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Cython: understanding what the html annotation file has to say?

After compiling the following Cython code, I get the html file that looks like this:
import numpy as np
cimport numpy as np
cpdef my_function(np.ndarray[np.double_t, ndim = 1] array_a,
np.ndarray[np.double_t, ndim = 1] array_b,
int n_rows,
int n_columns):
array_a[0:-1:n_columns] = 0
array_a[n_columns - 1:n_rows * n_columns:n_columns] = 0
array_a[0:n_columns] = 0
array_a[n_columns* (n_rows - 1):n_rows * n_columns] = 0
array_b[array_a == 3] = 0
return array_a, array_b
My question is that why those operations of my function are still yellow? Does this mean that the code is still not as fast as it could be using Cython?

As you already know, yellow lines mean that some interactions with python happen, i.e. python functionality and not raw c-functionality is used, and you can look into the produced code to see, what happens and if it can/should be fixed/avoided.
Not every interaction with python means a (measurable) slowdown.
Let's take a look at this simplified function:
%%cython
cimport numpy as np
def use_slices(np.ndarray[np.double_t] a):
a[0:len(a)]=0.0
When we look into the produced code we see (I kept only the important parts):
__pyx_t_1 = PyObject_Length(((PyObject *)__pyx_v_a));
__pyx_t_2 = PyInt_FromSsize_t(__pyx_t_1);
__pyx_t_3 = PySlice_New(__pyx_int_0, __pyx_t_2, Py_None);
PyObject_SetItem(((PyObject *)__pyx_v_a)
So basically we get a new slice (which is a numpy-array) and then use numpy's functionality (PyObject_SetItem) to set all elements to 0.0, which is C-code under the hood.
Let's take a look at version with hand-written for loop:
cimport numpy as np
def use_for(np.ndarray[np.double_t] a):
cdef int i
for i in range(len(a)):
a[i]=0.0
It still uses PyObject_Length (because of length) and bound-checking, but otherwise it is C-code. When we compare times:
>>> import numpy as np
>>> a=np.ones((500,))
>>> %timeit use_slices(a)
100000 loops, best of 3: 1.85 µs per loop
>>> %timeit use_for(a)
1000000 loops, best of 3: 1.42 µs per loop
>>> b=np.ones((250000,))
>>> %timeit use_slices(b)
10000 loops, best of 3: 104 µs per loop
>>> %timeit use_for(b)
1000 loops, best of 3: 364 µs per loop
You can see the additional overhead of creating a slice for small sizes, but the additional checks in the for-version means it has more overhead in the long run.
Let's disable these checks:
%%cython
cimport cython
cimport numpy as np
#cython.boundscheck(False)
#cython.wraparound(False)
def use_for_no_checks(np.ndarray[np.double_t] a):
cdef int i
for i in range(len(a)):
a[i]=0.0
In the produced html we can see, that a[i] gets as simple as it gets:
__pyx_t_3 = __pyx_v_i;
*__Pyx_BufPtrStrided1d(__pyx_t_5numpy_double_t *, __pyx_pybuffernd_a.rcbuffer->pybuffer.buf, __pyx_t_3, __pyx_pybuffernd_a.diminfo[0].strides) = 0.0;
}
__Pyx_BufPtrStrided1d(type, buf, i0, s0) is define for (type)((char*)buf + i0 * s0).
And now:
>>> %timeit use_for_no_checks(a)
1000000 loops, best of 3: 1.17 µs per loop
>>> %timeit use_for_no_checks(b)
1000 loops, best of 3: 246 µs per loop
We can improve it further by releasing gil in the for-loop:
%%cython
cimport cython
cimport numpy as np
#cython.boundscheck(False)
#cython.wraparound(False)
def use_for_no_checks_no_gil(np.ndarray[np.double_t] a):
cdef int i
cdef int n=len(a)
with nogil:
for i in range(n):
a[i]=0.0
and now:
>>> %timeit use_for_no_checks_no_gil(a)
1000000 loops, best of 3: 1.07 µs per loop
>>> %timeit use_for_no_checks_no_gil(b)
10000 loops, best of 3: 166 µs per loop
So it is somewhat faster, but still you cannot beat numpy for larger arrays.
In my opinion, there are two things to take from it:
Cython will not transform slices to an access through a for-loop and thus Python-functionality must be used.
There is small overhead, but it is only calling the numpy-functionality the most of work is done in numpy-code, and this cannot be speed up through Cython.
One last try using memset function:
%%cython
from libc.string cimport memset
cimport numpy as np
def use_memset(np.ndarray[np.double_t] a):
memset(&a[0], 0, len(a)*sizeof(np.double_t))
We get:
>>> %timeit use_memset(a)
1000000 loops, best of 3: 821 ns per loop
>>> %timeit use_memset(b)
10000 loops, best of 3: 102 µs per loop
It is also as fast as the numpy-code for large arrays.
As DavidW suggested, one could try to use memory-views:
%%cython
cimport numpy as np
def use_slices_memview(double[::1] a):
a[0:len(a)]=0.0
leads to a slightly faster code for small arrays but similar fast code fr large arrays (compared to numpy-slices):
>>> %timeit use_slices_memview(a)
1000000 loops, best of 3: 1.52 µs per loop
>>> %timeit use_slices_memview(b)
10000 loops, best of 3: 105 µs per loop
That means, that the memory-view slices have less overhead than the numpy-slices. Here is the produced code:
__pyx_t_1 = __Pyx_MemoryView_Len(__pyx_v_a);
__pyx_t_2.data = __pyx_v_a.data;
__pyx_t_2.memview = __pyx_v_a.memview;
__PYX_INC_MEMVIEW(&__pyx_t_2, 0);
__pyx_t_3 = -1;
if (unlikely(__pyx_memoryview_slice_memviewslice(
&__pyx_t_2,
__pyx_v_a.shape[0], __pyx_v_a.strides[0], __pyx_v_a.suboffsets[0],
0,
0,
&__pyx_t_3,
0,
__pyx_t_1,
0,
1,
1,
0,
1) < 0))
{
__PYX_ERR(0, 27, __pyx_L1_error)
}
{
double __pyx_temp_scalar = 0.0;
{
Py_ssize_t __pyx_temp_extent = __pyx_t_2.shape[0];
Py_ssize_t __pyx_temp_idx;
double *__pyx_temp_pointer = (double *) __pyx_t_2.data;
for (__pyx_temp_idx = 0; __pyx_temp_idx < __pyx_temp_extent; __pyx_temp_idx++) {
*((double *) __pyx_temp_pointer) = __pyx_temp_scalar;
__pyx_temp_pointer += 1;
}
}
}
__PYX_XDEC_MEMVIEW(&__pyx_t_2, 1);
__pyx_t_2.memview = NULL;
__pyx_t_2.data = NULL;
I think the most important part: this code doesn't create an additional temporary object - it reuses the existing memory view for the slice.
My compiler produces (at least for my machine) a slightly faster code if memory views are used. Not sure whether it is worth an investigation. At the first sight the difference in every iteration step is:
# created code for memview-slices:
*((double *) __pyx_temp_pointer) = __pyx_temp_scalar;
__pyx_temp_pointer += 1;
#created code for memview-for-loop:
__pyx_v_i = __pyx_t_3;
__pyx_t_4 = __pyx_v_i;
*((double *) ( /* dim=0 */ ((char *) (((double *) data) + __pyx_t_4)) )) = 0.0;
I would expect different compilers handle this code differently well. But clearly, the first version is easier to get optimized.
As Behzad Jamali pointed out, there is difference between double[:] a and double[::1] a. The second version using slices is about 20% faster on my machine. The difference is, that during the compile time it is know for the double[::1] version, that the memory-accesses will be consecutive and this can be used for optimization. In the version with double[:] we don't know anything about stride until the runtime.

Cython: why is size_t faster than int?

Changing certain Cython variables from type int to type size_t can significantly reduce some functions times (~30%), but I do not understand why.
For example:
cimport numpy as cnp
import numpy as np
def sum_int(cnp.int64_t[::1] A):
cdef unsigned long s = 0
cdef int k
for k in xrange(A.shape[0]):
s += A[k]
return s
def sum_size_t(cnp.int64_t[::1] A):
cdef unsigned long s = 0
cdef size_t k
for k in xrange(A.shape[0]):
s += A[k]
return s
a = np.array(range(1000000))
And the timing results:
In [17]: %timeit sum_int(a)
1000 loops, best of 3: 652 µs per loop
In [18]: %timeit sum_size_t(a)
1000 loops, best of 3: 427 µs per loop
I am new to Cython, and know Fortran better than C. Help me out. What is the important difference between these two variable types that causes such a performance difference? What is it that I don't grok about Cython?

You'd likely have to do a line by line profiling to find out exactly, but one thing stands out to me from the produced C file: int version is checked for wraparound to negative numbers, size_t is assumed ok.
In the int loop: (t_3 is assigned from k, they're the same type)
if (__pyx_t_3 < 0) {
__pyx_t_3 += __pyx_v_A.shape[0];
if (unlikely(__pyx_t_3 < 0)) __pyx_t_4 = 0;
} else if (unlikely(__pyx_t_3 >= __pyx_v_A.shape[0])) __pyx_t_4 = 0;
In the size_t loop:
if (unlikely(__pyx_t_3 >= (size_t)__pyx_v_A.shape[0])) __pyx_t_4 = 0;
So no wraparound test is needed because size_t is unsigned and guaranteed not to wrap around when indexing items in memory. The rest is virtually the same.
Update: regarding your unsigned int results - what's your size of int and size_t? Any chance they're different size, causing the change? In my case the C code for uint and size_t is identical. (since size_t is unsigned and specifically unsigned int on this system)

On a 64 bit system there seem to be two reasons:
Use an unsigned integer for the loop:
%%cython
cimport numpy as cnp
import numpy as np
def sum_int_unsigned(cnp.int64_t[::1] A):
cdef unsigned long s = 0
cdef unsigned k
for k in xrange(A.shape[0]):
s += A[k]
return s
Use a long instead of an int:
%%cython
cimport numpy as cnp
import numpy as np
def sum_int_unsigned_long(cnp.int64_t[::1] A):
cdef unsigned long s = 0
cdef unsigned long k
for k in xrange(A.shape[0]):
s += A[k]
return s
Timings:
%timeit sum_int(a)
1000 loops, best of 3: 1.52 ms per loop
%timeit sum_size_t(a)
1000 loops, best of 3: 671 µs per loop
Using unsigned brings us half way:
%timeit sum_int_unsigned(a)
1000 loops, best of 3: 1.09 ms per loop
Using long accounts for the rest:
%timeit sum_int_unsigned_long(a)
1000 loops, best of 3: 648 µs per loop

Loop over a Numpy array with Cython

Let a and b be two numpy.float arrays of length 1024, defined with
cdef numpy.ndarray a
cdef numpy.ndarray b
I notice that:
cdef int i
for i in range(1024):
b[i] += a[i]
is considerably slower than:
b += a
Why?
I really need to be able to loop manually over arrays.

The difference will be smaller if you tell Cython the data type and the number of dimensions for a and b:
cdef numpy.ndarray[np.float64_t, ndim=1] a, b
Although the difference will be smaller, you won't beat b += a because this is using NumPy's SIMD-boosted functions (which will perform depending if your CPU supports SIMD).

No speed gains from Cython again?

The following is my cython code, the purpose is to do a bootstrap.
def boots(int trial, np.ndarray[double, ndim=2] empirical, np.ndarray[double, ndim=2] expected):
cdef int length = len(empirical)
cdef np.ndarray[double, ndim=2] ret = np.empty((trial, 100))
cdef np.ndarray[long] choices
cdef np.ndarray[double] m
cdef np.ndarray[double] n
cdef long o
cdef int i
cdef int j
for i in range(trial):
choices = np.random.randint(0, length, length)
m = np.zeros(100)
n = np.zeros(100)
for j in range(length):
o = choices[j]
m.__iadd__(empirical[o])
n.__iadd__(expected[o])
empirical_boot = m / length
expected_boot = n / length
ret[i] = empirical_boot / expected_boot - 1
ret.sort(axis=0)
return ret[int(trial * 0.025)].reshape((10,10)), ret[int(trial * 0.975)].reshape((10,10))
# test code
empirical = np.ones((40000, 100))
expected = np.ones((40000, 100))
%prun -l 10 boots(100, empirical,expected)
It takes 11 seconds in pure python with fancy indexing, and no matter how hard I tuned in cython it stays the same.
np.random.randint(0, 40000, 40000) takes 1 ms, so 100x takes 0.1s.
np.sort(np.ones((40000, 100)) takes 0.2s.
Thus I feel there must be ways to improve boots.

The primary issue you are seeing is that Cython only optimizes single-item access for typed arrays. This means that each of the lines in your code where you are using vectorization from NumPy still involve creating and interacting with Python objects.
The code you have there wasn't faster than the pure Python version because it wasn't really doing any of the computation differently.
You will have to avoid this by writing out the looping operations explicitly.
Here is a modified version of your code that runs significantly faster.
from numpy cimport ndarray as ar
from numpy cimport int32_t as int32
from numpy import empty
from numpy.random import randint
cimport cython
ctypedef int
# Notice the use of these decorators to tell Cython to turn off
# some of the checking it does when accessing arrays.
#cython.boundscheck(False)
#cython.wraparound(False)
def boots(int32 trial, ar[double, ndim=2] empirical, ar[double, ndim=2] expected):
cdef:
int32 length = empirical.shape[0], i, j, k
int32 o
ar[double, ndim=2] ret = empty((trial, 100))
ar[int32] choices
ar[double] m = empty(100), n = empty(100)
for i in range(trial):
# Still calling Python on this line
choices = randint(0, length, length)
# It was faster to compute m and n separately.
# I suspect that has to do with cache management.
# Instead of allocating new arrays, I just filled the old ones with the new values.
o = choices[0]
for k in range(100):
m[k] = empirical[o,k]
for j in range(1, length):
o = choices[j]
for k in range(100):
m[k] += empirical[o,k]
o = choices[0]
for k in range(100):
n[k] = expected[o,k]
for j in range(1, length):
o = choices[j]
for k in range(100):
n[k] += expected[o,k]
# Here I simplified some of the math and got rid of temporary arrays
for k in range(100):
ret[i,k] = m[k] / n[k] - 1.
ret.sort(axis=0)
return ret[int(trial * 0.025)].reshape((10,10)), ret[int(trial * 0.975)].reshape((10,10))
If you want to have a look at which lines of your code involve Python calls, the Cython compiler can generate an html file showing which lines call Python.
This option is called annotation.
The way you use it depends on how you are compiling your cython code.
If you are using the IPython notebook, just add the --annotate flag to the Cython cell magic.
You may also be able to benefit from turning on the C compiler optimization flags.

C array vs NumPy array

In terms of performance (algebraic operations, lookup, caching, etc.), is there a difference between C arrays (which can be exposed as a C array, or a cython.view.array [Cython array], or a memoryview of the aforementioned two) and a NumPy arrays (which in Cython should have no Python overhead)
Edit:
I should mention that in the NumPy array is statically typed using Cython, and the dtypes are NumPy compile-time datypes (e.g. cdef np.int_t or cdef np.float32_t), and the types in the C case are the C equivalents (cdef int_t and cdef float)
Edit2:
Here is the example from the Cython Memoryview documentation to further illustrate my question:
from cython.view cimport array as cvarray
import numpy as np
# Memoryview on a NumPy array
narr = np.arange(27, dtype=np.dtype("i")).reshape((3, 3, 3))
cdef int [:, :, :] narr_view = narr
# Memoryview on a C array
cdef int carr[3][3][3]
cdef int [:, :, :] carr_view = carr
# Memoryview on a Cython array
cyarr = cvarray(shape=(3, 3, 3), itemsize=sizeof(int), format="i")
cdef int [:, :, :] cyarr_view = cyarr
Is there any difference between sticking with a C array vs a Cython array vs a NumPy array?

My knowledge on this is still imperfect, but this may be helpful.
I ran some informal benchmarks to show what each array type is good for and was intrigued by what I found.
Though these array types are different in many ways, if you are doing heavy computation with large arrays, you should be able to get similar performance out of any of them since item-by-item access should be roughly the same across the board.
A NumPy array is a Python object implemented using Python's C API.
NumPy arrays do provide an API at the C level, but they cannot be created independent from the Python interpreter.
They are especially useful because of all the different array manipulation routines available in NumPy and SciPy.
A Cython memory view is also a Python object, but it is made as a Cython extension type.
It does not appear to be designed for use in pure Python since it isn't a part of Cython that can be imported directly from Python, but you can return a view to Python from a Cython function.
You can look at the implementation at https://github.com/cython/cython/blob/master/Cython/Utility/MemoryView.pyx
A C array is a native type in the C language.
It is indexed like a pointer, but arrays and pointers are different.
There is some good discussion on this at http://c-faq.com/aryptr/index.html
They can be allocated on the stack and are easier for the C compiler to optimize, but they will be more difficult to access outside of Cython.
I know you can make a NumPy array from memory that has been dynamically allocated by other programs, but it seems a lot more difficult that way.
Travis Oliphant posted an example of this at http://blog.enthought.com/python/numpy-arrays-with-pre-allocated-memory/
If you are using C arrays or pointers for temporary storage within your program they should work very well for you.
They will not be as convenient for slicing or for any other sort of vectorized computation since you will have to do everything yourself with explicit looping, but they should allocate and deallocate faster and ought to provide a good baseline for speed.
Cython also provides an array class.
It looks like it is designed for internal use.
Instances are created when a memoryview is copied.
See http://docs.cython.org/src/userguide/memoryviews.html#view-cython-arrays
In Cython, you can also allocate memory and index a pointer to treat the allocated memory somewhat like an array.
See http://docs.cython.org/src/tutorial/memory_allocation.html
Here are some benchmarks that show somewhat similar performance for indexing large arrays.
This is the Cython file.
from numpy cimport ndarray as ar, uint64_t
cimport cython
import numpy as np
#cython.boundscheck(False)
#cython.wraparound(False)
def ndarr_time(uint64_t n=1000000, uint64_t size=10000):
cdef:
ar[uint64_t] A = np.empty(n, dtype=np.uint64)
uint64_t i, j
for i in range(n):
for j in range(size):
A[j] = n
def carr_time(uint64_t n=1000000):
cdef:
ar[uint64_t] A = np.empty(n, dtype=np.uint64)
uint64_t AC[10000]
uint64_t a
int i, j
for i in range(n):
for j in range(10000):
AC[j] = n
#cython.boundscheck(False)
#cython.wraparound(False)
def ptr_time(uint64_t n=1000000, uint64_t size=10000):
cdef:
ar[uint64_t] A = np.empty(n, dtype=np.uint64)
uint64_t* AP = &A[0]
uint64_t a
int i, j
for i in range(n):
for j in range(size):
AP[j] = n
#cython.boundscheck(False)
#cython.wraparound(False)
def view_time(uint64_t n=1000000, uint64_t size=10000):
cdef:
ar[uint64_t] A = np.empty(n, dtype=np.uint64)
uint64_t[:] AV = A
uint64_t i, j
for i in range(n):
for j in range(size):
AV[j] = n
Timing these using IPython we obtain
%timeit -n 10 ndarr_time()
%timeit -n 10 carr_time()
%timeit -n 10 ptr_time()
%timeit -n 10 view_time()
10 loops, best of 3: 6.33 s per loop
10 loops, best of 3: 3.12 s per loop
10 loops, best of 3: 6.26 s per loop
10 loops, best of 3: 3.74 s per loop
These results struck me as a little odd, considering that, as per Efficiency: arrays vs pointers , arrays are unlikely to be significantly faster than pointers.
It appears that some sort of compiler optimization is making the pure C arrays and the typed memory views faster.
I tried turning off all the optimization flags on my C compiler and got the timings
1 loops, best of 3: 25.1 s per loop
1 loops, best of 3: 25.5 s per loop
1 loops, best of 3: 32 s per loop
1 loops, best of 3: 28.4 s per loop
It looks to me like the item-by item access is pretty much the same across the board, except that C arrays and Cython memory views seem to be easier for the compiler to optimize.
More commentary on this can be seen at a these two blog posts I found some time ago:
http://jakevdp.github.io/blog/2012/08/08/memoryview-benchmarks/
http://jakevdp.github.io/blog/2012/08/16/memoryview-benchmarks-2/
In the second blog post he comments on how, if memory view slices are inlined, they can provide speeds similar to that of pointer arithmetic.
I have noticed in some of my own tests that explicitly inlining functions that use Memory View slices isn't always necessary.
As an example of this, I'll compute the inner product of every combination of two rows of an array.
from numpy cimport ndarray as ar
cimport cython
from numpy import empty
# An inlined dot product
#cython.boundscheck(False)
#cython.wraparound(False)
cdef inline double dot_product(double[:] a, double[:] b, int size):
cdef int i
cdef double tot = 0.
for i in range(size):
tot += a[i] * b[i]
return tot
# non-inlined dot-product
#cython.boundscheck(False)
#cython.wraparound(False)
cdef double dot_product_no_inline(double[:] a, double[:] b, int size):
cdef int i
cdef double tot = 0.
for i in range(size):
tot += a[i] * b[i]
return tot
# function calling inlined dot product
#cython.boundscheck(False)
#cython.wraparound(False)
def dot_rows_slicing(ar[double,ndim=2] A):
cdef:
double[:,:] Aview = A
ar[double,ndim=2] res = empty((A.shape[0], A.shape[0]))
int i, j
for i in range(A.shape[0]):
for j in range(A.shape[0]):
res[i,j] = dot_product(Aview[i], Aview[j], A.shape[1])
return res
# function calling non-inlined version
#cython.boundscheck(False)
#cython.wraparound(False)
def dot_rows_slicing_no_inline(ar[double,ndim=2] A):
cdef:
double[:,:] Aview = A
ar[double,ndim=2] res = empty((A.shape[0], A.shape[0]))
int i, j
for i in range(A.shape[0]):
for j in range(A.shape[0]):
res[i,j] = dot_product_no_inline(Aview[i], Aview[j], A.shape[1])
return res
# inlined dot product using numpy arrays
#cython.boundscheck(False)
#cython.boundscheck(False)
cdef inline double ndarr_dot_product(ar[double] a, ar[double] b):
cdef int i
cdef double tot = 0.
for i in range(a.size):
tot += a[i] * b[i]
return tot
# non-inlined dot product using numpy arrays
#cython.boundscheck(False)
#cython.boundscheck(False)
cdef double ndarr_dot_product_no_inline(ar[double] a, ar[double] b):
cdef int i
cdef double tot = 0.
for i in range(a.size):
tot += a[i] * b[i]
return tot
# function calling inlined numpy array dot product
#cython.boundscheck(False)
#cython.wraparound(False)
def ndarr_dot_rows_slicing(ar[double,ndim=2] A):
cdef:
ar[double,ndim=2] res = empty((A.shape[0], A.shape[0]))
int i, j
for i in range(A.shape[0]):
for j in range(A.shape[0]):
res[i,j] = ndarr_dot_product(A[i], A[j])
return res
# function calling nun-inlined version for numpy arrays
#cython.boundscheck(False)
#cython.wraparound(False)
def ndarr_dot_rows_slicing_no_inline(ar[double,ndim=2] A):
cdef:
ar[double,ndim=2] res = empty((A.shape[0], A.shape[0]))
int i, j
for i in range(A.shape[0]):
for j in range(A.shape[0]):
res[i,j] = ndarr_dot_product(A[i], A[j])
return res
# Version with explicit looping and item-by-item access.
#cython.boundscheck(False)
#cython.wraparound(False)
def dot_rows_loops(ar[double,ndim=2] A):
cdef:
ar[double,ndim=2] res = empty((A.shape[0], A.shape[0]))
int i, j, k
double tot
for i in range(A.shape[0]):
for j in range(A.shape[0]):
tot = 0.
for k in range(A.shape[1]):
tot += A[i,k] * A[j,k]
res[i,j] = tot
return res
Timing these we see
A = rand(1000, 1000)
%timeit dot_rows_slicing(A)
%timeit dot_rows_slicing_no_inline(A)
%timeit ndarr_dot_rows_slicing(A)
%timeit ndarr_dot_rows_slicing_no_inline(A)
%timeit dot_rows_loops(A)
1 loops, best of 3: 1.02 s per loop
1 loops, best of 3: 1.02 s per loop
1 loops, best of 3: 3.65 s per loop
1 loops, best of 3: 3.66 s per loop
1 loops, best of 3: 1.04 s per loop
The results were as fast with explicit inlining as they were without it.
In both cases, the typed memory views were comparable to a version of the function that was written without slicing.
In the blog post, he had to write a specific example to force the compiler to not inline a function.
It appears that a decent C compiler (I'm using MinGW) is able to take care of these optimizations without being told to inline certain functions.
Memoryviews can be faster for passing array slices between functions within a Cython module, even without explicit inlining.
In this particular case, however, even pushing the loops to C doesn't really reach a speed anywhere near what can be achieved through proper use of matrix multiplication.
The BLAS is still the best way to do things like this.
%timeit A.dot(A.T)
10 loops, best of 3: 25.7 ms per loop
There is also automatic conversion from NumPy arrays to memoryviews as in
cimport cython
#cython.boundscheck(False)
#cython.wraparound(False)
def cysum(double[:] A):
cdef tot = 0.
cdef int i
for i in range(A.size):
tot += A[i]
return tot
The one catch is that, if you want a function to return a NumPy array, you will have to use np.asarray to convert the memory view object to a NumPy array again.
This is a relatively inexpensive operation since memory views comply with http://www.python.org/dev/peps/pep-3118/
Conclusion
Typed memory views seem to be a viable alternative to NumPy arrays for internal use in a Cython module.
Array slicing will be faster with memory views, but there are not as many functions and methods written for memory views as there are for NumPy arrays.
If you don't need to call a bunch of the NumPy array methods and want easy array slicing, you can use memory views in place of NumPy arrays.
If you need both the array slicing and the NumPy functionality for a given array, you can make a memory view that points to the same memory as the NumPy array.
You can then use the view for passing slices between functions and the array for calling NumPy functions.
That approach is still somewhat limited, but it will work well if you are doing most of your processing with a single array.
C arrays and/or dynamically allocated blocks of memory could be useful for intermediate calculations, but they are not as easy to pass back to Python for use there.
In my opinion, it is also more cumbersome to dynamically allocate multidimensional C arrays.
The best approach I am aware of is to allocate a large block of memory and then use integer arithmetic to index it as if it were a multidimensional array.
This could be an issue if you want easy allocation of arrays on the fly.
On the other hand, allocation times are probably a good bit faster for C arrays.
The other array types are designed to be nearly as fast and much more convenient, so I would recommend using them unless there is a compelling reason to do otherwise.
Update: As mentioned in the answer by #Veedrac you can still pass Cython memory views to most NumPy functions.
When you do this, NumPy will usually have to create a new NumPy array object to work with the memory view anyway, so this will be somewhat slower.
For large arrays the effect will be negligible.
A call to np.asarray for a memory view will be relatively fast regardless of array size.
However, to demonstrate this effect, here is another benchmark:
Cython file:
def npy_call_on_view(npy_func, double[:] A, int n):
cdef int i
for i in range(n):
npy_func(A)
def npy_call_on_arr(npy_func, ar[double] A, int n):
cdef int i
for i in range(n):
npy_func(A)
in IPython:
from numpy.random import rand
A = rand(1)
%timeit npy_call_on_view(np.amin, A, 10000)
%timeit npy_call_on_arr(np.amin, A, 10000)
output:
10 loops, best of 3: 282 ms per loop
10 loops, best of 3: 35.9 ms per loop
I tried to choose an example that would show this effect well.
Unless many NumPy function calls on relatively small arrays are involved, this shouldn't change the time a whole lot.
Keep in mind that, regardless of which way we are calling NumPy, a Python function call still occurs.
This applies only to the functions in NumPy.
Most of the array methods are not available for memoryviews (some of the attributes still are, like size and shape and T).
For example A.dot(A.T) with NumPy arrays would become np.dot(A, A.T).

Don't use cython.view.array, use cpython.array.array.
See this answer of mine for details, although that only deals with speed. The recommendation is to treat cython.view.array as "demo" material, and cpython.array.array as an actual solid implementation. These arrays are very lightweight and better when just using them as scratch space.
Further, if you're ever tempted by malloc, raw access on these is no slower and instantiation takes only twice as long.
With regards to IanH's
If you need both the array slicing and the NumPy functionality for a given array, you can make a memory view that points to the same memory as the NumPy array.
It's worth noting that memoryviews have a "base" property and many Numpy functions can also take memoryviews, so these do not have to be separated variables.