<<TableOfContents>>
This tutorial is aimed at NumPy users who have no experience with Cython at all. If you have some knowledge of Cython you may want to skip to the Efficient indexing section which explains the new improvements made in summer 2008.
For passing (the data in) a numpy array to C or C++ code, see: tutorials/NumpyPointerToC
The main scenario considered is NumPy end-use rather than NumPy/SciPy development. The reason is that Cython is not (yet) able to support functions that are generic with respect to datatype and the number of dimensions in a high-level fashion. This restriction is much more severe for SciPy development than more specific, "end-user" functions. See the last section for more information on this.
The style of this tutorial will not fit everybody, so you can also consider:
Note: The fast array access documented below is a completely new feature, and there may be bugs waiting to be discovered. It might be a good idea to do a manual sanity check on the C code Cython generates before using this for serious purposes, at least until some months have passed.
Cython is a compiler which compiles Python-like code files to C code. Still, Cython is not a Python to C translator. That is, it doesn't take your full program and "turns it into C" -- rather, the result makes full use of the Python runtime environment. A way of looking at it may be that your code is still Python in that it runs within the Python runtime environment, but rather than compiling to interpreted Python bytecode one compiles to native machine code (but with the addition of extra syntax for easy embedding of faster C-like code).
This has two important consequences:
Some Python constructs are not yet supported, though making Cython compile all Python code is a stated goal (among the more important omissions are inner functions and generator functions).
Using Cython consists of these steps:
.pyx source fileHowever, there are several options to automate these steps:
Note: If using another interactive command line environment than SAGE, like IPython or Python itself, it is important that you restart the process when you recompile the module. It is not enough to issue an "import" statement again.
Unless you are used to some other automatic method: download Cython (0.9.8.1.1 or later), unpack it, and run the usual python setup.py install. This will install a cython executable on your system. It is also possible to use Cython from the source directory without installing (simply launch cython.py in the root directory).
As of this writing, SAGE comes with an older release of Cython than required for this tutorial. So if using SAGE you should download the newest Cython and then execute
$ cd path/to/cython-distro $ path-to-sage/sage -python setup.py install
This will install the newest Cython into SAGE.
As it is always important to know what is going on, I'll describe the manual method here. First Cython is run:
This creates yourmod.c which is the C source for a Python extension module. A useful additional switch is -a which will generate a document (yourmod.html) that shows which Cython code translates to which C code line by line.
Then we compile the C file. This may vary according to your system, but the C file should be built like Python was built. Python documentation for writing extensions should have some details. On Linux, this often means something like
$ gcc -shared -pthread -fPIC -fwrapv -O2 -Wall -fno-strict-aliasing -I/usr/include/python2.5 -o yourmod.so yourmod.c
gcc should have access to the NumPy C header files so if they are not installed at /usr/include/numpy or similar you may need to pass another option for those.
This creates yourmod.so in the same directory, which is importable by Python by using a normal import yourmod statement.
The code below does 2D discrete convolution of an image with a filter (and I'm sure you can do better! let it serve for demonstration purposes). It is both valid Python and valid Cython code. I'll refer to it as both convolve_py.py for the Python version and convolve1.pyx for the Cython version -- Cython uses ".pyx" as its file suffix.
#!python
from __future__ import division
import numpy as np
def naive_convolve(f, g):
# f is an image and is indexed by (v, w)
# g is a filter kernel and is indexed by (s, t),
# it needs odd dimensions
# h is the output image and is indexed by (x, y),
# it is not cropped
if g.shape[0] % 2 != 1 or g.shape[1] % 2 != 1:
raise ValueError("Only odd dimensions on filter supported")
# smid and tmid are number of pixels between the center pixel
# and the edge, ie for a 5x5 filter they will be 2.
#
# The output size is calculated by adding smid, tmid to each
# side of the dimensions of the input image.
vmax = f.shape[0]
wmax = f.shape[1]
smax = g.shape[0]
tmax = g.shape[1]
smid = smax // 2
tmid = tmax // 2
xmax = vmax + 2*smid
ymax = wmax + 2*tmid
# Allocate result image.
h = np.zeros([xmax, ymax], dtype=f.dtype)
# Do convolution
for x in range(xmax):
for y in range(ymax):
# Calculate pixel value for h at (x,y). Sum one component
# for each pixel (s, t) of the filter g.
s_from = max(smid - x, -smid)
s_to = min((xmax - x) - smid, smid + 1)
t_from = max(tmid - y, -tmid)
t_to = min((ymax - y) - tmid, tmid + 1)
value = 0
for s in range(s_from, s_to):
for t in range(t_from, t_to):
v = x - smid + s
w = y - tmid + t
value += g[smid - s, tmid - t] * f[v, w]
h[x, y] = value
return h
This should be compiled to produce yourmod.so (for Linux systems). We run a Python session to test both the Python version (imported from .py-file) and the compiled Cython module.
#!python
In[1]: import numpy as np
In[2]: import convolve_py
In[3]: convolve_py.naive_convolve(np.array([[1, 1, 1]], dtype=np.int), \
... np.array([[1],[2],[1]], dtype=np.int))
Out[3]:
array([[1, 1, 1],
[2, 2, 2],
[1, 1, 1]])
In[4]: import convolve1
In[4]: convolve1.naive_convolve(np.array([[1, 1, 1]], dtype=np.int), \
... np.array([[1],[2],[1]], dtype=np.int))
Out[4]:
array([[1, 1, 1],
[2, 2, 2],
[1, 1, 1]])
In [11]: N = 100
In [12]: f = np.arange(N*N, dtype=np.int).reshape((N,N))
In [13]: g = np.arange(81, dtype=np.int).reshape((9, 9))
In [19]: %timeit -n2 -r3 convolve_py.naive_convolve(f, g)
2 loops, best of 3: 1.86 s per loop
In [20]: %timeit -n2 -r3 convolve1.naive_convolve(f, g)
2 loops, best of 3: 1.41 s per loop
There's not such a huge difference yet; because the C code still does exactly what the Python interpreter does (meaning, for instance, that a new object is allocated for each number used). Look at the generated html file and see what is needed for even the simplest statements you get the point quickly. We need to give Cython more information; we need to add types.
To add types we use custom Cython syntax, so we are now breaking Python source compatibility. Here's convolve2.pyx. Read the comments!
#!python
from __future__ import division
import numpy as np
# "cimport" is used to import special compile-time information
# about the numpy module (this is stored in a file numpy.pxd which is
# currently part of the Cython distribution).
cimport numpy as np
# We now need to fix a datatype for our arrays. I've used the variable
# DTYPE for this, which is assigned to the usual NumPy runtime
# type info object.
DTYPE = np.int
# "ctypedef" assigns a corresponding compile-time type to DTYPE_t. For
# every type in the numpy module there's a corresponding compile-time
# type with a _t-suffix.
ctypedef np.int_t DTYPE_t
# The builtin min and max functions works with Python objects, and are
# so very slow. So we create our own.
# - "cdef" declares a function which has much less overhead than a normal
# def function (but it is not Python-callable)
# - "inline" is passed on to the C compiler which may inline the functions
# - The C type "int" is chosen as return type and argument types
# - Cython allows some newer Python constructs like "a if x else b", but
# the resulting C file compiles with Python 2.3 through to Python 3.0 beta.
cdef inline int int_max(int a, int b): return a if a >= b else b
cdef inline int int_min(int a, int b): return a if a <= b else b
# "def" can type its arguments but not have a return type. The type of the
# arguments for a "def" function is checked at run-time when entering the
# function.
#
# The arrays f, g and h is typed as "np.ndarray" instances. The only effect
# this has is to a) insert checks that the function arguments really are
# NumPy arrays, and b) make some attribute access like f.shape[0] much
# more efficient. (In this example this doesn't matter though.)
# The "not None" will enforce that an exception is raised if None is passed in.
# If the "not None" is left out, the function will accept either an ndarray or a None,
# but the None will cause problems later if not explicitly handled.
def naive_convolve(np.ndarray f not None, np.ndarray g not None):
if g.shape[0] % 2 != 1 or g.shape[1] % 2 != 1:
raise ValueError("Only odd dimensions on filter supported")
assert f.dtype == DTYPE and g.dtype == DTYPE
# The "cdef" keyword is also used within functions to type variables. It
# can only be used at the top indentation level (there are non-trivial
# problems with allowing them in other places, though we'd love to see
# good and thought out proposals for it).
#
# For the indices, the "int" type is used. This corresponds to a C int,
# other C types (like "unsigned int") could have been used instead.
# Purists could use "Py_ssize_t" which is the proper Python type for
# array indices.
cdef int vmax = f.shape[0]
cdef int wmax = f.shape[1]
cdef int smax = g.shape[0]
cdef int tmax = g.shape[1]
cdef int smid = smax // 2
cdef int tmid = tmax // 2
cdef int xmax = vmax + 2*smid
cdef int ymax = wmax + 2*tmid
cdef np.ndarray h = np.zeros([xmax, ymax], dtype=DTYPE)
cdef int x, y, s, t, v, w
# It is very important to type ALL your variables. You do not get any
# warnings if not, only much slower code (they are implicitly typed as
# Python objects).
cdef int s_from, s_to, t_from, t_to
# For the value variable, we want to use the same data type as is
# stored in the array, so we use "DTYPE_t" as defined above.
# NB! An important side-effect of this is that if "value" overflows its
# datatype size, it will simply wrap around like in C, rather than raise
# an error like in Python.
cdef DTYPE_t value
for x in range(xmax):
for y in range(ymax):
s_from = int_max(smid - x, -smid)
s_to = int_min((xmax - x) - smid, smid + 1)
t_from = int_max(tmid - y, -tmid)
t_to = int_min((ymax - y) - tmid, tmid + 1)
value = 0
for s in range(s_from, s_to):
for t in range(t_from, t_to):
v = x - smid + s
w = y - tmid + t
value += g[smid - s, tmid - t] * f[v, w]
h[x, y] = value
return h
At this point, have a look at the generated C code for convolve1.pyx and convolve2.pyx. Click on the lines to expand them and see corresponding C. (Note that this code annotation is currently experimental and especially "trailing" cleanup code for a block may stick to the last expression in the block and make it look worse than it is -- use some common sense).
Especially have a look at the for loops: In convolve1.c, these are ~20 lines of C code to set up while in convolve2.c a normal C for loop is used.
After building this and continuing my (very informal) benchmarks, I get
#!python In [21]: import convolve2 In [22]: %timeit -n2 -r3 convolve2.naive_convolve(f, g) 2 loops, best of 3: 828 ms per loop
This section is about the Cython's special features working with numpy arrays. The paper Fast numerical computations with Cython (2009) is a more up to date source on Cython's numpy specific features.
There's still a bottleneck killing performance, and that is the array lookups and assignments. The []-operator still uses full Python operations -- what we would like to do instead is to access the data buffer directly at C speed.
What we need to do then is to type the contents of the ndarray objects. We do this with a special "buffer" syntax which must be told the datatype (first argument) and number of dimensions ("ndim" keyword-only argument, if not provided then one-dimensional is assumed).
More information on this syntax can be found here.
Showing the changes needed to produce convolve3.pyx only:
#!python
...
def naive_convolve(np.ndarray[DTYPE_t, ndim=2] f not None,
np.ndarray[DTYPE_t, ndim=2] g not None):
...
cdef np.ndarray[DTYPE_t, ndim=2] h = ...
Usage:
#!python In [18]: import convolve3 In [19]: %timeit -n3 -r100 convolve3.naive_convolve(f, g) 3 loops, best of 100: 11.6 ms per loop
Note the importance of this change.
Gotcha: This efficient indexing only affects certain index operations, namely those with exactly ndim number of typed integer indices. So if v for instance isn't typed, then the lookup f[v, w] isn't optimized. On the other hand, this means that you can continue using Python objects for sophisticated dynamic slicing etc. just as when the array is not typed.
The array lookups are still slowed down by two factors:
2. Negative indices are checked for and handled correctly. The code above is explicitly coded so that it doesn't use negative indices, and it (hopefully) always access within bounds. We can add a decorator to disable bounds checking:
#!python
...
cimport cython
@cython.boundscheck(False) # turn of bounds-checking for entire function
def naive_convolve(np.ndarray[DTYPE_t, ndim=2] f not None,
np.ndarray[DTYPE_t, ndim=2] g not None):
...
Now bounds checking is not performed (and, as a side-effect, if you do happen to access out of bounds you will in the best case crash your program and in the worst case corrupt data). It is possible to switch bounds-checking mode in many ways, see compiler directives for more information.
Negative indices are dealt with by ensuring Cython that the indices will be positive, by casting the variables to unsigned integer types (if you do have negative values, then this casting will create a very large positive value instead and you will attempt to access out-of-bounds values). Casting is done with a special <>-syntax. The code below is changed to use either unsigned ints or casting as appropriate:
#!python
...
cdef int s, t # changed
cdef unsigned int x, y, v, w # changed
cdef int s_from, s_to, t_from, t_to
cdef DTYPE_t value
for x in range(xmax):
for y in range(ymax):
s_from = max(smid - x, -smid)
s_to = min((xmax - x) - smid, smid + 1)
t_from = max(tmid - y, -tmid)
t_to = min((ymax - y) - tmid, tmid + 1)
value = 0
for s in range(s_from, s_to):
for t in range(t_from, t_to):
v = <unsigned int>(x - smid + s) # changed
w = <unsigned int>(y - tmid + t) # changed
value += g[<unsigned int>(smid - s), <unsigned int>(tmid - t)] * f[v, w] # changed
h[x, y] = value
...
(In the next Cython release we will likely add a compiler directive or argument to the np.ndarray[]-type specifier to disable negative indexing so that casting so much isn't necessary; feedback on this is welcome.)
The function call overhead now starts to play a role, so we compare the latter two examples with larger N:
#!python In [11]: %timeit -n3 -r100 convolve4.naive_convolve(f, g) 3 loops, best of 100: 5.97 ms per loop In [12]: N = 1000 In [13]: f = np.arange(N*N, dtype=np.int).reshape((N,N)) In [14]: g = np.arange(81, dtype=np.int).reshape((9, 9)) In [17]: %timeit -n1 -r10 convolve3.naive_convolve(f, g) 1 loops, best of 10: 1.16 s per loop In [18]: %timeit -n1 -r10 convolve4.naive_convolve(f, g) 1 loops, best of 10: 597 ms per loop
(Also this is a mixed benchmark as the result array is allocated within the function call.)
Speed comes with some cost. Especially it can be dangerous to set typed objects (like f, g and h in our sample code) to None. Setting such objects to None is entirely legal, but all you can do with them is check whether they are None. All other use (attribute lookup or indexing) can potentially segfault or corrupt data (rather than raising exceptions as they would in Python).
The actual rules are a bit more complicated but the main message is clear: Do not use typed objects without knowing that they are not set to None.
Passing None (or not) into a functionBy default, when you declare a type in the function signature, Cython will raise an exception if the function is called with another type passed in. However, It will also except None, so that there is a way to signify "no value". However, if that None is not handled properly and used in an indexing operation, you can get a segfault or bus error or other hard crash. If you want your function not to be able to accept None, you can use the "not None" declaration, as so:
#!python
...
def naive_convolve(np.ndarray[DTYPE_t, ndim=2] f not None,
np.ndarray[DTYPE_t, ndim=2] g not None):
...
In this case, the function will raise an exception immediately if the argument is None. It's probably a good practice to always use "not None" unless you write code that explicitly checks for None.
It would be possible to do
def naive_convolve(object[DTYPE_t, ndim=2] f, ...):
i.e. use object rather than np.ndarray. Under Python 3.0 this can allow your algorithm to work with any libraries supporting the buffer interface; and support for e.g. the Python Imaging Library may easily be added if someone is interested also under Python 2.x.
There is some speed penalty to this, though (as one makes more assumptions compile-time if the type is set to np.ndarray, specifically it is assumed that the data is stored in pure strided mode and not in indirect mode).
If you're not writing a highly specialized function, you may want to write very general functions that work a lot like UFuncs. There are at least three generality features that numpy provides: handling of arbitrary dimensional arrays, array broadcasting, and array type casting. The simplest way to take advantage of these features is to use the Numpy C-API.
To access the C-API you must cimport numpy and call import_array. If you also wish to use Python numpy functions, you must import that separately.
#!python import numpy as np cimport numpy as np np.import_array()Dimensionally simple functions
If your function is dimensionally simple (no dimension is special), then it is easy to implement input array broadcasting by using multi-iterators to simultaneously iterate over multiple arrays. The API multi-iterator documentation is useful here.
The following example demonstrates how to do this.
#!python
import numpy as np
cimport numpy as np
np.import_array()
def prod2(a, b):
#generate a new output array of the correct shape by broadcasting input arrays together
out = np.empty(np.broadcast(a, b).shape, np.float)
#generate the iterator over the input and output arrays, does the same thing as
# PyArray_MultiIterNew
cdef np.broadcast it = np.broadcast(a, b, out)
while np.PyArray_MultiIter_NOTDONE(it):
#PyArray_MultiIter_DATA is used to access the pointers the iterator points to
aval = (<double*>np.PyArray_MultiIter_DATA(it, 0))[0]
bval = (<double*>np.PyArray_MultiIter_DATA(it, 1))[0]
(<double*>np.PyArray_MultiIter_DATA(it, 2))[0] = aval * bval
#PyArray_MultiIter_NEXT is used to advance the iterator
np.PyArray_MultiIter_NEXT(it)
return out Dimensionally complex functions
Sometimes your function will not be dimensionally simple. You may want to write general functions that can take an axis argument to treat a particular axis differently than other axes, for example for a moving average function.
One way to implement a dimensionally complex function is to use PyArray_IterAllButAxis to iterate over a single array but leave out iteration along the desired axis. The function then manually iterates over the distinct axis using pointer arithmetic. Manually iterating over the smallest axis also happens to be somewhat faster than using the iterator to iterate over it.
The following example demonstrates how to do this.
#!python
def ewma( a, double weight, int axis):
cdef int i
out = np.empty(a.shape, np.double)
cdef np.flatiter ita = np.PyArray_IterAllButAxis(a, &axis)
cdef np.flatiter ito = np.PyArray_IterAllButAxis(out, &axis)
cdef int a_axis_stride = a.strides[axis]
cdef int o_axis_stride = out.strides[axis]
cdef int axis_length = out.shape[axis]
cdef double avg
cdef double value
while np.PyArray_ITER_NOTDONE(ita):
avg = 0.0
for i in range(axis_length):
"""
PyArray_ITER_DATA is used to access the pointer of the iterator
(<char*>pointer) + i * a_axis_stride) moves ahead the correct
number of bytes to the next element. The result must then be
dereferenced and cast to the correct type (<double*>(char_pointer))[0]
"""
value = (<double*>((<char*>np.PyArray_ITER_DATA (ita)) + i*a_axis_stride))[0]
avg = value*weight + avg * (1.0 - weight)
#the result is written to the output array
(<double*>((<char*>np.PyArray_ITER_DATA (ito)) + i*o_axis_stride))[0] = avg
#PyArray_ITER_NEXT increments the iterators
np.PyArray_ITER_NEXT(ita)
np.PyArray_ITER_NEXT(ito)
return out
For the time being, it is not possible to use array broadcasting using this method of implementing dimensionally complex functions. It is also not possible to have multiple special dimensions using this method. However, the Optimizing Iterator/UFunc Performance NEP will soon provide a more unified and convenient iteration interface which will support both features simultaneously and allow multiple special dimensions.
Keep in mind that the Numpy C API has a strange convention where most entry points which take a "dtype" reference "steal" a reference to the dtype (they Py_DECREF the dtype), so if you wish to use one of those entry points you will need to do a Py_INCREF of the dtype before passing it into the entry point:
#!python
from python_ref cimport Py_INCREF, Py_DECREF
cimport numpy as np
np.import_array()
...
Py_INCREF(dtype)
return np.PyArray_FromArray(
instance, dtype, np.NPY_CARRAY|np.NPY_FORCECAST
)
These are some points to consider for further development. All points listed here has gone through a lot of thinking and planning already; still, they may or may not happen depending on available developer time and resources for Cython.
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