doc/python/static-persistence.rst
 author Dmitriy Morozov Fri, 11 May 2012 10:25:57 -0700 branch dev changeset 250 021030a8f97c parent 233 ce6d679368c6 permissions -rw-r--r--
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:class:`StaticPersistence` class
================================

.. class:: StaticPersistence

.. method:: __init__(filtration)

Initializes :class:`StaticPersistence` with the given
:class:`Filtration`. This operation effectively computes the boundary
matrix of the complex captured by the filtration with rows and columns
sorted with respect to the filtration ordering.

.. method:: pair_simplices(store_negative = False)

Pairs simplices using the [ELZ02]_ algorithm. `store_negative` indicates
whether to store the negative simplices in the cycles.

.. method:: __call__(i)

Given an SPNode in the internal representation, the method returns its
integer offset from the beginning of the filtration. This is useful to
lookup the actual name of the simplex in the complex.

.. method:: make_simplex_map(filtration)

Creates an auxilliary :class:`PersistenceSimplexMap` used to lookup the actual
simplices from the persistence indices. For example, the following
snippet prints out all the unpaired simplices::

smap = persistence.make_simplex_map(filtration)
for i in persistence:
if i.unpaired(): print smap[i]

.. method:: __iter__()

Iterator over the nodes (representing individual simplices). See
:class:`SPNode`.

.. method:: __len__()

Returns the number of nodes (i.e. the number of simplices).

.. class:: SPNode

The class represents nodes stored in :class:`StaticPersistence`. These nodes
are aware of their :meth:`sign` and :attr:`pair` (and :meth:`cycle` if
negative after :meth:`StaticPersistence.pair_simplices` has run).

.. method:: sign()

Returns the sign of the simplex: `True` for positive, `False` for
negative.

.. method:: pair()

Simplex's pair. The pair is set to self if the siplex is unpaired.

.. attribute:: cycle

If the simplex is negative, its cycle (that it kills) is non-empty, and
can be accessed using this method. The cycle itself is an iterable
container of :class:`SPNode`. For example, one can print the basis for
the (bounding) cycles::

smap = persistence.make_simplex_map(filtration)
for i in persistence:
for ii in i.cycle: print smap[ii]

.. method:: unpaired()

Indicates whether the simplex is unpaired.

.. class:: SPersistenceSimplexMap

.. method:: __getitem__(i)

Given a persistence index, i.e. an :class:`SPNode`, returns the
:class:`Simplex` it represents.

:class:`DynamicPersistenceChains` class
=======================================

.. class:: DynamicPersistenceChains

This class works exactly like :class:`StaticPersistence`, providing all the
same methods. The only difference is that when iterating over it, the
elements are of type :class:`DPCNode`, described below.

.. class:: DPCNode

This class works just like :class:`SPNode`, except it has an additional
attribute :attr:`chain`.

.. attribute:: chain

It allows one to retrieve the "chain" associated with the simplex.
(In terms of the :math:`R = DV` decomposition, it gives access to the
columns of the matrix :math:`V`.) In case of the positive simplex, this
is a cycle created by the addition of this simplex.  This access is
particularly useful for the unpaired positive simplices, allowing one to
recover the cycles they create. In case of the negative simplex, this chain's
boundary is exactly what's stored in the :attr:`~SPNode.cycle` attribute.

For example, to print out all the essential cycles of the complex, one
can run the following loop::

smap = persistence.make_simplex_map(filtration)
for i in persistence:
if i.unpaired()
for ii in i.chain: print smap[ii]

```