Advanced NumPy

Author: Pauli Virtanen

NumPy is at the base of Python’s scientific stack of tools. Its purpose to implement efficient operations on many items in a block of memory. Understanding how it works in detail helps in making efficient use of its flexibility, taking useful shortcuts.

This section covers:

• Anatomy of NumPy arrays, and its consequences. Tips and tricks.

• Universal functions: what, why, and what to do if you want a new one.

• Integration with other tools: NumPy offers several ways to wrap any data in an ndarray, without unnecessary copies.

• Recently added features, and what’s in them: PEP 3118 buffers, generalized ufuncs, …

Prerequisites

• NumPy

• Cython

• Pillow (Python imaging library, used in a couple of examples)

# Import Numpy module.
import numpy as np
# Import Matplotlib (for later).
import matplotlib.pyplot as plt

Life of ndarray

It’s…

What is an ndarray

An ndarray is:

• A block of memory and

• an indexing scheme and

• a data type descriptor.

Put another way, an ndarray has raw data, and algorithms to:

• locate an element

• interpret an element

typedef struct PyArrayObject {
        PyObject_HEAD

        /* Block of memory */
        char *data;

        /* Data type descriptor */
        PyArray_Descr *descr;

        /* Indexing scheme */
        int nd;
        npy_intp *dimensions;
        npy_intp *strides;

        /* Other stuff */
        PyObject *base;
        int flags;
        PyObject *weakreflist;
} PyArrayObject;

Block of memory

x = np.array([1, 2, 3], dtype=np.int32)
x.data

<memory at 0x11391d3c0>

bytes(x.data)

b'\x01\x00\x00\x00\x02\x00\x00\x00\x03\x00\x00\x00'

Memory address of the data:

x.__array_interface__['data'][0]

105553143800288

The whole __array_interface__:

x.__array_interface__

{'data': (105553143800288, False),
 'strides': None,
 'descr': [('', '<i4')],
 'typestr': '<i4',
 'shape': (3,),
 'version': 3}

Reminder: two ndarrays may share the same memory:

x = np.array([1, 2, 3, 4])
y = x[:-1]
x[0] = 9
y

array([9, 2, 3])

Memory does not need to be owned by an ndarray:

x = b'1234'

x is a string (in Python 3 a bytes), we can represent its data as an array of ints:

y = np.frombuffer(x, dtype=np.int8)
y.data

<memory at 0x11391d600>

y.base is x

True

y.flags

  C_CONTIGUOUS : True
  F_CONTIGUOUS : True
  OWNDATA : False
  WRITEABLE : False
  ALIGNED : True
  WRITEBACKIFCOPY : False

The owndata and writeable flags indicate status of the memory block.

See also

array interface

Data types

The descriptor

dtype describes a single item in the array:

Table 6 Dtypes

type	scalar type of the data, one of: int8, int16, float64, et al. (fixed size) str, unicode, void (flexible size)
itemsize	size of the data block
byteorder	byte order: big-endian > little-endian < not applicable |
fields	sub-dtypes, if it’s a structured data type
shape	shape of the array, if it’s a sub-array

np.dtype(int).type

numpy.int64

np.dtype(int).itemsize

8

np.dtype(int).byteorder

'='

Example: reading .wav files

The .wav file header:

chunk_id	"RIFF"
chunk_size	4-byte unsigned little-endian integer
format	"WAVE"
fmt_id	"fmt "
fmt_size	4-byte unsigned little-endian integer
audio_fmt	2-byte unsigned little-endian integer
num_channels	2-byte unsigned little-endian integer
sample_rate	4-byte unsigned little-endian integer
byte_rate	4-byte unsigned little-endian integer
block_align	2-byte unsigned little-endian integer
bits_per_sample	2-byte unsigned little-endian integer
data_id	"data"
data_size	4-byte unsigned little-endian integer

• 44-byte block of raw data (in the beginning of the file)

• … followed by data_size bytes of actual sound data.

The .wav file header as a NumPy structured data type:

wav_header_dtype = np.dtype([
    ("chunk_id", (bytes, 4)), # flexible-sized scalar type, item size 4
    ("chunk_size", "<u4"),    # little-endian unsigned 32-bit integer
    ("format", "S4"),         # 4-byte string
    ("fmt_id", "S4"),
    ("fmt_size", "<u4"),
    ("audio_fmt", "<u2"),     #
    ("num_channels", "<u2"),  # .. more of the same ...
    ("sample_rate", "<u4"),   #
    ("byte_rate", "<u4"),
    ("block_align", "<u2"),
    ("bits_per_sample", "<u2"),
    ("data_id", ("S1", (2, 2))), # sub-array, just for fun!
    ("data_size", "u4"),
    #
    # the sound data itself cannot be represented here:
    # it does not have a fixed size
   ])

See also

wavreader.py

wav_header_dtype['format']

dtype('S4')

wav_header_dtype.fields

mappingproxy({'chunk_id': (dtype('S4'), 0),
              'chunk_size': (dtype('uint32'), 4),
              'format': (dtype('S4'), 8),
              'fmt_id': (dtype('S4'), 12),
              'fmt_size': (dtype('uint32'), 16),
              'audio_fmt': (dtype('uint16'), 20),
              'num_channels': (dtype('uint16'), 22),
              'sample_rate': (dtype('uint32'), 24),
              'byte_rate': (dtype('uint32'), 28),
              'block_align': (dtype('uint16'), 32),
              'bits_per_sample': (dtype('uint16'), 34),
              'data_id': (dtype(('S1', (2, 2))), 36),
              'data_size': (dtype('uint32'), 40)})

wav_header_dtype.fields['format']

(dtype('S4'), 8)

• The first element is the sub-dtype in the structured data, corresponding to the name format

• The second one is its offset (in bytes) from the beginning of the item

Exercise 43

Make a “sparse” dtype by using offsets, and only some of the fields:

wav_header_dtype = np.dtype(dict(
  names=['format', 'sample_rate', 'data_id'],
  offsets=[offset_1, offset_2, offset_3], # counted from start of structure in bytes
  formats=list of dtypes for each of the fields,
))

  Cell In[18], line 4
    formats=list of dtypes for each of the fields,
            ^
SyntaxError: invalid syntax. Perhaps you forgot a comma?

and use that to read the sample rate, and data_id (as sub-array).

f = open('data/test.wav', 'r')
wav_header = np.fromfile(f, dtype=wav_header_dtype, count=1)
f.close()
print(wav_header)

[(b'RIFF', 17402, b'WAVE', b'fmt ', 16, 1, 1, 16000, 32000, 2, 16, [[b'd', b'a'], [b't', b'a']], 17366)]

wav_header['sample_rate']

array([16000], dtype=uint32)

Let’s try accessing the sub-array:

wav_header['data_id']

array([[[b'd', b'a'],
        [b't', b'a']]], dtype='|S1')

wav_header.shape

(1,)

wav_header['data_id'].shape

(1, 2, 2)

When accessing sub-arrays, the dimensions get added to the end!

Note

There are existing modules such as wavfile, audiolab, etc. for loading sound data…

Casting and re-interpretation/views

casting

• on assignment

• on array construction

• on arithmetic

• etc.

• and manually: .astype(dtype)

data re-interpretation

• manually: .view(dtype)

Casting

• Casting in arithmetic, in nutshell:

  • only type (not value!) of operands matters

  • largest “safe” type able to represent both is picked

  • scalars can “lose” to arrays in some situations

• Casting in general copies data:

x = np.array([1, 2, 3, 4], dtype=float)
x

array([1., 2., 3., 4.])

y = x.astype(np.int8)
y

array([1, 2, 3, 4], dtype=int8)

y + 1

array([2, 3, 4, 5], dtype=int8)

y + 256

---------------------------------------------------------------------------
OverflowError                             Traceback (most recent call last)
Cell In[27], line 1
----> 1 y + 256

OverflowError: Python integer 256 out of bounds for int8

y + 256.0

array([257., 258., 259., 260.])

y + np.array([256], dtype=np.int32)

array([257, 258, 259, 260], dtype=int32)

• Casting on setitem: dtype of the array is not changed on item assignment:

y[:] = y + 1.5
y

array([2, 3, 4, 5], dtype=int8)

Note

Exact rules: see NumPy documentation

Re-interpretation / viewing

Let’s say we have a data block in memory (4 bytes). For the moment (as indicated by the bars between the values), we’ll consider this to be four unit8 values:

0x01	│	0x02	│	0x03	│	0x04

However, we can interpret this block as:

• 4 of uint8 (as here), OR,

• 4 of int8, OR,

• 2 of int16, OR,

• 1 of int32, OR,

• 1 of float32, OR,

• …

How to switch from one to another?

Option 1: Switch the dtype

x = np.array([1, 2, 3, 4], dtype=np.uint8)
x.dtype = "<i2"
x

array([ 513, 1027], dtype=int16)

0x01	0x02	│	0x03	0x04

Note

little-endian: least significant byte is on the left in memory

Option 2: Create a new view of type uint32, shorthand i4

y = x.view("<i4")
y

array([67305985], dtype=int32)

0x01	0x02	0x03	0x04

On Views

• .view() makes views, does not copy (or alter) the memory block

• it only changes the dtype (and adjusts array shape):

x[1] = 5
y

array([328193], dtype=int32)

y.base is x

True

Mini-exercise: data re-interpretation

See also

view-colors.py

Exercise 44

You have RGBA data in an array:

x = np.zeros((10, 10, 4), dtype=np.int8)
x[:, :, 0] = 1
x[:, :, 1] = 2
x[:, :, 2] = 3
x[:, :, 3] = 4

where the last three dimensions are the R, B, and G, and alpha channels.

How would you make a (10, 10) structured array with field names ‘r’, ‘g’, ‘b’, ‘a’ without copying data?

y = ...

assert (y['r'] == 1).all()
assert (y['g'] == 2).all()
assert (y['b'] == 3).all()
assert (y['a'] == 4).all()

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In[37], line 1
----> 1 assert (y['r'] == 1).all()
      2 assert (y['g'] == 2).all()
      3 assert (y['b'] == 3).all()

TypeError: 'ellipsis' object is not subscriptable

Solution to Exercise 44

y = x.view([('r', 'i1'),
            ('g', 'i1'),
            ('b', 'i1'),
            ('a', 'i1')]
             )[:, :, 0]

A puzzle

Another two arrays, each occupying exactly 4 bytes of memory:

x = np.array([[1, 3], [2, 4]], dtype=np.uint8)
x

array([[1, 3],
       [2, 4]], dtype=uint8)

y = x.transpose()
y

array([[1, 2],
       [3, 4]], dtype=uint8)

We view the elements of x (1 byte each) as int16 (2 bytes each):

x.view(np.int16)

array([[ 769],
       [1026]], dtype=int16)

What is happening here? Take a look at the bytes stored in memory by x:

x.tobytes()

b'\x01\x03\x02\x04'

The \x stands for heXadecimal, so what we are seeing is:

0x01 0x03 0x02 0x04

We ask NumPy to interpret these bytes as elements of dtype int16—each of which occupies two bytes in memory. Therefore, 0x01 0x03 becomes the first uint16 and 0x02 0x04 the second.

You may then expect to see 0x0103 (259, when converting from hexadecimal to decimal) as the first result. But your computer likely stores most significant bytes first, and as such reads the number as 0x0301 or 769 (go on and type 0x0301 into your Python terminal to verify).

We can do the same on a copy of y (why doesn’t it work on y directly?):

y.copy().view(np.int16)

array([[ 513],
       [1027]], dtype=int16)

Can you explain these numbers, 513 and 1027, as well as the output shape of the resulting array?

Indexing scheme: strides

Main point

The question:

x = np.array([[1, 2, 3],
              [4, 5, 6],
              [7, 8, 9]], dtype=np.int8)
x.tobytes('A')

b'\x01\x02\x03\x04\x05\x06\x07\x08\t'

The answer (in NumPy)

• strides: the number of bytes to jump to find the next element

• 1 stride per dimension

x.strides

(3, 1)

byte_offset = 3 * 1 + 1 * 2  # to find x[1, 2]
x.flat[byte_offset]

np.int8(6)

x[1, 2]

np.int8(6)

simple, flexible

C and Fortran order

Note

The Python built-in bytes returns bytes in C-order by default which can cause confusion when trying to inspect memory layout. We use numpy.ndarray.tobytes() with order=A instead, which preserves the C or F ordering of the bytes in memory.

x = np.array([[1, 2, 3],
              [4, 5, 6]], dtype=np.int16, order='C')
x.strides

(6, 2)

x.tobytes('A')

b'\x01\x00\x02\x00\x03\x00\x04\x00\x05\x00\x06\x00'

• Need to jump 6 bytes to find the next row

• Need to jump 2 bytes to find the next column

y = np.array(x, order='F')
y.strides

(2, 4)

y.tobytes('A')

b'\x01\x00\x04\x00\x02\x00\x05\x00\x03\x00\x06\x00'

• Need to jump 2 bytes to find the next row

• Need to jump 4 bytes to find the next column

Similarly for higher dimensions:

• C: last dimensions vary fastest (= smaller strides)

• F: first dimensions vary fastest

\[\begin{split} \begin{align} \mathrm{shape} &= (d_1, d_2, ..., d_n) \\ \mathrm{strides} &= (s_1, s_2, ..., s_n) \\ s_j^C &= d_{j+1} d_{j+2} ... d_{n} \times \mathrm{itemsize} \\ s_j^F &= d_{1} d_{2} ... d_{j-1} \times \mathrm{itemsize} \end{align} \end{split}\]

Now we can understand the behavior of .view()

y = np.array([[1, 3], [2, 4]], dtype=np.uint8).transpose()
x = y.copy()

Transposition does not affect the memory layout of the data, only strides

x.strides

(2, 1)

y.strides

(1, 2)

x.tobytes('A')

b'\x01\x02\x03\x04'

y.tobytes('A')

b'\x01\x03\x02\x04'

• the results are different when interpreted as 2 of int16

• .copy() creates new arrays in the C order (by default)

Slicing with integers

• Everything can be represented by changing only shape, strides, and possibly adjusting the data pointer!

• Never makes copies of the data

x = np.array([1, 2, 3, 4, 5, 6], dtype=np.int32)
y = x[::-1]
y

array([6, 5, 4, 3, 2, 1], dtype=int32)

y.strides

(-4,)

y = x[2:]
y.__array_interface__['data'][0] - x.__array_interface__['data'][0]

8

x = np.zeros((10, 10, 10), dtype=float)
x.strides

(800, 80, 8)

x[::2,::3,::4].strides

(1600, 240, 32)

Similarly, transposes never make copies (it just swaps strides):

x = np.zeros((10, 10, 10), dtype=float)
x.strides

(800, 80, 8)

x.T.strides

(8, 80, 800)

But: not all reshaping operations can be represented by playing with strides:

a = np.arange(6, dtype=np.int8).reshape(3, 2)
b = a.T
b.strides

(1, 2)

So far, so good. However:

bytes(a.data)

b'\x00\x01\x02\x03\x04\x05'

b

array([[0, 2, 4],
       [1, 3, 5]], dtype=int8)

c = b.reshape(3*2)
c

array([0, 2, 4, 1, 3, 5], dtype=int8)

Here, there is no way to represent the array c given one stride and the block of memory for a. Therefore, the reshape operation needs to make a copy here.

Example: fake dimensions with strides

Stride manipulation

from numpy.lib.stride_tricks import as_strided
help(as_strided)

Help on function as_strided in module numpy.lib.stride_tricks:

as_strided(x, shape=None, strides=None, subok=False, writeable=True)
    Create a view into the array with the given shape and strides.

    .. warning:: This function has to be used with extreme care, see notes.

    Parameters
    ----------
    x : ndarray
        Array to create a new.
    shape : sequence of int, optional
        The shape of the new array. Defaults to ``x.shape``.
    strides : sequence of int, optional
        The strides of the new array. Defaults to ``x.strides``.
    subok : bool, optional
        If True, subclasses are preserved.
    writeable : bool, optional
        If set to False, the returned array will always be readonly.
        Otherwise it will be writable if the original array was. It
        is advisable to set this to False if possible (see Notes).

    Returns
    -------
    view : ndarray

    See also
    --------
    broadcast_to : broadcast an array to a given shape.
    reshape : reshape an array.
    lib.stride_tricks.sliding_window_view :
        userfriendly and safe function for a creation of sliding window views.

    Notes
    -----
    ``as_strided`` creates a view into the array given the exact strides
    and shape. This means it manipulates the internal data structure of
    ndarray and, if done incorrectly, the array elements can point to
    invalid memory and can corrupt results or crash your program.
    It is advisable to always use the original ``x.strides`` when
    calculating new strides to avoid reliance on a contiguous memory
    layout.

    Furthermore, arrays created with this function often contain self
    overlapping memory, so that two elements are identical.
    Vectorized write operations on such arrays will typically be
    unpredictable. They may even give different results for small, large,
    or transposed arrays.

    Since writing to these arrays has to be tested and done with great
    care, you may want to use ``writeable=False`` to avoid accidental write
    operations.

    For these reasons it is advisable to avoid ``as_strided`` when
    possible.

Warning

as_strided does not check that you stay inside the memory block bounds…

x = np.array([1, 2, 3, 4], dtype=np.int16)
as_strided(x, strides=(2*2, ), shape=(2, ))

array([1, 3], dtype=int16)

x[::2]

array([1, 3], dtype=int16)

See also

stride-fakedims.py

Exercise 45

Convert this:

in_arr = np.array([1, 2, 3, 4], dtype=np.int8)
in_arr

array([1, 2, 3, 4], dtype=int8)

to this:

array([[1, 2, 3, 4],
       [1, 2, 3, 4],
       [1, 2, 3, 4]], dtype=np.int8)

using only as_strided.:

Hint: byte_offset = stride[0]*index[0] + stride[1]*index[1] + ...

Spoiler for strides exercise

Stride can also be 0:

Solution to Exercise 45

x = np.array([1, 2, 3, 4], dtype=np.int8)
y = as_strided(x, strides=(0, 1), shape=(3, 4))
y

array([[1, 2, 3, 4],
       [1, 2, 3, 4],
       [1, 2, 3, 4]], dtype=int8)

y.base.base is x

True

Broadcasting

• Doing something useful with it: outer product of [1, 2, 3, 4] and [5, 6, 7]

x = np.array([1, 2, 3, 4], dtype=np.int16)
x2 = as_strided(x, strides=(0, 1*2), shape=(3, 4))
x2

array([[1, 2, 3, 4],
       [1, 2, 3, 4],
       [1, 2, 3, 4]], dtype=int16)

y = np.array([5, 6, 7], dtype=np.int16)
y2 = as_strided(y, strides=(1*2, 0), shape=(3, 4))
y2

array([[5, 5, 5, 5],
       [6, 6, 6, 6],
       [7, 7, 7, 7]], dtype=int16)

x2 * y2

array([[ 5, 10, 15, 20],
       [ 6, 12, 18, 24],
       [ 7, 14, 21, 28]], dtype=int16)

… seems somehow familiar …

x = np.array([1, 2, 3, 4], dtype=np.int16)
y = np.array([5, 6, 7], dtype=np.int16)
x[np.newaxis,:] * y[:,np.newaxis]

array([[ 5, 10, 15, 20],
       [ 6, 12, 18, 24],
       [ 7, 14, 21, 28]], dtype=int16)

• Internally, array broadcasting is indeed implemented using 0-strides.

More tricks: diagonals

See also

stride-diagonals.py

Exercise 46

Pick diagonal entries of the matrix: (assume C memory order):

x = np.array([[1, 2, 3],
              [4, 5, 6],
              [7, 8, 9]], dtype=np.int32)

Your task is to work out the correct strides for to get the diagonal of the array, as in:

x_diag = as_strided(x, shape=(3,), strides=(...,))

Next:

• Pick the first super-diagonal entries [2, 6].

• And the sub-diagonals?

Hint to the last two: slicing first moves the point where striding starts from.

Solution to Exercise 46

Pick diagonals:

x_diag = as_strided(x, shape=(3, ), strides=((3+1)*x.itemsize,))
x_diag

array([1, 5, 9], dtype=int32)

Slice first, to adjust the data pointer:

as_strided(x[0, 1:], shape=(2, ), strides=((3+1)*x.itemsize, ))

array([2, 6], dtype=int32)

as_strided(x[1:, 0], shape=(2, ), strides=((3+1)*x.itemsize, ))

array([4, 8], dtype=int32)

Using np.diag

y = np.diag(x, k=1)
y

array([2, 6], dtype=int32)

However,

y.flags.owndata

False

Challenge

Exercise 47

Compute the tensor trace:

x = np.arange(5*5*5*5).reshape(5, 5, 5, 5)
s = 0
for i in range(5):
    for j in range(5):
        s += x[j, i, j, i]

by striding, and using sum() on the result.

y = as_strided(x, shape=(5, 5), strides=(..., ...))
s2 = ...
assert s == s2

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In[85], line 1
----> 1 y = as_strided(x, shape=(5, 5), strides=(..., ...))
      2 s2 = ...
      3 assert s == s2

File /Volumes/zorg/mb312/.virtualenvs/sp-lectures/lib/python3.12/site-packages/numpy/lib/_stride_tricks_impl.py:101, in as_strided(x, shape, strides, subok, writeable)
     98 if strides is not None:
     99     interface['strides'] = tuple(strides)
--> 101 array = np.asarray(DummyArray(interface, base=x))
    102 # The route via `__interface__` does not preserve structured
    103 # dtypes. Since dtype should remain unchanged, we set it explicitly.
    104 array.dtype = x.dtype

TypeError: 'ellipsis' object cannot be interpreted as an integer

Solution to Exercise 47

y = as_strided(x, shape=(5, 5), strides=((5*5*5 + 5)*x.itemsize,
                                         (5*5 + 1)*x.itemsize))
s2 = y.sum()
s2

np.int64(7800)

CPU cache effects

Memory layout can affect performance:

x = np.zeros((20000,))
y = np.zeros((20000*67,))[::67]

x.shape, y.shape

((20000,), (20000,))

%timeit np.median(x)

87.2 μs ± 6.22 μs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)

%timeit np.median(y)

146 μs ± 5.81 μs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)

x.strides, y.strides

((8,), (536,))

Note

** Are smaller strides faster**

• CPU pulls data from main memory to its cache in blocks

• If many array items consecutively operated on fit in a single block (small stride):

  • \(\Rightarrow\) fewer transfers needed

  • \(\Rightarrow\) faster

See also

• numexpr is designed to mitigate cache effects when evaluating array expressions.

• numba is a compiler for Python code, that is aware of numpy arrays.

Findings in dissection

• memory block: may be shared, .base, .data

• data type descriptor: structured data, sub-arrays, byte order, casting, viewing, .astype(), .view()

• strided indexing: strides, C/F-order, slicing w/ integers, as_strided, broadcasting, stride tricks, diag, CPU cache coherence

Universal functions

What are they?

• Ufunc performs an elementwise operation on all elements of an array.

  Examples: np.add, np.subtract, scipy.special.*, …

• Automatically support: broadcasting, casting, …

• The author of an ufunc only has to supply the elementwise operation, NumPy takes care of the rest.

• The elementwise operation needs to be implemented in C (or, e.g., Cython)

Parts of an Ufunc

Part 1: provided by user

void ufunc_loop(void **args, int *dimensions, int *steps, void *data)
{
    /*
    * int8 output = elementwise_function(int8 input_1, int8 input_2)
    *
    * This function must compute the ufunc for many values at once,
    * in the way shown below.
    */
    char *input_1 = (char*)args[0];
    char *input_2 = (char*)args[1];
    char *output = (char*)args[2];
    int i;

    for (i = 0; i < dimensions[0]; ++i) {
        *output = elementwise_function(*input_1, *input_2);
        input_1 += steps[0];
        input_2 += steps[1];
        output += steps[2];
    }
}

Part 2. The NumPy part, built by

char types[3]

types[0] = NPY_BYTE   /* type of first input arg */
types[1] = NPY_BYTE   /* type of second input arg */
types[2] = NPY_BYTE   /* type of third input arg */

PyObject *python_ufunc = PyUFunc_FromFuncAndData(
    ufunc_loop,
    NULL,
    types,
    1, /* ntypes */
    2, /* num_inputs */
    1, /* num_outputs */
    identity_element,
    name,
    docstring,
    unused)

A ufunc can also support multiple different input-output type combinations.

Making it easier

ufunc_loop is of very generic form, and NumPy provides pre-made ones

PyUfunc_f_f	float elementwise_func(float input_1)
PyUfunc_ff_f	float elementwise_func(float input_1, float input_2)
PyUfunc_d_d	double elementwise_func(double input_1)
PyUfunc_dd_d	double elementwise_func(double input_1, double input_2)
PyUfunc_D_D	elementwise_func(npy_cdouble *input, npy_cdouble* output)
PyUfunc_DD_D	elementwise_func(npy_cdouble *in1, npy_cdouble *in2, npy_cdouble* out)

• Only elementwise_func needs to be supplied

• … except when your elementwise function is not in one of the above forms

Exercise: building an ufunc from scratch

Exercise 48

The Mandelbrot fractal is defined by the iteration

\[ z \leftarrow z^2 + c \]

where \(c = x + i y\) is a complex number. This iteration is repeated – if \(z\) stays finite no matter how long the iteration runs, \(c\) belongs to the Mandelbrot set.

First — make a ufunc called mandel(z0, c) that computes:

z = z0
for k in range(iterations):
    z = z*z + c

Run for, say, 100 iterations or until z.real**2 + z.imag**2 > 1000. Use it to determine which c are in the Mandelbrot set.

Our function is a simple one, so make use of the PyUFunc_* helpers.

Write it in Cython

See also

mandel.pyx, mandelplot.py

Reminder: some pre-made Ufunc loops:

PyUfunc_f_f	float elementwise_func(float input_1)
PyUfunc_ff_f	float elementwise_func(float input_1, float input_2)
PyUfunc_d_d	double elementwise_func(double input_1)
PyUfunc_dd_d	double elementwise_func(double input_1, double input_2)
PyUfunc_D_D	elementwise_func(complex_double *input, complex_double* output)
PyUfunc_DD_D	elementwise_func(complex_double *in1, complex_double *in2, complex_double* out)

Type codes:

NPY_BOOL, NPY_BYTE, NPY_UBYTE, NPY_SHORT, NPY_USHORT, NPY_INT, NPY_UINT,
NPY_LONG, NPY_ULONG, NPY_LONGLONG, NPY_ULONGLONG, NPY_FLOAT, NPY_DOUBLE,
NPY_LONGDOUBLE, NPY_CFLOAT, NPY_CDOUBLE, NPY_CLONGDOUBLE, NPY_DATETIME,
NPY_TIMEDELTA, NPY_OBJECT, NPY_STRING, NPY_UNICODE, NPY_VOID

Solution to Exercise 48

# The elementwise function
# ------------------------

cdef void mandel_single_point(double complex *z_in,
                              double complex *c_in,
                              double complex *z_out) noexcept nogil:
    #
    # The Mandelbrot iteration
    #

    #
    # Some points of note:
    #
    # - It's *NOT* allowed to call any Python functions here.
    #
    #   The Ufunc loop runs with the Python Global Interpreter Lock released.
    #   Hence, the ``nogil``.
    #
    # - And so all local variables must be declared with ``cdef``
    #
    # - Note also that this function receives *pointers* to the data;
    #   the "traditional" solution to passing complex variables around
    #

    cdef double complex z = z_in[0]
    cdef double complex c = c_in[0]
    cdef int k  # the integer we use in the for loop

    # Straightforward iteration

    for k in range(100):
        z = z*z + c
        if z.real**2 + z.imag**2 > 1000:
            break

    # Return the answer for this point
    z_out[0] = z


# Boilerplate Cython definitions
#
# Pulls definitions from the NumPy C headers.
# -------------------------------------------

from numpy cimport import_array, import_ufunc
from numpy cimport (PyUFunc_FromFuncAndData,
                    PyUFuncGenericFunction)
from numpy cimport NPY_CDOUBLE
from numpy cimport PyUFunc_DD_D

# Required module initialization
# ------------------------------

import_array()
import_ufunc()


# The actual ufunc declaration
# ----------------------------

cdef PyUFuncGenericFunction loop_func[1]
cdef char input_output_types[3]
cdef void *elementwise_funcs[1]

loop_func[0] = PyUFunc_DD_D

input_output_types[0] = NPY_CDOUBLE
input_output_types[1] = NPY_CDOUBLE
input_output_types[2] = NPY_CDOUBLE

elementwise_funcs[0] = <void*>mandel_single_point

mandel = PyUFunc_FromFuncAndData(
    loop_func,
    elementwise_funcs,
    input_output_types,
    1, # number of supported input types
    2, # number of input args
    1, # number of output args
    0, # `identity` element, never mind this
    "mandel", # function name
    "mandel(z, c) -> computes iterated z*z + c", # docstring
    0 # unused
    )

"""
Plot Mandelbrot
================

Plot the Mandelbrot ensemble.

"""

import numpy as np
import mandel

x = np.linspace(-1.7, 0.6, 1000)
y = np.linspace(-1.4, 1.4, 1000)
c = x[None, :] + 1j * y[:, None]
z = mandel.mandel(c, c)

import matplotlib.pyplot as plt

plt.imshow(abs(z) ** 2 < 1000, extent=[-1.7, 0.6, -1.4, 1.4])
plt.gray()
plt.show()

Note

Most of the boilerplate could be automated by these Cython modules:

cython/cython

Several accepted input types

E.g. supporting both single- and double-precision versions

cdef void mandel_single_point(double complex *z_in,
                              double complex *c_in,
                              double complex *z_out) nogil:
   ...

cdef void mandel_single_point_singleprec(float complex *z_in,
                                         float complex *c_in,
                                         float complex *z_out) nogil:
   ...

cdef PyUFuncGenericFunction loop_funcs[2]
cdef char input_output_types[3*2]
cdef void *elementwise_funcs[1*2]

loop_funcs[0] = PyUFunc_DD_D
input_output_types[0] = NPY_CDOUBLE
input_output_types[1] = NPY_CDOUBLE
input_output_types[2] = NPY_CDOUBLE
elementwise_funcs[0] = <void*>mandel_single_point

loop_funcs[1] = PyUFunc_FF_F
input_output_types[3] = NPY_CFLOAT
input_output_types[4] = NPY_CFLOAT
input_output_types[5] = NPY_CFLOAT
elementwise_funcs[1] = <void*>mandel_single_point_singleprec

mandel = PyUFunc_FromFuncAndData(
    loop_func,
    elementwise_funcs,
    input_output_types,
    2, # number of supported input types   <----------------
    2, # number of input args
    1, # number of output args
    0, # `identity` element, never mind this
    "mandel", # function name
    "mandel(z, c) -> computes iterated z*z + c", # docstring
    0 # unused
    )

Generalized ufuncs

ufunc

output = elementwise_function(input)

Both output and input can be a single array element only.

generalized ufunc

output and input can be arrays with a fixed number of dimensions

For example, matrix trace (sum of diag elements):

input shape = (n, n)
output shape = ()    #  i.e.  scalar

(n, n) -> ()

Matrix product:

input_1 shape = (m, n)
input_2 shape = (n, p)
output shape  = (m, p)

(m, n), (n, p) -> (m, p)

• This is called the “signature” of the generalized ufunc

• The dimensions on which the g-ufunc acts, are “core dimensions”

Status in NumPy

• g-ufuncs are in NumPy already …

• new ones can be created with PyUFunc_FromFuncAndDataAndSignature

• most linear-algebra functions are implemented as g-ufuncs to enable working with stacked arrays:

import numpy as np
rng = np.random.default_rng(27446968)
np.linalg.det(rng.random((3, 5, 5)))

array([ 0.01829761, -0.0077266 , -0.05336566])

np.linalg._umath_linalg.det.signature

'(m,m)->()'

• matrix multiplication this way could be useful for operating on many small matrices at once

• Also see tensordot and einsum

Generalized ufunc loop

Matrix multiplication (m,n),(n,p) -> (m,p)

void gufunc_loop(void **args, int *dimensions, int *steps, void *data)
{
    char *input_1 = (char*)args[0];  /* these are as previously */
    char *input_2 = (char*)args[1];
    char *output = (char*)args[2];

    int input_1_stride_m = steps[3];  /* strides for the core dimensions */
    int input_1_stride_n = steps[4];  /* are added after the non-core */
    int input_2_strides_n = steps[5]; /* steps */
    int input_2_strides_p = steps[6];
    int output_strides_n = steps[7];
    int output_strides_p = steps[8];

    int m = dimension[1]; /* core dimensions are added after */
    int n = dimension[2]; /* the main dimension; order as in */
    int p = dimension[3]; /* signature */

    int i;

    for (i = 0; i < dimensions[0]; ++i) {
        matmul_for_strided_matrices(input_1, input_2, output,
                                    strides for each array...);

        input_1 += steps[0];
        input_2 += steps[1];
        output += steps[2];
    }
}

Interoperability features

Sharing multidimensional, typed data

Suppose you

1. Write a library than handles (multidimensional) binary data,

2. Want to make it easy to manipulate the data with NumPy, or whatever other library,

3. … but would not like to have NumPy as a dependency.

Currently, 3 solutions:

1. the “old” buffer interface

2. the array interface

3. the “new” buffer interface (PEP 3118)

The old buffer protocol

• Only 1-D buffers

• No data type information

• C-level interface; PyBufferProcs tp_as_buffer in the type object

• But it’s integrated into Python (e.g. strings support it)

Mini-exercise using Pillow (Python Imaging Library):

See also

pilbuffer.py

Exercise 49

from PIL import Image
data = np.zeros((200, 200, 4), dtype=np.uint8)
data[:, :] = [255, 0, 0, 255] # Red
# In PIL, RGBA images consist of 32-bit integers whose bytes are [RR,GG,BB,AA]
data = data.view(np.int32).squeeze()
img = Image.frombuffer("RGBA", (200, 200), data, "raw", "RGBA", 0, 1)
img.save('test.png')

The question

What happens if data is now modified, and img saved again?

The old buffer protocol

Show how to exchange data between numpy and a library that only knows the buffer interface:

# Make a sample image, RGBA format
x = np.zeros((200, 200, 4), dtype=np.uint8)
x[:, :, 0] = 255  # red
x[:, :, 3] = 255  # opaque

data_i32 = x.view(np.int32)  # Check that you understand why this is OK!

img = Image.frombuffer("RGBA", (200, 200), data_i32)
img.save("test_red.png")

# Modify the original data, and save again.
x[:, :, 1] = 255
img.save("test_recolored.png")

Array interface protocol

• Multidimensional buffers

• Data type information present

• NumPy-specific approach; slowly deprecated (but not going away)

• Not integrated in Python otherwise

See also

Documentation: https://numpy.org/doc/stable/reference/arrays.interface.html

x = np.array([[1, 2], [3, 4]])
x.__array_interface__

{'data': (105553141891808, False),
 'strides': None,
 'descr': [('', '<i8')],
 'typestr': '<i8',
 'shape': (2, 2),
 'version': 3}

Show code cell source

Hide code cell source

import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import os
if not os.path.exists('data'): os.mkdir('data')
plt.imsave('data/test.png', data)

from PIL import Image
img = Image.open('data/test.png')
img.__array_interface__

{'version': 3,
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ff',
 'shape': (200, 200, 4),
 'typestr': '|u1'}

x = np.asarray(img)
x.shape

(200, 200, 4)

Note

A more C-friendly variant of the array interface is also defined.

Array siblings: chararray, MaskedArray

chararray: vectorized string operations

x = np.char.asarray(['a', '  bbb', '  ccc'])
x

chararray(['a', '  bbb', '  ccc'], dtype='<U5')

x.upper()

chararray(['A', '  BBB', '  CCC'], dtype='<U5')

MaskedArray missing data

Masked arrays are arrays that may have missing or invalid entries.

For example, suppose we have an array where the fourth entry is invalid:

x = np.array([1, 2, 3, -99, 5])

One way to describe this is to create a masked array:

mx = np.ma.MaskedArray(x, mask=[0, 0, 0, 1, 0])
mx

masked_array(data=[1, 2, 3, --, 5],
             mask=[False, False, False,  True, False],
       fill_value=999999)

Masked mean ignores masked data:

mx.mean()

np.float64(2.75)

np.mean(mx)

np.float64(2.75)

Warning

Not all NumPy functions respect masks, for instance np.dot, so check the return types.

The MaskedArray returns a view to the original array:

mx[1] = 9
x

array([  1,   9,   3, -99,   5])

The mask

You can modify the mask by assigning:

mx[1] = np.ma.masked
mx

masked_array(data=[1, --, 3, --, 5],
             mask=[False,  True, False,  True, False],
       fill_value=999999)

The mask is cleared on assignment:

mx[1] = 9
mx

masked_array(data=[1, 9, 3, --, 5],
             mask=[False, False, False,  True, False],
       fill_value=999999)

The mask is also available directly:

mx.mask

array([False, False, False,  True, False])

The masked entries can be filled with a given value to get an usual array back:

x2 = mx.filled(-1)
x2

array([ 1,  9,  3, -1,  5])

The mask can also be cleared:

mx.mask = np.ma.nomask
mx

masked_array(data=[1, 9, 3, -99, 5],
             mask=[False, False, False, False, False],
       fill_value=999999)

Domain-aware functions

The masked array package also contains domain-aware functions:

np.ma.log(np.array([1, 2, -1, -2, 3, -5]))

masked_array(data=[0.0, 0.6931471805599453, --, --, 1.0986122886681098,
                   --],
             mask=[False, False,  True,  True, False,  True],
       fill_value=1e+20)

Note

Streamlined and more seamless support for dealing with missing data in arrays is making its way into NumPy 1.7. Stay tuned!

Example: Masked statistics

Canadian rangers were distracted when counting hares and lynxes in 1903-1910 and 1917-1918, and got the numbers are wrong. (Carrot farmers stayed alert, though.) Compute the mean populations over time, ignoring the invalid numbers.

data = np.loadtxt('data/populations.txt')
populations = np.ma.MaskedArray(data[:,1:])
year = data[:, 0]

bad_years = (((year >= 1903) & (year <= 1910))
           | ((year >= 1917) & (year <= 1918)))
# '&' means 'and' and '|' means 'or'
populations[bad_years, 0] = np.ma.masked
populations[bad_years, 1] = np.ma.masked

populations.mean(axis=0)

masked_array(data=[40472.72727272727, 18627.272727272728, 42400.0],
             mask=[False, False, False],
       fill_value=1e+20)

populations.std(axis=0)

masked_array(data=[21087.656489006717, 15625.799814240254,
                   3322.5062255844787],
             mask=[False, False, False],
       fill_value=1e+20)

Note that Matplotlib knows about masked arrays:

plt.plot(year, populations, 'o-')

[<matplotlib.lines.Line2D at 0x113af5a30>,
 <matplotlib.lines.Line2D at 0x113b776b0>,
 <matplotlib.lines.Line2D at 0x113b77740>]

np.recarray: purely convenience

arr = np.array([('a', 1), ('b', 2)], dtype=[('x', 'S1'), ('y', int)])
arr2 = arr.view(np.recarray)
arr2.x

array([b'a', b'b'], dtype='|S1')

arr2.y

array([1, 2])

Summary

• Anatomy of the ndarray: data, dtype, strides.

• Universal functions: elementwise operations, how to make new ones

• Ndarray subclasses

• Various buffer interfaces for integration with other tools

• Recent additions: PEP 3118, generalized ufuncs

Contributing to NumPy/SciPy

Get this tutorial: https://www.euroscipy.org/talk/882

Why

• “There’s a bug?”

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  • No replies in a week or so? Just file a bug ticket.

Good bug report

Title: numpy.random.permutations fails for non-integer arguments

I'm trying to generate random permutations, using numpy.random.permutations

When calling numpy.random.permutation with non-integer arguments
it fails with a cryptic error message::

    >>> rng.permutation(12)
    array([ 2,  6,  4,  1,  8, 11, 10,  5,  9,  3,  7,  0])
    >>> rng.permutation(12.)
    Traceback (most recent call last):
    File "<stdin>", line 1, in <module>
    File "_generator.pyx", line 4844, in numpy.random._generator.Generator.permutation
    numpy.exceptions.AxisError: axis 0 is out of bounds for array of dimension 0

This also happens with long arguments, and so
np.random.permutation(X.shape[0]) where X is an array fails on 64
bit windows (where shape is a tuple of longs).

It would be great if it could cast to integer or at least raise a
proper error for non-integer types.

I'm using NumPy 1.4.1, built from the official tarball, on Windows
64 with Visual studio 2008, on Python.org 64-bit Python.

0. What are you trying to do?

1. Small code snippet reproducing the bug (if possible)

   • What actually happens

   • What you’d expect

2. Platform (Windows / Linux / OSX, 32/64 bits, x86/PPC, …)

3. Version of NumPy/SciPy

print(np.__version__)

2.2.5

Check that the following is what you expect

print(np.__file__)

/Volumes/zorg/mb312/.virtualenvs/sp-lectures/lib/python3.12/site-packages/numpy/__init__.py

In case you have old/broken NumPy installations lying around.

If unsure, try to remove existing NumPy installations, and reinstall…

Contributing to documentation

1. Documentation editor

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     • Send a mail @ scipy-dev mailing list; ask for activation:

       To: scipy-dev@scipy.org
       
       Hi,
       
       I'd like to edit NumPy/SciPy docstrings. My account is XXXXX
       
       Cheers,
       N. N.
       

   • Check the style guide:

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     • Don’t be intimidated; to fix a small thing, just fix it

   • Edit

2. Edit sources and send patches (as for bugs)

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Contributing features

The contribution of features is documented on https://numpy.org/doc/stable/dev/

How to help, in general

• Bug fixes always welcome!

  • What irks you most

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• Documentation work

  • API docs: improvements to docstrings

    • Know some SciPy module well?

  • User guide

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• Ask on communication channels:

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  • scipy-dev list