap      arr.dump          arr.nbytes        arr.setasflat     arr.tostring
     arr.choose        arr.dumps         arr.ndim          arr.setfield      arr.trace
     arr.clip          arr.fill          arr.newbyteorder  arr.setflags      arr.transpose
     arr.compress      arr.flags         arr.nonzero       arr.shape         arr.var
     arr.conj          arr.flat          arr.prod          arr.size          arr.view
     arr.conjugate     arr.flatten       arr.ptp           arr.sort          
 
 
 Operating with arrays
 ---------------------
 
 Arrays support all regular arithmetic operators, and the numpy library
 also contains a complete collection of basic mathematical functions that
 operate on arrays. It is important to remember that in general, all
 operations with arrays are applied *element-wise*, i.e., are applied to
 all the elements of the array at the same time. Consider for example:
 
 In[40]:
 
 .. code:: python
 
     arr1 = np.arange(4)
     arr2 = np.arange(10, 14)
     print arr1, '+', arr2, '=', arr1+arr2
 
 .. parsed-literal::
 
     [0 1 2 3] + [10 11 12 13] = [10 12 14 16]
 
 
 Importantly, you must remember that even the multiplication operator is
 by default applied element-wise, it is *not* the matrix multiplication
 from linear algebra (as is the case in Matlab, for example):
 
 In[41]:
 
 .. code:: python
 
     print arr1, '*', arr2, '=', arr1*arr2
 
 .. parsed-literal::
 
     [0 1 2 3] * [10 11 12 13] = [ 0 11 24 39]
 
 
 While this means that in principle arrays must always match in their
 dimensionality in order for an operation to be valid, numpy will
 *broadcast* dimensions when possible. For example, suppose that you want
 to add the number 1.5 to ``arr1``; the following would be a valid way to
 do it:
 
 In[42]:
 
 .. code:: python
 
     arr1 + 1.5*np.ones(4)
 
 Out[42]:
 
 .. parsed-literal::
 
     array([ 1.5,  2.5,  3.5,  4.5])
 
 But thanks to numpy's broadcasting rules, the following is equally
 valid:
 
 In[43]:
 
 .. code:: python
 
     arr1 + 1.5
 
 Out[43]:
 
 .. parsed-literal::
 
     array([ 1.5,  2.5,  3.5,  4.5])
 
 In this case, numpy looked at both operands and saw that the first
 (``arr1``) was a one-dimensional array of length 4 and the second was a
 scalar, considered a zero-dimensional object. The broadcasting rules
 allow numpy to:
 
 -  *create* new dimensions of length 1 (since this doesn't change the
    size of the array)
 -  'stretch' a dimension of length 1 that needs to be matched to a
    dimension of a different size.
 
 So in the above example, the scalar 1.5 is effectively:
 
 -  first 'promoted' to a 1-dimensional array of length 1
 -  then, this array is 'stretched' to length 4 to match the dimension of
    ``arr1``.
 
 After these two operations are complete, the addition can proceed as now
 both operands are one-dimensional arrays of length 4.
 
 This broadcasting behavior is in practice enormously powerful,
 especially because when numpy broadcasts to create new dimensions or to
 'stretch' existing ones, it doesn't actually replicate the data. In the
 example above the operation is carried *as if* the 1.5 was a 1-d array
 with 1.5 in all of its entries, but no actual array was ever created.
 This can save lots of memory in cases when the arrays in question are
 large and can have significant performance implications.
 
 The general rule is: when operating on two arrays, NumPy compares their
 shapes element-wise. It starts with the trailing dimensions, and works
 its way forward, creating dimensions of length 1 as needed. Two
 dimensions are considered compatible when
 
 -  they are equal to begin with, or
 -  one of them is 1; in this case numpy will do the 'stretching' to make
    them equal.
 
 If these conditions are not met, a
 ``ValueError: frames are not aligned`` exception is thrown, indicating
 that the arrays have incompatible shapes. The size of the resulting
 array is the maximum size along each dimension of the input arrays.
 
 This shows how the broadcasting rules work in several dimensions:
 
 In[44]:
 
 .. code:: python
 
     b = np.array([2, 3, 4, 5])
     print arr, '\n\n+', b , '\n----------------\n', arr + b
 
 .. parsed-literal::
 
     [[0 1 2 3]
      [4 5 6 7]] 
     
     + [2 3 4 5] 
     ----------------
     [[ 2  4  6  8]
      [ 6  8 10 12]]
 
 
 Now, how could you use broadcasting to say add ``[4, 6]`` along the rows
 to ``arr`` above? Simply performing the direct addition will produce the
 error we previously mentioned:
 
 In[45]:
 
 .. code:: python
 
     c = np.array([4, 6])
     arr + c
 
 ::
 
     ---------------------------------------------------------------------------
     ValueError                                Traceback (most recent call last)
     /home/fperez/teach/book-math-labtool/<ipython-input-45-62aa20ac1980> in <module>()
           1 c = np.array([4, 6])
     ----> 2 arr + c
     
     ValueError: operands could not be broadcast together with shapes (2,4) (2) 
 
 According to the rules above, the array ``c`` would need to have a
 *trailing* dimension of 1 for the broadcasting to work. It turns out
 that numpy allows you to 'inject' new dimensions anywhere into an array
 on the fly, by indexing it with the special object ``np.newaxis``:
 
 In[46]:
 
 .. code:: python
 
     (c[:, np.newaxis]).shape
 
 Out[46]:
 
 .. parsed-literal::
 
     (2, 1)
 
 This is exactly what we need, and indeed it works:
 
 In[47]:
 
 .. code:: python
 
     arr + c[:, np.newaxis]
 
 Out[47]:
 
 .. parsed-literal::
 
     array([[ 4,  5,  6,  7],
            [10, 11, 12, 13]])
 
 For the full broadcasting rules, please see the official Numpy docs,
 which describe them in detail and with more complex examples.
 
 As we mentioned before, Numpy ships with a full complement of
 mathematical functions that work on entire arrays, including logarithms,
 exponentials, trigonometric and hyperbolic trigonometric functions, etc.
 Furthermore, scipy ships a rich special function library in the
 ``scipy.special`` module that includes Bessel, Airy, Fresnel, Laguerre
 and other classical special functions. For example, sampling the sine
 function at 100 points between :math:`0` and :math:`2\pi` is as simple
 as:
 
 In[48]:
 
 .. code:: python
 
     x = np.linspace(0, 2*np.pi, 100)
     y = np.sin(x)
 
 Linear algebra in numpy
 -----------------------
 
 Numpy ships with a basic linear algebra library, and all arrays have a
 ``dot`` method whose behavior is that of the scalar dot product when its
 arguments are vectors (one-dimensional arrays) and the traditional
 matrix multiplication when one or both of its arguments are
 two-dimensional arrays:
 
 In[49]:
 
 .. code:: python
 
     v1 = np.array([2, 3, 4])
     v2 = np.array([1, 0, 1])
     print v1, '.', v2, '=', v1.dot(v2)
 
 .. parsed-literal::
 
     [2 3 4] . [1 0 1] = 6
 
 
 Here is a regular matrix-vector multiplication, note that the array
 ``v1`` should be viewed as a *column* vector in traditional linear
 algebra notation; numpy makes no distinction between row and column
 vectors and simply verifies that the dimensions match the required rules
 of matrix multiplication, in this case we have a :math:`2 \times 3`
 matrix multiplied by a 3-vector, which produces a 2-vector:
 
 In[50]:
 
 .. code:: python
 
     A = np.arange(6).reshape(2, 3)
     print A, 'x', v1, '=', A.dot(v1)
 
 .. parsed-literal::
 
     [[0 1 2]
      [3 4 5]] x [2 3 4] = [11 38]
 
 
 For matrix-matrix multiplication, the same dimension-matching rules must
 be satisfied, e.g. consider the difference between :math:`A \times A^T`:
 
 In[51]:
 
 .. code:: python
 
     print A.dot(A.T)
 
 .. parsed-literal::
 
     [[ 5 14]
      [14 50]]
 
 
 and :math:`A^T \times A`:
 
 In[52]:
 
 .. code:: python
 
     print A.T.dot(A)
 
 .. parsed-literal::
 
     [[ 9 12 15]
      [12 17 22]
      [15 22 29]]
 
 
 Furthermore, the ``numpy.linalg`` module includes additional
 functionality such as determinants, matrix norms, Cholesky, eigenvalue
 and singular value decompositions, etc. For even more linear algebra
 tools, ``scipy.linalg`` contains the majority of the tools in the
 classic LAPACK libraries as well as functions to operate on sparse
 matrices. We refer the reader to the Numpy and Scipy documentations for
 additional details on these.
 
 Reading and writing arrays to disk
 ----------------------------------
 
 Numpy lets you read and write arrays into files in a number of ways. In
 order to use these tools well, it is critical to understand the
 difference between a *text* and a *binary* file containing numerical
 data. In a text file, the number :math:`\pi` could be written as
 "3.141592653589793", for example: a string of digits that a human can
 read, with in this case 15 decimal digits. In contrast, that same number
 written to a binary file would be encoded as 8 characters (bytes) that
 are not readable by a human but which contain the exact same data that
 the variable ``pi`` had in the computer's memory.
 
 The tradeoffs between the two modes are thus:
 
 -  Text mode: occupies more space, precision can be lost (if not all
    digits are written to disk), but is readable and editable by hand
    with a text editor. Can *only* be used for one- and two-dimensional
    arrays.
 
 -  Binary mode: compact and exact representation of the data in memory,
    can't be read or edited by hand. Arrays of any size and
    dimensionality can be saved and read without loss of information.
 
 First, let's see how to read and write arrays in text mode. The
 ``np.savetxt`` function saves an array to a text file, with options to
 control the precision, separators and even adding a header:
 
 In[53]:
 
 .. code:: python
 
     arr = np.arange(10).reshape(2, 5)
     np.savetxt('test.out', arr, fmt='%.2e', header="My dataset")
     !cat test.out
 
 .. parsed-literal::
 
     # My dataset
     0.00e+00 1.00e+00 2.00e+00 3.00e+00 4.00e+00
     5.00e+00 6.00e+00 7.00e+00 8.00e+00 9.00e+00
 
 
 And this same type of file can then be read with the matching
 ``np.loadtxt`` function:
 
 In[54]:
 
 .. code:: python
 
     arr2 = np.loadtxt('test.out')
     print arr2
 
 .. parsed-literal::
 
     [[ 0.  1.  2.  3.  4.]
      [ 5.  6.  7.  8.  9.]]
 
 
 For binary data, Numpy provides the ``np.save`` and ``np.savez``
 routines. The first saves a single array to a file with ``.npy``
 extension, while the latter can be used to save a *group* of arrays into
 a single file with ``.npz`` extension. The files created with these
 routines can then be read with the ``np.load`` function.
 
 Let us first see how to use the simpler ``np.save`` function to save a
 single array:
 
 In[55]:
 
 .. code:: python
 
     np.save('test.npy', arr2)
     # Now we read this back
     arr2n = np.load('test.npy')
     # Let's see if any element is non-zero in the difference.
     # A value of True would be a problem.
     print 'Any differences?', np.any(arr2-arr2n)
 
 .. parsed-literal::
 
     Any differences? False
 
 
 Now let us see how the ``np.savez`` function works. You give it a
 filename and either a sequence of arrays or a set of keywords. In the
 first mode, the function will auotmatically name the saved arrays in the
 archive as ``arr_0``, ``arr_1``, etc:
 
 In[56]:
 
 .. code:: python
 
     np.savez('test.npz', arr, arr2)
     arrays = np.load('test.npz')
     arrays.files
 
 Out[56]:
 
 .. parsed-literal::
 
     ['arr_1', 'arr_0']
 
 Alternatively, we can explicitly choose how to name the arrays we save:
 
 In[57]:
 
 .. code:: python
 
     np.savez('test.npz', array1=arr, array2=arr2)
     arrays = np.load('test.npz')
     arrays.files
 
 Out[57]:
 
 .. parsed-literal::
 
     ['array2', 'array1']
 
 The object returned by ``np.load`` from an ``.npz`` file works like a
 dictionary, though you can also access its constituent files by
 attribute using its special ``.f`` field; this is best illustrated with
 an example with the ``arrays`` object from above:
 
 In[58]:
 
 .. code:: python
 
     print 'First row of first array:', arrays['array1'][0]
     # This is an equivalent way to get the same field
     print 'First row of first array:', arrays.f.array1[0]
 
 .. parsed-literal::
 
     First row of first array: [0 1 2 3 4]
     First row of first array: [0 1 2 3 4]
 
 
 This ``.npz`` format is a very convenient way to package compactly and
 without loss of information, into a single file, a group of related
 arrays that pertain to a specific problem. At some point, however, the
 complexity of your dataset may be such that the optimal approach is to
 use one of the standard formats in scientific data processing that have
 been designed to handle complex datasets, such as NetCDF or HDF5.
 
 Fortunately, there are tools for manipulating these formats in Python,
 and for storing data in other ways such as databases. A complete
 discussion of the possibilities is beyond the scope of this discussion,
 but of particular interest for scientific users we at least mention the
 following:
 
 -  The ``scipy.io`` module contains routines to read and write Matlab
    files in ``.mat`` format and files in the NetCDF format that is
    widely used in certain scientific disciplines.
 
 -  For manipulating files in the HDF5 format, there are two excellent
    options in Python: The PyTables project offers a high-level, object
    oriented approach to manipulating HDF5 datasets, while the h5py
    project offers a more direct mapping to the standard HDF5 library
    interface. Both are excellent tools; if you need to work with HDF5
    datasets you should read some of their documentation and examples and
    decide which approach is a better match for your needs.
 
 
 
 High quality data visualization with Matplotlib
 ===============================================
 
 The `matplotlib <http://matplotlib.sf.net>`_ library is a powerful tool
 capable of producing complex publication-quality figures with fine
 layout control in two and three dimensions; here we will only provide a
 minimal self-contained introduction to its usage that covers the
 functionality needed for the rest of the book. We encourage the reader
 to read the tutorials included with the matplotlib documentation as well
 as to browse its extensive gallery of examples that include source code.
 
 Just as we typically use the shorthand ``np`` for Numpy, we will use
 ``plt`` for the ``matplotlib.pyplot`` module where the easy-to-use
 plotting functions reside (the library contains a rich object-oriented
 architecture that we don't have the space to discuss here):
 
 In[59]:
 
 .. code:: python
 
     import matplotlib.pyplot as plt
 
 The most frequently used function is simply called ``plot``, here is how
 you can make a simple plot of :math:`\sin(x)` for
 :math:`x \in [0, 2\pi]` with labels and a grid (we use the semicolon in
 the last line to suppress the display of some information that is
 unnecessary right now):
 
 In[60]:
 
 .. code:: python
 
     x = np.linspace(0, 2*np.pi)
     y = np.sin(x)
     plt.plot(x,y, label='sin(x)')
     plt.legend()
     plt.grid()
     plt.title('Harmonic')
     plt.xlabel('x')
     plt.ylabel('y');
 
-.. image:: tests/ipynbref/IntroNumPy.orig_files/IntroNumPy.orig_fig_01.svg
+.. image:: tests/ipynbref/IntroNumPy_orig_files/IntroNumPy_orig_fig_01.svg
 
 You can control the style, color and other properties of the markers,
 for example:
 
 In[61]:
 
 .. code:: python
 
     plt.plot(x, y, linewidth=2);
 
-.. image:: tests/ipynbref/IntroNumPy.orig_files/IntroNumPy.orig_fig_02.svg
+.. image:: tests/ipynbref/IntroNumPy_orig_files/IntroNumPy_orig_fig_02.svg
 
 In[62]:
 
 .. code:: python
 
     plt.plot(x, y, 'o', markersize=5, color='r');
 
-.. image:: tests/ipynbref/IntroNumPy.orig_files/IntroNumPy.orig_fig_03.svg
+.. image:: tests/ipynbref/IntroNumPy_orig_files/IntroNumPy_orig_fig_03.svg
 
 We will now see how to create a few other common plot types, such as a
 simple error plot:
 
 In[63]:
 
 .. code:: python
 
     # example data
     x = np.arange(0.1, 4, 0.5)
     y = np.exp(-x)
     
     # example variable error bar values
     yerr = 0.1 + 0.2*np.sqrt(x)
     xerr = 0.1 + yerr
     
     # First illustrate basic pyplot interface, using defaults where possible.
     plt.figure()
     plt.errorbar(x, y, xerr=0.2, yerr=0.4)
     plt.title("Simplest errorbars, 0.2 in x, 0.4 in y");
 
-.. image:: tests/ipynbref/IntroNumPy.orig_files/IntroNumPy.orig_fig_04.svg
+.. image:: tests/ipynbref/IntroNumPy_orig_files/IntroNumPy_orig_fig_04.svg
 
 A simple log plot
 
 In[64]:
 
 .. code:: python
 
     x = np.linspace(-5, 5)
     y = np.exp(-x**2)
     plt.semilogy(x, y);
 
-.. image:: tests/ipynbref/IntroNumPy.orig_files/IntroNumPy.orig_fig_05.svg
+.. image:: tests/ipynbref/IntroNumPy_orig_files/IntroNumPy_orig_fig_05.svg
 
 A histogram annotated with text inside the plot, using the ``text``
 function:
 
 In[65]:
 
 .. code:: python
 
     mu, sigma = 100, 15
     x = mu + sigma * np.random.randn(10000)
     
     # the histogram of the data
     n, bins, patches = plt.hist(x, 50, normed=1, facecolor='g', alpha=0.75)
     
     plt.xlabel('Smarts')
     plt.ylabel('Probability')
     plt.title('Histogram of IQ')
     # This will put a text fragment at the position given:
     plt.text(55, .027, r'$\mu=100,\ \sigma=15$', fontsize=14)
     plt.axis([40, 160, 0, 0.03])
     plt.grid(True)
 
-.. image:: tests/ipynbref/IntroNumPy.orig_files/IntroNumPy.orig_fig_06.svg
+.. image:: tests/ipynbref/IntroNumPy_orig_files/IntroNumPy_orig_fig_06.svg
 
 Image display
 -------------
 
 The ``imshow`` command can display single or multi-channel images. A
 simple array of random numbers, plotted in grayscale:
 
 In[66]:
 
 .. code:: python
 
     from matplotlib import cm
     plt.imshow(np.random.rand(5, 10), cmap=cm.gray, interpolation='nearest');
 
-.. image:: tests/ipynbref/IntroNumPy.orig_files/IntroNumPy.orig_fig_07.svg
+.. image:: tests/ipynbref/IntroNumPy_orig_files/IntroNumPy_orig_fig_07.svg
 
 A real photograph is a multichannel image, ``imshow`` interprets it
 correctly:
 
 In[67]:
 
 .. code:: python
 
     img = plt.imread('stinkbug.png')
     print 'Dimensions of the array img:', img.shape
     plt.imshow(img);
 
 .. parsed-literal::
 
     Dimensions of the array img: (375, 500, 3)
 
 
-.. image:: tests/ipynbref/IntroNumPy.orig_files/IntroNumPy.orig_fig_08.svg
+.. image:: tests/ipynbref/IntroNumPy_orig_files/IntroNumPy_orig_fig_08.svg
 
 Simple 3d plotting with matplotlib
 ----------------------------------
 
 Note that you must execute at least once in your session:
 
 In[68]:
 
 .. code:: python
 
     from mpl_toolkits.mplot3d import Axes3D
 
 One this has been done, you can create 3d axes with the
 ``projection='3d'`` keyword to ``add_subplot``:
 
 ::
 
     fig = plt.figure()
     fig.add_subplot(<other arguments here>, projection='3d')
 
 
 A simple surface plot:
 
 In[72]:
 
 .. code:: python
 
     from mpl_toolkits.mplot3d.axes3d import Axes3D
     from matplotlib import cm
     
     fig = plt.figure()
     ax = fig.add_subplot(1, 1, 1, projection='3d')
     X = np.arange(-5, 5, 0.25)
     Y = np.arange(-5, 5, 0.25)
     X, Y = np.meshgrid(X, Y)
     R = np.sqrt(X**2 + Y**2)
     Z = np.sin(R)
     surf = ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=cm.jet,
             linewidth=0, antialiased=False)
     ax.set_zlim3d(-1.01, 1.01);
 
-.. image:: tests/ipynbref/IntroNumPy.orig_files/IntroNumPy.orig_fig_09.svg
+.. image:: tests/ipynbref/IntroNumPy_orig_files/IntroNumPy_orig_fig_09.svg
 
 IPython: a powerful interactive environment
 ===========================================
 
 A key component of the everyday workflow of most scientific computing
 environments is a good interactive environment, that is, a system in
 which you can execute small amounts of code and view the results
 immediately, combining both printing out data and opening graphical
 visualizations. All modern systems for scientific computing, commercial
 and open source, include such functionality.
 
 Out of the box, Python also offers a simple interactive shell with very
 limited capabilities. But just like the scientific community built Numpy
 to provide arrays suited for scientific work (since Pytyhon's lists
 aren't optimal for this task), it has also developed an interactive
 environment much more sophisticated than the built-in one. The `IPython
 project <http://ipython.org>`_ offers a set of tools to make productive
 use of the Python language, all the while working interactively and with
 immedate feedback on your results. The basic tools that IPython provides
 are:
 
 1. A powerful terminal shell, with many features designed to increase
    the fluidity and productivity of everyday scientific workflows,
    including:
 
    -  rich introspection of all objects and variables including easy
       access to the source code of any function
    -  powerful and extensible tab completion of variables and filenames,
    -  tight integration with matplotlib, supporting interactive figures
       that don't block the terminal,
    -  direct access to the filesystem and underlying operating system,
    -  an extensible system for shell-like commands called 'magics' that
       reduce the work needed to perform many common tasks,
    -  tools for easily running, timing, profiling and debugging your
       codes,
    -  syntax highlighted error messages with much more detail than the
       default Python ones,
    -  logging and access to all previous history of inputs, including
       across sessions
 
 2. A Qt console that provides the look and feel of a terminal, but adds
    support for inline figures, graphical calltips, a persistent session
    that can survive crashes (even segfaults) of the kernel process, and
    more.
 
 3. A web-based notebook that can execute code and also contain rich text
    and figures, mathematical equations and arbitrary HTML. This notebook
    presents a document-like view with cells where code is executed but
    that can be edited in-place, reordered, mixed with explanatory text
    and figures, etc.
 
 4. A high-performance, low-latency system for parallel computing that
    supports the control of a cluster of IPython engines communicating
    over a network, with optimizations that minimize unnecessary copying
    of large objects (especially numpy arrays).
 
 We will now discuss the highlights of the tools 1-3 above so that you
 can make them an effective part of your workflow. The topic of parallel
 computing is beyond the scope of this document, but we encourage you to
 read the extensive
 `documentation <http://ipython.org/ipython-doc/rel-0.12.1/parallel/index.html>`_
 and `tutorials <http://minrk.github.com/scipy-tutorial-2011/>`_ on this
 available on the IPython website.
 
 The IPython terminal
 --------------------
 
 You can start IPython at the terminal simply by typing:
 
 ::
 
     $ ipython
 
 which will provide you some basic information about how to get started
 and will then open a prompt labeled ``In [1]:`` for you to start typing.
 Here we type :math:`2^{64}` and Python computes the result for us in
 exact arithmetic, returning it as ``Out[1]``:
 
 ::
 
     $ ipython
     Python 2.7.2+ (default, Oct  4 2011, 20:03:08) 
     Type "copyright", "credits" or "license" for more information.
 
     IPython 0.13.dev -- An enhanced Interactive Python.
     ?         -> Introduction and overview of IPython's features.
     %quickref -> Quick reference.
     help      -> Python's own help system.
     object?   -> Details about 'object', use 'object??' for extra details.
 
     In [1]: 2**64
     Out[1]: 18446744073709551616L
 
 The first thing you should know about IPython is that all your inputs
 and outputs are saved. There are two variables named ``In`` and ``Out``
 which are filled as you work with your results. Furthermore, all outputs
 are also saved to auto-created variables of the form ``_NN`` where
 ``NN`` is the prompt number, and inputs to ``_iNN``. This allows you to
 recover quickly the result of a prior computation by referring to its
 number even if you forgot to store it as a variable. For example, later
 on in the above session you can do:
 
 ::
 
     In [6]: print _1
     18446744073709551616
 
 
 We strongly recommend that you take a few minutes to read at least the
 basic introduction provided by the ``?`` command, and keep in mind that
 the ``%quickref`` command at all times can be used as a quick reference
 "cheat sheet" of the most frequently used features of IPython.
 
 At the IPython prompt, any valid Python code that you type will be
 executed similarly to the default Python shell (though often with more
 informative feedback). But since IPython is a *superset* of the default
 Python shell; let's have a brief look at some of its additional
 functionality.
 
 **Object introspection**
 
 A simple ``?`` command provides a general introduction to IPython, but
 as indicated in the banner above, you can use the ``?`` syntax to ask
 for details about any object. For example, if we type ``_1?``, IPython
 will print the following details about this variable:
 
 ::
 
     In [14]: _1?
     Type:       long
     Base Class: <type 'long'>
     String Form:18446744073709551616
     Namespace:  Interactive
     Docstring:
     long(x[, base]) -> integer
 
     Convert a string or number to a long integer, if possible.  A floating
 
     [etc... snipped for brevity]
 
 If you add a second ``?`` and for any oobject ``x`` type ``x??``,
 IPython will try to provide an even more detailed analsysi of the
 object, including its syntax-highlighted source code when it can be
 found. It's possible that ``x??`` returns the same information as
 ``x?``, but in many cases ``x??`` will indeed provide additional
 details.
 
 Finally, the ``?`` syntax is also useful to search *namespaces* with
 wildcards. Suppose you are wondering if there is any function in Numpy
 that may do text-related things; with ``np.*txt*?``, IPython will print
 all the names in the ``np`` namespace (our Numpy shorthand) that have
 'txt' anywhere in their name:
 
 ::
 
     In [17]: np.*txt*?
     np.genfromtxt
     np.loadtxt
     np.mafromtxt
     np.ndfromtxt
     np.recfromtxt
     np.savetxt
 
 
 **Tab completion**
 
 IPython makes the tab key work extra hard for you as a way to rapidly
 inspect objects and libraries. Whenever you have typed something at the
 prompt, by hitting the ``<tab>`` key IPython will try to complete the
 rest of the line. For this, IPython will analyze the text you had so far
 and try to search for Python data or files that may match the context
 you have already provided.
 
 For example, if you type ``np.load`` and hit the key, you'll see:
 
 ::
 
     In [21]: np.load<TAB HERE>
     np.load     np.loads    np.loadtxt  
 
 so you can quickly find all the load-related functionality in numpy. Tab
 completion works even for function arguments, for example consider this
 function definition:
 
 ::
 
     In [20]: def f(x, frobinate=False):
        ....:     if frobinate:
        ....:         return x**2
        ....:     
 
 If you now use the ``<tab>`` key after having typed 'fro' you'll get all
 valid Python completions, but those marked with ``=`` at the end are
 known to be keywords of your function:
 
 ::
 
     In [21]: f(2, fro<TAB HERE>
     frobinate=    frombuffer    fromfunction  frompyfunc    fromstring    
     from          fromfile      fromiter      fromregex     frozenset     
 
 at this point you can add the ``b`` letter and hit ``<tab>`` once more,
 and IPython will finish the line for you:
 
 ::
 
     In [21]: f(2, frobinate=
 
 As a beginner, simply get into the habit of using ``<tab>`` after most
 objects; it should quickly become second nature as you will see how
 helps keep a fluid workflow and discover useful information. Later on
 you can also customize this behavior by writing your own completion
 code, if you so desire.
 
 **Matplotlib integration**
 
 One of the most useful features of IPython for scientists is its tight
 integration with matplotlib: at the terminal IPython lets you open
 matplotlib figures without blocking your typing (which is what happens
 if you try to do the same thing at the default Python shell), and in the
 Qt console and notebook you can even view your figures embedded in your
 workspace next to the code that created them.
 
 The matplotlib support can be either activated when you start IPython by
 passing the ``--pylab`` flag, or at any point later in your session by
 using the ``%pylab`` command. If you start IPython with ``--pylab``,
 you'll see something like this (note the extra message about pylab):
 
 ::
 
     $ ipython --pylab
     Python 2.7.2+ (default, Oct  4 2011, 20:03:08) 
     Type "copyright", "credits" or "license" for more information.
 
     IPython 0.13.dev -- An enhanced Interactive Python.
     ?         -> Introduction and overview of IPython's features.
     %quickref -> Quick reference.
     help      -> Python's own help system.
     object?   -> Details about 'object', use 'object??' for extra details.
 
     Welcome to pylab, a matplotlib-based Python environment [backend: Qt4Agg].
     For more information, type 'help(pylab)'.
 
     In [1]: 
 
 Furthermore, IPython will import ``numpy`` with the ``np`` shorthand,
 ``matplotlib.pyplot`` as ``plt``, and it will also load all of the numpy
 and pyplot top-level names so that you can directly type something like:
 
 ::
 
     In [1]: x = linspace(0, 2*pi, 200)
 
     In [2]: plot(x, sin(x))
     Out[2]: [<matplotlib.lines.Line2D at 0x9e7c16c>]
 
 instead of having to prefix each call with its full signature (as we
 have been doing in the examples thus far):
 
 ::
 
     In [3]: x = np.linspace(0, 2*np.pi, 200)
 
     In [4]: plt.plot(x, np.sin(x))
     Out[4]: [<matplotlib.lines.Line2D at 0x9e900ac>]
 
 This shorthand notation can be a huge time-saver when working
 interactively (it's a few characters but you are likely to type them
 hundreds of times in a session). But we should note that as you develop
 persistent scripts and notebooks meant for reuse, it's best to get in
 the habit of using the longer notation (known as *fully qualified names*
 as it's clearer where things come from and it makes for more robust,
 readable and maintainable code in the long run).
 
 **Access to the operating system and files**
 
 In IPython, you can type ``ls`` to see your files or ``cd`` to change
 directories, just like you would at a regular system prompt:
 
 ::
 
     In [2]: cd tests
     /home/fperez/ipython/nbconvert/tests
 
     In [3]: ls test.*
     test.aux  test.html  test.ipynb  test.log  test.out  test.pdf  test.rst  test.tex
 
 Furthermore, if you use the ``!`` at the beginning of a line, any
 commands you pass afterwards go directly to the operating system:
 
 ::
 
     In [4]: !echo "Hello IPython"
     Hello IPython
 
 IPython offers a useful twist in this feature: it will substitute in the
 command the value of any *Python* variable you may have if you prepend
 it with a ``$`` sign:
 
 ::
 
     In [5]: message = 'IPython interpolates from Python to the shell'
 
     In [6]: !echo $message
     IPython interpolates from Python to the shell
 
 This feature can be extremely useful, as it lets you combine the power
 and clarity of Python for complex logic with the immediacy and
 familiarity of many shell commands. Additionally, if you start the line
 with *two* ``$$`` signs, the output of the command will be automatically
 captured as a list of lines, e.g.:
 
 ::
 
     In [10]: !!ls test.*
     Out[10]: 
     ['test.aux',
      'test.html',
      'test.ipynb',
      'test.log',
      'test.out',
      'test.pdf',
      'test.rst',
      'test.tex']
 
 As explained above, you can now use this as the variable ``_10``. If you
 directly want to capture the output of a system command to a Python
 variable, you can use the syntax ``=!``:
 
 ::
 
     In [11]: testfiles =! ls test.*
 
     In [12]: print testfiles
     ['test.aux', 'test.html', 'test.ipynb', 'test.log', 'test.out', 'test.pdf', 'test.rst', 'test.tex']
 
 Finally, the special ``%alias`` command lets you define names that are
 shorthands for system commands, so that you can type them without having
 to prefix them via ``!`` explicitly (for example, ``ls`` is an alias
 that has been predefined for you at startup).
 
 **Magic commands**
 
 IPython has a system for special commands, called 'magics', that let you
 control IPython itself and perform many common tasks with a more
 shell-like syntax: it uses spaces for delimiting arguments, flags can be
 set with dashes and all arguments are treated as strings, so no
 additional quoting is required. This kind of syntax is invalid in the
 Python language but very convenient for interactive typing (less
 parentheses, commans and quoting everywhere); IPython distinguishes the
 two by detecting lines that start with the ``%`` character.
 
 You can learn more about the magic system by simply typing ``%magic`` at
 the prompt, which will give you a short description plus the
 documentation on *all* available magics. If you want to see only a
 listing of existing magics, you can use ``%lsmagic``:
 
 ::
 
     In [4]: lsmagic
     Available magic functions:
     %alias  %autocall  %autoindent  %automagic  %bookmark  %c  %cd  %colors  %config  %cpaste
     %debug  %dhist  %dirs  %doctest_mode  %ds  %ed  %edit  %env  %gui  %hist  %history
     %install_default_config  %install_ext  %install_profiles  %load_ext  %loadpy  %logoff  %logon  
     %logstart  %logstate  %logstop  %lsmagic  %macro  %magic  %notebook  %page  %paste  %pastebin  
     %pd  %pdb  %pdef  %pdoc  %pfile  %pinfo  %pinfo2  %pop  %popd  %pprint  %precision  %profile  
     %prun  %psearch  %psource  %pushd  %pwd  %pycat  %pylab  %quickref  %recall  %rehashx  
     %reload_ext  %rep  %rerun  %reset  %reset_selective  %run  %save  %sc  %stop  %store  %sx  %tb
     %time  %timeit  %unalias  %unload_ext  %who  %who_ls  %whos  %xdel  %xmode
 
     Automagic is ON, % prefix NOT needed for magic functions.
 
 Note how the example above omitted the eplicit ``%`` marker and simply
 uses ``lsmagic``. As long as the 'automagic' feature is on (which it is
 by default), you can omit the ``%`` marker as long as there is no
 ambiguity with a Python variable of the same name.
 
 **Running your code**
 
 While it's easy to type a few lines of code in IPython, for any
 long-lived work you should keep your codes in Python scripts (or in
 IPython notebooks, see below). Consider that you have a script, in this
 case trivially simple for the sake of brevity, named ``simple.py``:
 
 ::
 
     In [12]: !cat simple.py
     import numpy as np
 
     x = np.random.normal(size=100)
 
     print 'First elment of x:', x[0]
 
 The typical workflow with IPython is to use the ``%run`` magic to
 execute your script (you can omit the .py extension if you want). When
 you run it, the script will execute just as if it had been run at the
 system prompt with ``python simple.py`` (though since modules don't get
 re-executed on new imports by Python, all system initialization is
 essentially free, which can have a significant run time impact in some
 cases):
 
 ::
 
     In [13]: run simple
     First elment of x: -1.55872256289
 
 Once it completes, all variables defined in it become available for you
 to use interactively:
 
 ::
 
     In [14]: x.shape
     Out[14]: (100,)
 
 This allows you to plot data, try out ideas, etc, in a
 ``%run``/interact/edit cycle that can be very productive. As you start
 understanding your problem better you can refine your script further,
 incrementally improving it based on the work you do at the IPython
 prompt. At any point you can use the ``%hist`` magic to print out your
 history without prompts, so that you can copy useful fragments back into
 the script.
 
 By default, ``%run`` executes scripts in a completely empty namespace,
 to better mimic how they would execute at the system prompt with plain
 Python. But if you use the ``-i`` flag, the script will also see your
 interactively defined variables. This lets you edit in a script larger
 amounts of code that still behave as if you had typed them at the
 IPython prompt.
 
 You can also get a summary of the time taken by your script with the
 ``-t`` flag; consider a different script ``randsvd.py`` that takes a bit
 longer to run:
 
 ::
 
     In [21]: run -t randsvd.py
 
     IPython CPU timings (estimated):
       User   :       0.38 s.
       System :       0.04 s.
     Wall time:       0.34 s.
 
 ``User`` is the time spent by the computer executing your code, while
 ``System`` is the time the operating system had to work on your behalf,
 doing things like memory allocation that are needed by your code but
 that you didn't explicitly program and that happen inside the kernel.
 The ``Wall time`` is the time on a 'clock on the wall' between the start
 and end of your program.
 
 If ``Wall > User+System``, your code is most likely waiting idle for
 certain periods. That could be waiting for data to arrive from a remote
 source or perhaps because the operating system has to swap large amounts
 of virtual memory. If you know that your code doesn't explicitly wait
 for remote data to arrive, you should investigate further to identify
 possible ways of improving the performance profile.
 
 If you only want to time how long a single statement takes, you don't
 need to put it into a script as you can use the ``%timeit`` magic, which
 uses Python's ``timeit`` module to very carefully measure timig data;
 ``timeit`` can measure even short statements that execute extremely
 fast:
 
 ::
 
     In [27]: %timeit a=1
     10000000 loops, best of 3: 23 ns per loop
 
 and for code that runs longer, it automatically adjusts so the overall
 measurement doesn't take too long:
 
 ::
 
     In [28]: %timeit np.linalg.svd(x)
     1 loops, best of 3: 310 ms per loop
 
 The ``%run`` magic still has more options for debugging and profiling
 data; you should read its documentation for many useful details (as
 always, just type ``%run?``).
 
 The graphical Qt console
 ------------------------
 
 If you type at the system prompt (see the IPython website for
 installation details, as this requires some additional libraries):
 
 ::
 
     $ ipython qtconsole
 
 instead of opening in a terminal as before, IPython will start a
 graphical console that at first sight appears just like a terminal, but
 which is in fact much more capable than a text-only terminal. This is a
 specialized terminal designed for interactive scientific work, and it
 supports full multi-line editing with color highlighting and graphical
 calltips for functions, it can keep multiple IPython sessions open
 simultaneously in tabs, and when scripts run it can display the figures
 inline directly in the work area.
 
 .. raw:: html
 
    <center>
 
 .. raw:: html
 
    </center>
 
 
 % This cell is for the pdflatex output only
 \begin{figure}[htbp]
 \centering
 \includegraphics[width=3in]{ipython_qtconsole2.png}
 \caption{The IPython Qt console: a lightweight terminal for scientific exploration, with code, results and graphics in a soingle environment.}
 \end{figure}
 The Qt console accepts the same ``--pylab`` startup flags as the
 terminal, but you can additionally supply the value ``--pylab inline``,
 which enables the support for inline graphics shown in the figure. This
 is ideal for keeping all the code and figures in the same session, given
 that the console can save the output of your entire session to HTML or
 PDF.
 
 Since the Qt console makes it far more convenient than the terminal to
 edit blocks of code with multiple lines, in this environment it's worth
 knowing about the ``%loadpy`` magic function. ``%loadpy`` takes a path
 to a local file or remote URL, fetches its contents, and puts it in the
 work area for you to further edit and execute. It can be an extremely
 fast and convenient way of loading code from local disk or remote
 examples from sites such as the `Matplotlib
 gallery <http://matplotlib.sourceforge.net/gallery.html>`_.
 
 Other than its enhanced capabilities for code and graphics, all of the
 features of IPython we've explained before remain functional in this
 graphical console.
 
 The IPython Notebook
 --------------------
 
 The third way to interact with IPython, in addition to the terminal and
 graphical Qt console, is a powerful web interface called the "IPython
 Notebook". If you run at the system console (you can omit the ``pylab``
 flags if you don't need plotting support):
 
 ::
 
     $ ipython notebook --pylab inline
 
 IPython will start a process that runs a web server in your local
 machine and to which a web browser can connect. The Notebook is a
 workspace that lets you execute code in blocks called 'cells' and
 displays any results and figures, but which can also contain arbitrary
 text (including LaTeX-formatted mathematical expressions) and any rich
 media that a modern web browser is capable of displaying.
 
 .. raw:: html
 
    <center>
 
 .. raw:: html
 
    </center>
 
 
 % This cell is for the pdflatex output only
 \begin{figure}[htbp]
 \centering
 \includegraphics[width=3in]{ipython-notebook-specgram-2.png}
 \caption{The IPython Notebook: text, equations, code, results, graphics and other multimedia in an open format for scientific exploration and collaboration}
 \end{figure}
 In fact, this document was written as a Notebook, and only exported to
 LaTeX for printing. Inside of each cell, all the features of IPython
 that we have discussed before remain functional, since ultimately this
 web client is communicating with the same IPython code that runs in the
 terminal. But this interface is a much more rich and powerful
 environment for maintaining long-term "live and executable" scientific
 documents.
 
 Notebook environments have existed in commercial systems like
 Mathematica(TM) and Maple(TM) for a long time; in the open source world
 the `Sage <http://sagemath.org>`_ project blazed this particular trail
 starting in 2006, and now we bring all the features that have made
 IPython such a widely used tool to a Notebook model.
 
 Since the Notebook runs as a web application, it is possible to
 configure it for remote access, letting you run your computations on a
 persistent server close to your data, which you can then access remotely
 from any browser-equipped computer. We encourage you to read the
 extensive documentation provided by the IPython project for details on
 how to do this and many more features of the notebook.
 
 Finally, as we said earlier, IPython also has a high-level and easy to
 use set of libraries for parallel computing, that let you control
 (interactively if desired) not just one IPython but an entire cluster of
 'IPython engines'. Unfortunately a detailed discussion of these tools is
 beyond the scope of this text, but should you need to parallelize your
 analysis codes, a quick read of the tutorials and examples provided at
 the IPython site may prove fruitful.