Introduction

In the last chapter of the usermod documentation, I mentioned that I have another story, for another day. The day has come, so here is my story.

Enter ulab

ulab is a numpy-like module for micropython, meant to simplify and speed up common mathematical operations on arrays. The primary goal was to implement a small subset of numpy that might be useful in the context of a microcontroller. This means low-level data processing of linear (array) and two-dimensional (matrix) data.

Purpose

Of course, the first question that one has to answer is, why on Earth one would need a fast math library on a microcontroller. After all, it is not expected that heavy number crunching is going to take place on bare metal. It is not meant to. On a PC, the main reason for writing fast code is the sheer amount of data that one wants to process. On a microcontroller, the data volume is probably small, but it might lead to catastrophic system failure, if these data are not processed in time, because the microcontroller is supposed to interact with the outside world in a timely fashion. In fact, this latter objective was the initiator of this project: I needed the Fourier transform of the ADC signal, and all the available options were simply too slow.

In addition to speed, another issue that one has to keep in mind when working with embedded systems is the amount of available RAM: I believe, everything here could be implemented in pure python with relatively little effort, but the price we would have to pay for that is not only speed, but RAM, too. python code, if is not frozen, and compiled into the firmware, has to be compiled at runtime, which is not exactly a cheap process. On top of that, if numbers are stored in a list or tuple, which would be the high-level container, then they occupy 8 bytes, no matter, whether they are all smaller than 100, or larger than one hundred million. This is obviously a waste of resources in an environment, where resources are scarce.

Finally, there is a reason for using micropython in the first place. Namely, that a microcontroller can be programmed in a very elegant, and pythonic way. But if it is so, why should we not extend this idea to other tasks and concepts that might come up in this context? If there was no other reason than this elegance, I would find that convincing enough.

Based on the above-mentioned considerations, all functions are implemented in a way that

  1. conforms to numpy as much as possible
  2. is so frugal with RAM as possible,
  3. and yet, fast. Much faster than pure python.

The main points of ulab are

  • compact, iterable and slicable containers of numerical data in 1, and 2 dimensions (arrays and matrices). These containers support all the relevant unary and binary operators (e.g., len, ==, +, *, etc.)
  • vectorised computations on micropython iterables and numerical arrays/matrices (in numpy-speak, universal functions)
  • basic linear algebra routines (matrix inversion, multiplication, reshaping, transposition, determinant, and eigenvalues)
  • polynomial fits to numerical data
  • fast Fourier transforms

At the time of writing this manual (for version 0.32), the library adds approximately 30 kB of extra compiled code to the micropython (pyboard.v.11) firmware. However, if you are tight with flash space, you can easily shave off a couple of kB. See the section on customising ulab.

Friendly request

If you use ulab, and bump into a bug, or think that a particular function is missing, or its behaviour does not conform to numpy, please, raise an issue on github, so that the community can profit from your experiences.

Even better, if you find the project useful, and think that it could be made better, faster, tighter, and shinier, please, consider contributing, and issue a pull request with the implementation of your improvements and new features. ulab can only become successful, if it offers what the community needs.

These last comments apply to the documentation, too. If, in your opinion, the documentation is obscure, misleading, or not detailed enough, please, let me know, so that we can fix it.

Supported functions and methods

ulab supports a number of array operators, which are listed here. I tried to follow the specifications of the numpy interface as closely as possible, though, it was not always practical to implement verbatim behaviour. The differences, if any, are in each case small (e.g., a function cannot take all possible keyword arguments), and should not hinder everyday use. In the list below, a single asterisk denotes slight deviations from numpy’s nomenclature, and a double asterisk denotes those cases, where a bit more caution should be exercised, though this usually means functions that are not supported by numpy.

The detailed discussion of the various functions always contains a link to the corresponding numpy documentation. However, before going down the rabbit hole, the module also defines a constant, the version, which can always be queried as

# code to be run in micropython

import ulab as np

print('you are running ulab version', np.__version__)
you are running ulab version 0.24

If you find a bug, please, include this number in your report!

Customising ulab

ulab implements a great number of functions, and it is quite possible that you do not need all of them in a particular application. If you want to save some flash space, you can easily exclude arbitrary functions from the firmware. The https://github.com/v923z/micropython-ulab/blob/master/code/ulab.h header file contains a pre-processor flag for all functions in ulab. The default setting is 1 for each of them, but if you change that to 0, the corresponding function will not be part of the compiled firmware.

The first couple of lines of the file look like this

// vectorise (all functions) takes approx. 3 kB of flash space
#define ULAB_VECTORISE_ACOS (1)
#define ULAB_VECTORISE_ACOSH (1)
#define ULAB_VECTORISE_ASIN (1)
#define ULAB_VECTORISE_ASINH (1)
#define ULAB_VECTORISE_ATAN (1)
#define ULAB_VECTORISE_ATANH (1)

In order to simplify navigation in the file, each flag begins with ULAB_, continues with the sub-module, where the function itself is implemented, and ends with the function’s name. Each section displays a hint as to how much space you can save by un-setting the flag.

Basic ndarray operations

Unary operators

Binary operators

Indexing and slicing

ndarray iterators

Comparison operators*

Universal functions (also support function calls on general iterables)

Methods of ndarrays

.shape

.reshape

.transpose

.flatten**

Matrix methods

size

inv

dot

det

roll

flip

Array initialisation functions

eye

ones

zeros

linspace

Statistical and other properties of arrays

min

argmin

max

argmax

sum

std

mean

diff

sort

argsort

Manipulation of polynomials

polyval

polyfit

FFT routines

fft**

ifft**

spectrum**

Filter functions

convolve

ndarray, the basic container

The ndarray is the underlying container of numerical data. It is derived from micropython’s own array object, but has a great number of extra features starting with how it can be initialised, how operations can be done on it, and which functions can accept it as an argument.

Since the ndarray is a binary container, it is also compact, meaning that it takes only a couple of bytes of extra RAM in addition to what is required for storing the numbers themselves. ndarrays are also type-aware, i.e., one can save RAM by specifying a data type, and using the smallest reasonable one. Five such types are defined, namely uint8, int8, which occupy a single byte of memory per datum, uint16, and int16, which occupy two bytes per datum, and float, which occupies four or eight bytes per datum. The precision/size of the float type depends on the definition of mp_float_t. Some platforms, e.g., the PYBD, implement doubles, but some, e.g., the pyboard.v.11, don’t. You can find out, what type of float your particular platform implements by looking at the output of the .rawsize class method.

On the following pages, we will see how one can work with ndarrays. Those familiar with numpy should find that the nomenclature and naming conventions of numpy are adhered to as closely as possible. I will point out the few differences, where necessary.

For the sake of comparison, in addition to ulab code snippets, sometimes the equivalent numpy code is also presented. You can find out, where the snippet is supposed to run by looking at its first line, the header.

Hint: you can easily port existing numpy code, if you import ulab as np.

Initialising an array

A new array can be created by passing either a standard micropython iterable, or another ndarray into the constructor.

Initialising by passing iterables

If the iterable is one-dimensional, i.e., one whose elements are numbers, then a row vector will be created and returned. If the iterable is two-dimensional, i.e., one whose elements are again iterables, a matrix will be created. If the lengths of the iterables is not consistent, a ValueError will be raised. Iterables of different types can be mixed in the initialisation function.

If the dtype keyword with the possible uint8/int8/uint16/int16/float values is supplied, the new ndarray will have that type, otherwise, it assumes float as default.

# code to be run in micropython

import ulab as np

a = [1, 2, 3, 4, 5, 6, 7, 8]
b = np.array(a)

print("a:\t", a)
print("b:\t", b)

# a two-dimensional array with mixed-type initialisers
c = np.array([range(5), range(20, 25, 1), [44, 55, 66, 77, 88]], dtype=np.uint8)
print("\nc:\t", c)

# and now we throw an exception
d = np.array([range(5), range(10), [44, 55, 66, 77, 88]], dtype=np.uint8)
print("\nd:\t", d)
a:   [1, 2, 3, 4, 5, 6, 7, 8]
b:   array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)

c:   array([[0, 1, 2, 3, 4],
     [20, 21, 22, 23, 24],
     [44, 55, 66, 77, 88]], dtype=uint8)

Traceback (most recent call last):
  File "/dev/shm/micropython.py", line 15, in <module>
ValueError: iterables are not of the same length

ndarrays are pretty-printed, i.e., if the length is larger than 10, then only the first and last three entries will be printed. Also note that, as opposed to numpy, the printout always contains the dtype.

# code to be run in micropython

import ulab as np

a = np.array(range(200))
print("a:\t", a)
a:   array([0.0, 1.0, 2.0, ..., 197.0, 198.0, 199.0], dtype=float)

Initialising by passing arrays

An ndarray can be initialised by supplying another array. This statement is almost trivial, since ndarrays are iterables themselves, though it should be pointed out that initialising through arrays should be faster, because simply a new copy is created, without inspection, iteration etc.

# code to be run in micropython

import ulab as np

a = [1, 2, 3, 4, 5, 6, 7, 8]
b = np.array(a)
c = np.array(b)

print("a:\t", a)
print("b:\t", b)
print("\nc:\t", c)
a:   [1, 2, 3, 4, 5, 6, 7, 8]
b:   array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)

c:   array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)

Methods of ndarrays

.shape

The .shape method (property) returns a 2-tuple with the number of rows, and columns.

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3, 4], dtype=np.int8)
print("a:\n", a)
print("shape of a:", a.shape)

b= np.array([[1, 2], [3, 4]], dtype=np.int8)
print("\nb:\n", b)
print("shape of b:", b.shape)
a:
 array([1, 2, 3, 4], dtype=int8)
shape of a: (1, 4)

b:
 array([[1, 2],
     [3, 4]], dtype=int8)
shape of b: (2, 2)

.size

The .size method (property) returns an integer with the number of elements in the array.

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3], dtype=np.int8)
print("a:\n", a)
print("size of a:", a.size)

b= np.array([[1, 2], [3, 4]], dtype=np.int8)
print("\nb:\n", b)
print("size of b:", b.size)
a:
 array([1, 2, 3], dtype=int8)
size of a: 3

b:
 array([[1, 2],
     [3, 4]], dtype=int8)
size of b: 4

.itemsize

The .itemsize method (property) returns an integer with the siz enumber of elements in the array.

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3], dtype=np.int8)
print("a:\n", a)
print("itemsize of a:", a.itemsize)

b= np.array([[1, 2], [3, 4]], dtype=np.float)
print("\nb:\n", b)
print("itemsize of b:", b.itemsize)
a:
 array([1, 2, 3], dtype=int8)
itemsize of a: 1

b:
 array([[1.0, 2.0],
     [3.0, 4.0]], dtype=float)
itemsize of b: 8

.reshape

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.reshape.html

reshape re-writes the shape properties of an ndarray, but the array will not be modified in any other way. The function takes a single 2-tuple with two integers as its argument. The 2-tuple should specify the desired number of rows and columns. If the new shape is not consistent with the old, a ValueError exception will be raised.

# code to be run in micropython

import ulab as np

a = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]], dtype=np.uint8)
print('a (4 by 4):', a)
print('a (2 by 8):', a.reshape((2, 8)))
print('a (1 by 16):', a.reshape((1, 16)))
a (4 by 4): array([[1, 2, 3, 4],
     [5, 6, 7, 8],
     [9, 10, 11, 12],
     [13, 14, 15, 16]], dtype=uint8)
a (2 by 8): array([[1, 2, 3, 4, 5, 6, 7, 8],
     [9, 10, 11, 12, 13, 14, 15, 16]], dtype=uint8)
a (1 by 16): array([1, 2, 3, ..., 14, 15, 16], dtype=uint8)

.flatten

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.ndarray.flatten.htm

.flatten returns the flattened array. The array can be flattened in C style (i.e., moving horizontally in the matrix), or in fortran style (i.e., moving vertically in the matrix). The C-style flattening is the default, and it is also fast, because this is just a verbatim copy of the contents.

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3, 4], dtype=np.int8)
print("a: \t\t", a)
print("a flattened: \t", a.flatten())

b = np.array([[1, 2, 3], [4, 5, 6]], dtype=np.int8)
print("\nb:", b)

print("b flattened (C): \t", b.flatten())
print("b flattened (F): \t", b.flatten(order='F'))
a:           array([1, 2, 3, 4], dtype=int8)
a flattened:         array([1, 2, 3, 4], dtype=int8)

b: array([[1, 2, 3],
     [4, 5, 6]], dtype=int8)
b flattened (C):     array([1, 2, 3, 4, 5, 6], dtype=int8)
b flattened (F):     array([1, 4, 2, 5, 3, 6], dtype=int8)

.transpose

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.transpose.html

Note that only square matrices can be transposed in place, and in general, an internal copy of the matrix is required. If RAM is a concern, plan accordingly!

# code to be run in micropython

import ulab as np

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12]], dtype=np.uint8)
print('a:\n', a)
print('shape of a:', a.shape())
a.transpose()
print('\ntranspose of a:\n', a)
print('shape of a:', a.shape())
a:
 array([[1, 2, 3],
     [4, 5, 6],
     [7, 8, 9],
     [10, 11, 12]], dtype=uint8)
shape of a: (4, 3)

transpose of a:
 array([[1, 4, 7, 10],
     [2, 5, 8, 11],
     [3, 6, 9, 12]], dtype=uint8)
shape of a: (3, 4)

.sort

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.sort.html

In-place sorting of an ndarray. For a more detailed exposition, see sort.

# code to be run in micropython

import ulab as np

a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.uint8)
print('\na:\n', a)
a.sort(axis=0)
print('\na sorted along vertical axis:\n', a)

a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.uint8)
a.sort(a, axis=1)
print('\na sorted along horizontal axis:\n', a)

a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.uint8)
a.sort(a, axis=None)
print('\nflattened a sorted:\n', a)
a:
 array([[1, 12, 3, 0],
     [5, 3, 4, 1],
     [9, 11, 1, 8],
     [7, 10, 0, 1]], dtype=uint8)

a sorted along vertical axis:
 array([[1, 3, 0, 0],
     [5, 10, 1, 1],
     [7, 11, 3, 1],
     [9, 12, 4, 8]], dtype=uint8)

a sorted along horizontal axis:
 array([[0, 1, 3, 12],
     [1, 3, 4, 5],
     [1, 8, 9, 11],
     [0, 1, 7, 10]], dtype=uint8)

flattened a sorted:
 array([0, 0, 1, ..., 10, 11, 12], dtype=uint8)

Unary operators

With the exception of len, which returns a single number, all unary operators manipulate the underlying data element-wise.

len

This operator takes a single argument, and returns either the length (for row vectors), or the number of rows (for matrices) of its argument.

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3, 4, 5], dtype=np.uint8)
b = np.array([range(5), range(5), range(5), range(5)], dtype=np.uint8)

print("a:\t", a)
print("length of a: ", len(a))
print("shape of a: ", a.shape())
print("\nb:\t", b)
print("length of b: ", len(b))
print("shape of b: ", b.shape())
a:   array([1, 2, 3, 4, 5], dtype=uint8)
length of a:  5
shape of a:  (1, 5)

b:   array([[0, 1, 2, 3, 4],
     [0, 1, 2, 3, 4],
     [0, 1, 2, 3, 4],
     [0, 1, 2, 3, 4]], dtype=uint8)
length of b:  4
shape of b:  (4, 5)

The number returned by len is also the length of the iterations, when the array supplies the elements for an iteration (see later).

invert

The function function is defined for integer data types (uint8, int8, uint16, and int16) only, takes a single argument, and returns the element-by-element, bit-wise inverse of the array. If a float is supplied, the function raises a ValueError exception.

With signed integers (int8, and int16), the results might be unexpected, as in the example below:

# code to be run in micropython

import ulab as np

a = np.array([0, -1, -100], dtype=np.int8)
print("a:\t\t", a)
print("inverse of a:\t", ~a)

a = np.array([0, 1, 254, 255], dtype=np.uint8)
print("\na:\t\t", a)
print("inverse of a:\t", ~a)
a:           array([0, -1, -100], dtype=int8)
inverse of a:        array([-1, 0, 99], dtype=int8)

a:           array([0, 1, 254, 255], dtype=uint8)
inverse of a:        array([255, 254, 1, 0], dtype=uint8)

abs

This function takes a single argument, and returns the element-by-element absolute value of the array. When the data type is unsigned (uint8, or uint16), a copy of the array will be returned immediately, and no calculation takes place.

# code to be run in micropython

import ulab as np

a = np.array([0, -1, -100], dtype=np.int8)
print("a:\t\t\t ", a)
print("absolute value of a:\t ", abs(a))
a:                    array([0, -1, -100], dtype=int8)
absolute value of a:          array([0, 1, 100], dtype=int8)

neg

This operator takes a single argument, and changes the sign of each element in the array. Unsigned values are wrapped.

# code to be run in micropython

import ulab as np

a = np.array([10, -1, 1], dtype=np.int8)
print("a:\t\t", a)
print("negative of a:\t", -a)

b = np.array([0, 100, 200], dtype=np.uint8)
print("\nb:\t\t", b)
print("negative of b:\t", -b)
a:           array([10, -1, 1], dtype=int8)
negative of a:       array([-10, 1, -1], dtype=int8)

b:           array([0, 100, 200], dtype=uint8)
negative of b:       array([0, 156, 56], dtype=uint8)

pos

This function takes a single argument, and simply returns a copy of the array.

# code to be run in micropython

import ulab as np

a = np.array([10, -1, 1], dtype=np.int8)
print("a:\t\t", a)
print("positive of a:\t", +a)
a:           array([10, -1, 1], dtype=int8)
positive of a:       array([10, -1, 1], dtype=int8)

Binary operators

All binary operators work element-wise. This also means that the operands either must have the same shape, or one of them must be a scalar.

WARNING: numpy also allows operations between a matrix, and a row vector, if the row vector has exactly as many elements, as many columns the matrix has. This feature will be added in future versions of ulab.

# code to be run in CPython

a = array([[1, 2, 3], [4, 5, 6], [7, 8, 6]])
b = array([10, 20, 30])
a+b
array([[11, 22, 33],
       [14, 25, 36],
       [17, 28, 36]])

Upcasting

Binary operations require special attention, because two arrays with different typecodes can be the operands of an operation, in which case it is not trivial, what the typecode of the result is. This decision on the result’s typecode is called upcasting. Since the number of typecodes in ulab is significantly smaller than in numpy, we have to define new upcasting rules. Where possible, I followed numpy’s conventions.

ulab observes the following upcasting rules:

  1. Operations with two ndarrays of the same dtype preserve their dtype, even when the results overflow.
  2. if either of the operands is a float, the result is automatically a float
  3. When the right hand side of a binary operator is a micropython variable, mp_obj_int, or mp_obj_float, then the result will be promoted to dtype float. This is necessary, because a micropython integer can be 31 bites wide. Other micropython types (e.g., lists, tuples, etc.) raise a TypeError exception.
left hand side right hand side ulab result numpy result
uint8 int8 int16 int16
uint8 int16 int16 int16
uint8 uint16 uint16 uint16
int8 int16 int16 int16
int8 uint16 uint16 int32
uint16 int16 float int32

Note that the last two operations are promoted to int32 in numpy.

WARNING: Due to the lower number of available data types, the upcasting rules of ulab are slightly different to those of numpy. Watch out for this, when porting code!

Upcasting can be seen in action in the following snippet:

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3, 4], dtype=np.uint8)
b = np.array([1, 2, 3, 4], dtype=np.int8)
print("a:\t", a)
print("b:\t", b)
print("a+b:\t", a+b)

c = np.array([1, 2, 3, 4], dtype=np.float)
print("\na:\t", a)
print("c:\t", c)
print("a*c:\t", a*c)
a:   array([1, 2, 3, 4], dtype=uint8)
b:   array([1, 2, 3, 4], dtype=int8)
a+b:         array([2, 4, 6, 8], dtype=int16)

a:   array([1, 2, 3, 4], dtype=uint8)
c:   array([1.0, 2.0, 3.0, 4.0], dtype=float)
a*c:         array([1.0, 4.0, 9.0, 16.0], dtype=float)

WARNING: If a binary operation involves an ndarray and a micropython type (integer, or float), then the array must be on the left hand side.

# code to be run in micropython

import ulab as np

# this is going to work
a = np.array([1, 2, 3, 4], dtype=np.uint8)
b = 12
print("a:\t", a)
print("b:\t", b)
print("a+b:\t", a+b)

# but this will spectacularly fail
print("b+a:\t", b+a)
a:   array([1, 2, 3, 4], dtype=uint8)
b:   12
a+b:         array([13, 14, 15, 16], dtype=uint8)

Traceback (most recent call last):
  File "/dev/shm/micropython.py", line 12, in <module>
TypeError: unsupported types for __add__: 'int', 'ndarray'

The reason for this lies in how micropython resolves binary operators, and this means that a fix can only be implemented, if micropython itself changes the corresponding function(s). Till then, keep ndarrays on the left hand side.

Benchmarks

The following snippet compares the performance of binary operations to a possible implementation in python. For the time measurement, we will take the following snippet from the micropython manual:

# code to be run in micropython

def timeit(f, *args, **kwargs):
    func_name = str(f).split(' ')[1]
    def new_func(*args, **kwargs):
        t = utime.ticks_us()
        result = f(*args, **kwargs)
        print('execution time: ', utime.ticks_diff(utime.ticks_us(), t), ' us')
        return result
    return new_func
# code to be run in micropython

import ulab as np

@timeit
def py_add(a, b):
    return [a[i]+b[i] for i in range(1000)]

@timeit
def py_multiply(a, b):
    return [a[i]*b[i] for i in range(1000)]

@timeit
def ulab_add(a, b):
    return a + b

@timeit
def ulab_multiply(a, b):
    return a * b

a = [0.0]*1000
b = range(1000)

print('python add:')
py_add(a, b)

print('\npython multiply:')
py_multiply(a, b)

a = np.linspace(0, 10, num=1000)
b = np.ones(1000)

print('\nulab add:')
ulab_add(a, b)

print('\nulab multiply:')
ulab_multiply(a, b)
python add:
execution time:  10051  us

python multiply:
execution time:  14175  us

ulab add:
execution time:  222  us

ulab multiply:
execution time:  213  us

I do not claim that the python implementation above is perfect, and certainly, there is much room for improvement. However, the factor of 50 difference in execution time is very spectacular. This is nothing but a consequence of the fact that the ulab functions run C code, with very little python overhead. The factor of 50 appears to be quite universal: the FFT routine obeys similar scaling (see Speed of FFTs), and this number came up with font rendering, too: fast font rendering on graphical displays.

Comparison operators

The smaller than, greater than, smaller or equal, and greater or equal operators return a vector of Booleans indicating the positions (True), where the condition is satisfied.

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3, 4, 5, 6, 7, 8], dtype=np.uint8)
print(a < 5)
[True, True, True, True, False, False, False, False]

WARNING: Note that numpy returns an array of Booleans. For most use cases this fact should not make a difference.

# code to be run in CPython

a = array([1, 2, 3, 4, 5, 6, 7, 8])
a < 5
array([ True,  True,  True,  True, False, False, False, False])

These operators work with matrices, too, in which case a list of lists of Booleans will be returned:

# code to be run in micropython

import ulab as np

a = np.array([range(0, 5, 1), range(1, 6, 1), range(2, 7, 1)], dtype=np.uint8)
print(a)
print(a < 5)
array([[0, 1, 2, 3, 4],
     [1, 2, 3, 4, 5],
     [2, 3, 4, 5, 6]], dtype=uint8)
[[True, True, True, True, True], [True, True, True, True, False], [True, True, True, False, False]]

Iterating over arrays

ndarrays are iterable, which means that their elements can also be accessed as can the elements of a list, tuple, etc. If the array is one-dimensional, the iterator returns scalars, otherwise a new one-dimensional ndarray, which is simply a copy of the corresponding row of the matrix, i.e, its data type will be inherited.

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3, 4, 5], dtype=np.uint8)
b = np.array([range(5), range(10, 15, 1), range(20, 25, 1), range(30, 35, 1)], dtype=np.uint8)

print("a:\t", a)

for i, _a in enumerate(a):
    print("element %d in a:"%i, _a)

print("\nb:\t", b)

for i, _b in enumerate(b):
    print("element %d in b:"%i, _b)
a:   array([1, 2, 3, 4, 5], dtype=uint8)
element 0 in a: 1
element 1 in a: 2
element 2 in a: 3
element 3 in a: 4
element 4 in a: 5

b:   array([[0, 1, 2, 3, 4],
     [10, 11, 12, 13, 14],
     [20, 21, 22, 23, 24],
     [30, 31, 32, 33, 34]], dtype=uint8)
element 0 in b: array([0, 1, 2, 3, 4], dtype=uint8)
element 1 in b: array([10, 11, 12, 13, 14], dtype=uint8)
element 2 in b: array([20, 21, 22, 23, 24], dtype=uint8)
element 3 in b: array([30, 31, 32, 33, 34], dtype=uint8)

Slicing and indexing

Copies of sub-arrays can be created by indexing, and slicing.

Indexing

The simplest form of indexing is specifying a single integer between the square brackets as in

# code to be run in micropython

import ulab as np

a = np.array(range(10), dtype=np.uint8)
print("a:\t\t\t\t\t\t", a)
print("the first, and first from right element of a:\t", a[0], a[-1])
print("the second, and second from right element of a:\t", a[1], a[-2])
a:                                           array([0, 1, 2, ..., 7, 8, 9], dtype=uint8)
the first, and first from right element of a:        0 9
the second, and second from right element of a:      1 8

Indices are (not necessarily non-negative) integers, or a list of Booleans. By using a Boolean list, we can select those elements of an array that satisfy a specific condition. At the moment, such indexing is defined for row vectors only, for matrices the function raises a ValueError exception, though this will be rectified in a future version of ulab.

# code to be run in micropython

import ulab as np

a = np.array(range(9), dtype=np.float)
print("a:\t", a)
print("a < 5:\t", a[a < 5])
a:   array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
a < 5:       array([0.0, 1.0, 2.0, 3.0, 4.0], dtype=float)

Indexing with Boolean arrays can take more complicated expressions. This is a very concise way of comparing two vectors, e.g.:

# code to be run in micropython

import ulab as np

a = np.array(range(9), dtype=np.uint8)
b = np.array([4, 4, 4, 3, 3, 3, 13, 13, 13], dtype=np.uint8)
print("a:\t", a)
print("\na**2:\t", a*a)
print("\nb:\t", b)
print("\n100*sin(b):\t", np.sin(b)*100.0)
print("\na[a*a > np.sin(b)*100.0]:\t", a[a*a > np.sin(b)*100.0])
a:   array([0, 1, 2, 3, 4, 5, 6, 7, 8], dtype=uint8)

a**2:        array([0, 1, 4, 9, 16, 25, 36, 49, 64], dtype=uint8)

b:   array([4, 4, 4, 3, 3, 3, 13, 13, 13], dtype=uint8)

100*sin(b):  array([-75.68025, -75.68025, -75.68025, 14.112, 14.112, 14.112, 42.01671, 42.01671, 42.01671], dtype=float)

a[a*a > np.sin(b)*100.0]:    array([0, 1, 2, 4, 5, 7, 8], dtype=uint8)

Slicing and assigning to slices

You can also generate sub-arrays by specifying slices as the index of an array. Slices are special python objects of the form

slice = start:end:stop

where start, end, and stop are (not necessarily non-negative) integers. Not all of these three numbers must be specified in an index, in fact, all three of them can be missing. The interpreter takes care of filling in the missing values. (Note that slices cannot be defined in this way, only there, where an index is expected.) For a good explanation on how slices work in python, you can read the stackoverflow question https://stackoverflow.com/questions/509211/understanding-slice-notation.

Slices work on both axes:

# code to be run in micropython

import ulab as np

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=np.uint8)
print('a:\n', a)

# the first row
print('\na[0]:\n', a[0])

# the first two elements of the first row
print('\na[0,:2]:\n', a[0,:2])

# the zeroth element in each row (also known as the zeroth column)
print('\na[:,0]:\n', a[:,0])

# the last but one row
print('\na[-1]:\n', a[-1])

# the last two rows backwards
print('\na[::1]:\n', a[::-1])
a:
 array([[1, 2, 3],
     [4, 5, 6],
     [7, 8, 9]], dtype=uint8)

a[0]:
 array([1, 2, 3], dtype=uint8)

a[0,:2]:
 array([1, 2], dtype=uint8)

a[:,0]:
 array([1, 4, 7], dtype=uint8)

a[-1]:
 array([7, 8, 9], dtype=uint8)

a[::1]:
 array([[7, 8, 9],
     [4, 5, 6]], dtype=uint8)

Assignment to slices can be done for the whole slice, per row, and per column. A couple of examples should make these statements clearer:

# code to be run in micropython

import ulab as np

zero_list = [0, 0, 0]
a = np.array([zero_list, zero_list, zero_list], dtype=np.uint8)
print('a:\n', a)

# assigning to the whole row
a[0] = 1
print('\na[0] = 1\n', a)

# assigning to the whole row
a[0] = np.array([1, 2, -333], dtype=np.float)
print('\na[0] = np.array([1, 2, 3])\n', a)

# assigning to a column
a[:,2] = 3.0
print('\na[:,0]:\n', a)
a:
 array([[0, 0, 0],
     [0, 0, 0],
     [0, 0, 0]], dtype=uint8)

a[0] = 1
 array([[1, 1, 1],
     [0, 0, 0],
     [0, 0, 0]], dtype=uint8)

a[0] = np.array([1, 2, 3])
 array([[1, 2, 179],
     [0, 0, 0],
     [0, 0, 0]], dtype=uint8)

a[:,0]:
 array([[1, 2, 3],
     [0, 0, 3],
     [0, 0, 3]], dtype=uint8)

Universal functions

Standard mathematical functions can be calculated on any scalar-valued iterable (ranges, lists, tuples containing numbers), and on ndarrays without having to change the call signature. In all cases the functions return a new ndarray of typecode float (since these functions usually generate float values, anyway). The functions execute faster with ndarray arguments than with iterables, because the values of the input vector can be extracted faster.

At present, the following functions are supported:

acos, acosh, asin, asinh, atan, atanh, ceil, cos, erf, erfc, exp, expm1, floor, tgamma, lgamma, log, log10, log2, sin, sinh, sqrt, tan, tanh.

These functions are applied element-wise to the arguments, thus, e.g., the exponential of a matrix cannot be calculated in this way.

# code to be run in micropython

import ulab as np

a = range(9)
b = np.array(a)

# works with ranges, lists, tuples etc.
print('a:\t', a)
print('exp(a):\t', np.exp(a))

# with 1D arrays
print('\nb:\t', b)
print('exp(b):\t', np.exp(b))

# as well as with matrices
c = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print('\nc:\t', c)
print('exp(c):\t', np.exp(c))
a:   range(0, 9)
exp(a):      array([1.0, 2.718282, 7.389056, 20.08554, 54.59816, 148.4132, 403.4288, 1096.633, 2980.958], dtype=float)

b:   array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
exp(b):      array([1.0, 2.718282, 7.389056, 20.08554, 54.59816, 148.4132, 403.4288, 1096.633, 2980.958], dtype=float)

c:   array([[1.0, 2.0, 3.0],
     [4.0, 5.0, 6.0],
     [7.0, 8.0, 9.0]], dtype=float)
exp(c):      array([[2.718282, 7.389056, 20.08554],
     [54.59816, 148.4132, 403.4288],
     [1096.633, 2980.958, 8103.084]], dtype=float)

Computation expenses

The overhead for calculating with micropython iterables is quite significant: for the 1000 samples below, the difference is more than 800 microseconds, because internally the function has to create the ndarray for the output, has to fetch the iterable’s items of unknown type, and then convert them to floats. All these steps are skipped for ndarrays, because these pieces of information are already known.

# code to be run in micropython

import ulab as np

a = [0]*1000
b = np.array(a)

@timeit
def measure_run_time(x):
    return np.exp(x)

measure_run_time(a)

measure_run_time(b)
execution time:  1259  us
execution time:  408  us

Of course, such a time saving is reasonable only, if the data are already available as an ndarray. If one has to initialise the ndarray from the list, then there is no gain, because the iterator was simply pushed into the initialisation function.

Numerical

linspace

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.linspace.html

This function returns an array, whose elements are uniformly spaced between the start, and stop points. The number of intervals is determined by the num keyword argument, whose default value is 50. With the endpoint keyword argument (defaults to True) one can include stop in the sequence. In addition, the dtype keyword can be supplied to force type conversion of the output. The default is float. Note that, when dtype is of integer type, the sequence is not necessarily evenly spaced. This is not an error, rather a consequence of rounding. (This is also the numpy behaviour.)

# code to be run in micropython

import ulab as np

# generate a sequence with defaults
print('default sequence:\t', np.linspace(0, 10))

# num=5
print('num=5:\t\t\t', np.linspace(0, 10, num=5))

# num=5, endpoint=False
print('num=5:\t\t\t', np.linspace(0, 10, num=5, endpoint=False))

# num=5, endpoint=False, dtype=uint8
print('num=5:\t\t\t', np.linspace(0, 5, num=7, endpoint=False, dtype=np.uint8))
default sequence:    array([0.0, 0.2040816396474838, 0.4081632792949677, ..., 9.591833114624023, 9.795914649963379, 9.999996185302734], dtype=float)
num=5:                       array([0.0, 2.5, 5.0, 7.5, 10.0], dtype=float)
num=5:                       array([0.0, 2.0, 4.0, 6.0, 8.0], dtype=float)
num=5:                       array([0, 0, 1, 2, 2, 3, 4], dtype=uint8)

min, argmin, max, argmax

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.min.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmax.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.max.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmax.html

WARNING: Difference to numpy: the out keyword argument is not implemented.

These functions follow the same pattern, and work with generic iterables, and ndarrays. min, and max return the minimum or maximum of a sequence. If the input array is two-dimensional, the axis keyword argument can be supplied, in which case the minimum/maximum along the given axis will be returned. If axis=None (this is also the default value), the minimum/maximum of the flattened array will be determined.

argmin/argmax return the position (index) of the minimum/maximum in the sequence.

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 0, 1, 10])
print('a:', a)
print('min of a:', np.min(a))
print('argmin of a:', np.argmin(a))

b = np.array([[1, 2, 0], [1, 10, -1]])
print('\nb:\n', b)
print('min of b (flattened):', np.min(b))
print('min of b (axis=0):', np.min(b, axis=0))
print('min of b (axis=1):', np.min(b, axis=1))
a: array([1.0, 2.0, 0.0, 1.0, 10.0], dtype=float)
min of a: 0.0
argmin of a: 2

b:
 array([[1.0, 2.0, 0.0],
     [1.0, 10.0, -1.0]], dtype=float)
min of b (flattened): -1.0
min of b (axis=0): array([1.0, 2.0, -1.0], dtype=float)
min of b (axis=1): array([0.0, -1.0], dtype=float)

sum, std, mean

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.sum.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.std.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.mean.html

These three functions follow the same pattern: if the axis keyword is not specified, it assumes the default value of None, and returns the result of the computation for the flattened array. Otherwise, the calculation is along the given axis.

# code to be run in micropython

import ulab as np

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print('a: \n', a)

print('sum, flat array: ', np.sum(a))

print('mean, horizontal: ', np.mean(a, axis=1))

print('std, vertical: ', np.std(a, axis=0))
a:
 array([[1.0, 2.0, 3.0],
     [4.0, 5.0, 6.0],
     [7.0, 8.0, 9.0]], dtype=float)
sum, flat array:  45.0
mean, horizontal:  array([2.0, 5.0, 8.0], dtype=float)
std, vertical:  array([2.44949, 2.44949, 2.44949], dtype=float)

roll

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.roll.html

The roll function shifts the content of a vector by the positions given as the second argument. If the axis keyword is supplied, the shift is applied to the given axis.

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3, 4, 5, 6, 7, 8])
print("a:\t\t\t", a)

np.roll(a, 2)
print("a rolled to the left:\t", a)

# this should be the original vector
np.roll(a, -2)
print("a rolled to the right:\t", a)
a:                   array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
a rolled to the left:        array([3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 1.0, 2.0], dtype=float)
a rolled to the right:       array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)

Rolling works with matrices, too. If the axis keyword is 0, the matrix is rolled along its vertical axis, otherwise, horizontally.

Horizontal rolls are faster, because they require fewer steps, and larger memory chunks are copied, however, they also require more RAM: basically the whole row must be stored internally. Most expensive are the None keyword values, because with axis = None, the array is flattened first, hence the row’s length is the size of the whole matrix.

Vertical rolls require two internal copies of single columns.

# code to be run in micropython

import ulab as np

a = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])
print("a:\n", a)

np.roll(a, 2)
print("\na rolled to the left:\n", a)

np.roll(a, -1, axis=1)
print("\na rolled up:\n", a)

np.roll(a, 1, axis=None)
print("\na rolled with None:\n", a)
a:
 array([[1.0, 2.0, 3.0, 4.0],
     [5.0, 6.0, 7.0, 8.0]], dtype=float)

a rolled to the left:
 array([[3.0, 4.0, 5.0, 6.0],
     [7.0, 8.0, 1.0, 2.0]], dtype=float)

a rolled up:
 array([[6.0, 3.0, 4.0, 5.0],
     [2.0, 7.0, 8.0, 1.0]], dtype=float)

a rolled with None:
 array([[3.0, 4.0, 5.0, 2.0],
     [7.0, 8.0, 1.0, 6.0]], dtype=float)

Simple running weighted average

As a demonstration of the conciseness of ulab/numpy operations, we will calculate an exponentially weighted running average of a measurement vector in just a couple of lines. I chose this particular example, because I think that this can indeed be used in real-life applications.

# code to be run in micropython

import ulab as np

def dummy_adc():
    # dummy adc function, so that the results are reproducible
    return 2

n = 10
# These are the normalised weights; the last entry is the most dominant
weight = np.exp([1, 2, 3, 4, 5])
weight = weight/np.sum(weight)

print(weight)
# initial array of samples
samples = np.array([0]*n)

for i in range(n):
    # a new datum is inserted on the right hand side. This simply overwrites whatever was in the last slot
    samples[-1] = dummy_adc()
    print(np.mean(samples[-5:]*weight))
    print(samples[-5:])
    # the data are shifted by one position to the left
    np.roll(samples, 1)
array([0.01165623031556606, 0.03168492019176483, 0.08612854033708572, 0.234121635556221, 0.6364086270332336], dtype=float)
0.2545634508132935
array([0.0, 0.0, 0.0, 0.0, 2.0], dtype=float)
0.3482121050357819
array([0.0, 0.0, 0.0, 2.0, 2.0], dtype=float)
0.3826635211706161
array([0.0, 0.0, 2.0, 2.0, 2.0], dtype=float)
0.3953374892473221
array([0.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)

flip

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.flip.html

The flip function takes one positional, an ndarray, and one keyword argument, axis = None, and reverses the order of elements along the given axis. If the keyword argument is None, the matrix’ entries are flipped along all axes. flip returns a new copy of the array.

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3, 4, 5])
print("a: \t", a)
print("a flipped:\t", np.flip(a))

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=np.uint8)
print("\na flipped horizontally\n", np.flip(a, axis=1))
print("\na flipped vertically\n", np.flip(a, axis=0))
print("\na flipped horizontally+vertically\n", np.flip(a))
a:   array([1.0, 2.0, 3.0, 4.0, 5.0], dtype=float)
a flipped:   array([5.0, 4.0, 3.0, 2.0, 1.0], dtype=float)

a flipped horizontally
 array([[3, 2, 1],
     [6, 5, 4],
     [9, 8, 7]], dtype=uint8)

a flipped vertically
 array([[7, 8, 9],
     [4, 5, 6],
     [1, 2, 3]], dtype=uint8)

a flipped horizontally+vertically
 array([[9, 8, 7],
     [6, 5, 4],
     [3, 2, 1]], dtype=uint8)

diff

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.diff.html

The diff function returns the numerical derivative of the forward scheme, or more accurately, the differences of an ndarray along a given axis. The order of derivative can be stipulated with the n keyword argument, which should be between 0, and 9. Default is 1. If higher order derivatives are required, they can be gotten by repeated calls to the function. The axis keyword argument should be -1 (last axis, in ulab equivalent to the second axis, and this also happens to be the default value), 0, or 1.

Beyond the output array, the function requires only a couple of bytes of extra RAM for the differentiation stencil. (The stencil is an int8 array, one byte longer than n. This also explains, why the highest order is 9: the coefficients of a ninth-order stencil all fit in signed bytes, while 10 would require int16.) Note that as usual in numerical differentiation (and also in numpy), the length of the respective axis will be reduced by n after the operation. If n is larger than, or equal to the length of the axis, an empty array will be returned.

WARNING: the diff function does not implement the prepend and append keywords that can be found in numpy.

# code to be run in micropython

import ulab as np

a = np.array(range(9), dtype=np.uint8)
print('a:\n', a)

print('\nfirst derivative:\n', np.diff(a, n=1))
print('\nsecond derivative:\n', np.diff(a, n=2))

c = np.array([[1, 2, 3, 4], [4, 3, 2, 1], [1, 4, 9, 16], [0, 0, 0, 0]])
print('\nc:\n', c)
print('\nfirst derivative, first axis:\n', np.diff(c, axis=0))
print('\nfirst derivative, second axis:\n', np.diff(c, axis=1))
a:
 array([0, 1, 2, 3, 4, 5, 6, 7, 8], dtype=uint8)

first derivative:
 array([1, 1, 1, 1, 1, 1, 1, 1], dtype=uint8)

second derivative:
 array([0, 0, 0, 0, 0, 0, 0], dtype=uint8)

c:
 array([[1.0, 2.0, 3.0, 4.0],
     [4.0, 3.0, 2.0, 1.0],
     [1.0, 4.0, 9.0, 16.0],
     [0.0, 0.0, 0.0, 0.0]], dtype=float)

first derivative, first axis:
 array([[3.0, 1.0, -1.0, -3.0],
     [-3.0, 1.0, 7.0, 15.0],
     [-1.0, -4.0, -9.0, -16.0]], dtype=float)

first derivative, second axis:
 array([[1.0, 1.0, 1.0],
     [-1.0, -1.0, -1.0],
     [3.0, 5.0, 7.0],
     [0.0, 0.0, 0.0]], dtype=float)

sort

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.sort.html

The sort function takes an ndarray, and sorts its elements in ascending order along the specified axis using a heap sort algorithm. As opposed to the .sort() method discussed earlier, this function creates a copy of its input before sorting, and at the end, returns this copy. Sorting takes place in place, without auxiliary storage. The axis keyword argument takes on the possible values of -1 (the last axis, in ulab equivalent to the second axis, and this also happens to be the default value), 0, 1, or None. The first three cases are identical to those in diff, while the last one flattens the array before sorting.

If descending order is required, the result can simply be flipped, see flip.

WARNING: numpy defines the kind, and order keyword arguments that are not implemented here. The function in ulab always uses heap sort, and since ulab does not have the concept of data fields, the order keyword argument would have no meaning.

# code to be run in micropython

import ulab as np

a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.float)
print('\na:\n', a)
b = np.sort(a, axis=0)
print('\na sorted along vertical axis:\n', b)

c = np.sort(a, axis=1)
print('\na sorted along horizontal axis:\n', c)

c = np.sort(a, axis=None)
print('\nflattened a sorted:\n', c)
a:
 array([[1.0, 12.0, 3.0, 0.0],
     [5.0, 3.0, 4.0, 1.0],
     [9.0, 11.0, 1.0, 8.0],
     [7.0, 10.0, 0.0, 1.0]], dtype=float)

a sorted along vertical axis:
 array([[1.0, 3.0, 0.0, 0.0],
     [5.0, 10.0, 1.0, 1.0],
     [7.0, 11.0, 3.0, 1.0],
     [9.0, 12.0, 4.0, 8.0]], dtype=float)

a sorted along horizontal axis:
 array([[0.0, 1.0, 3.0, 12.0],
     [1.0, 3.0, 4.0, 5.0],
     [1.0, 8.0, 9.0, 11.0],
     [0.0, 1.0, 7.0, 10.0]], dtype=float)

flattened a sorted:
 array([0.0, 0.0, 1.0, ..., 10.0, 11.0, 12.0], dtype=float)

Heap sort requires \(\sim N\log N\) operations, and notably, the worst case costs only 20% more time than the average. In order to get an order-of-magnitude estimate, we will take the sine of 1000 uniformly spaced numbers between 0, and two pi, and sort them:

# code to be run in micropython

import ulab as np

@timeit
def sort_time(array):
    return np.sort(array)

b = np.sin(np.linspace(0, 6.28, num=1000))
print('b: ', b)
sort_time(b)
print('\nb sorted:\n', b)

argsort

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argsort.html

Similarly to sort, argsort takes a positional, and a keyword argument, and returns an unsigned short index array of type ndarray with the same dimensions as the input, or, if axis=None, as a row vector with length equal to the number of elements in the input (i.e., the flattened array). The indices in the output sort the input in ascending order. The routine in argsort is the same as in sort, therefore, the comments on computational expenses (time and RAM) also apply. In particular, since no copy of the original data is required, virtually no RAM beyond the output array is used.

Since the underlying container of the output array is of type uint16_t, neither of the output dimensions should be larger than 65535.

# code to be run in micropython

import ulab as np

a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.float)
print('\na:\n', a)
b = np.argsort(a, axis=0)
print('\na sorted along vertical axis:\n', b)

c = np.argsort(a, axis=1)
print('\na sorted along horizontal axis:\n', c)

c = np.argsort(a, axis=None)
print('\nflattened a sorted:\n', c)
a:
 array([[1.0, 12.0, 3.0, 0.0],
     [5.0, 3.0, 4.0, 1.0],
     [9.0, 11.0, 1.0, 8.0],
     [7.0, 10.0, 0.0, 1.0]], dtype=float)

a sorted along vertical axis:
 array([[0, 1, 3, 0],
     [1, 3, 2, 1],
     [3, 2, 0, 3],
     [2, 0, 1, 2]], dtype=uint16)

a sorted along horizontal axis:
 array([[3, 0, 2, 1],
     [3, 1, 2, 0],
     [2, 3, 0, 1],
     [2, 3, 0, 1]], dtype=uint16)

flattened a sorted:
 array([3, 14, 0, ..., 13, 9, 1], dtype=uint16)

Since during the sorting, only the indices are shuffled, argsort does not modify the input array, as one can verify this by the following example:

# code to be run in micropython

import ulab as np

a = np.array([0, 5, 1, 3, 2, 4], dtype=np.uint8)
print('\na:\n', a)
b = np.argsort(a, axis=1)
print('\nsorting indices:\n', b)
print('\nthe original array:\n', a)
a:
 array([0, 5, 1, 3, 2, 4], dtype=uint8)

sorting indices:
 array([0, 2, 4, 3, 5, 1], dtype=uint16)

the original array:
 array([0, 5, 1, 3, 2, 4], dtype=uint8)

Linalg

size

size takes a single argument, the axis, whose size is to be returned. Depending on the value of the argument, the following information will be returned:

  1. argument is 0: the number of elements of the array
  2. argument is 1: the number of rows
  3. argument is 2: the number of columns
# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3, 4], dtype=np.int8)
print("a:\n", a)
print("size of a:", np.size(a, axis=None), ",", np.size(a, axis=0))

b= np.array([[1, 2], [3, 4]], dtype=np.int8)
print("\nb:\n", b)
print("size of b:", np.size(b, axis=None), ",", np.size(b, axis=0), ",", np.size(b, axis=1))
a:
 array([1, 2, 3, 4], dtype=int8)
size of a: 4 , 4

b:
 array([[1, 2],
     [3, 4]], dtype=int8)
size of b: 4 , 2 , 2

ones, zeros

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.zeros.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.ones.html

A couple of special arrays and matrices can easily be initialised by calling one of the ones, or zeros functions. ones and zeros follow the same pattern, and have the call signature

ones(shape, dtype=float)
zeros(shape, dtype=float)

where shape is either an integer, or a 2-tuple.

# code to be run in micropython

import ulab as np

print(np.ones(6, dtype=np.uint8))
print(np.zeros((6, 4)))
array([1, 1, 1, 1, 1, 1], dtype=uint8)
array([[0.0, 0.0, 0.0, 0.0],
     [0.0, 0.0, 0.0, 0.0],
     [0.0, 0.0, 0.0, 0.0],
     [0.0, 0.0, 0.0, 0.0],
     [0.0, 0.0, 0.0, 0.0],
     [0.0, 0.0, 0.0, 0.0]], dtype=float)

eye

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.eye.html

Another special array method is the eye function, whose call signature is

eye(N, M, k=0, dtype=float)

where N (M) specify the dimensions of the matrix (if only N is supplied, then we get a square matrix, otherwise one with M rows, and N columns), and k is the shift of the ones (the main diagonal corresponds to k=0). Here are a couple of examples.

With a single argument

# code to be run in micropython

import ulab as np

print(np.eye(5))
array([[1.0, 0.0, 0.0, 0.0, 0.0],
     [0.0, 1.0, 0.0, 0.0, 0.0],
     [0.0, 0.0, 1.0, 0.0, 0.0],
     [0.0, 0.0, 0.0, 1.0, 0.0],
     [0.0, 0.0, 0.0, 0.0, 1.0]], dtype=float)

Specifying the dimensions of the matrix

# code to be run in micropython

import ulab as np

print(np.eye(4, M=6, dtype=np.int8))
array([[1, 0, 0, 0],
     [0, 1, 0, 0],
     [0, 0, 1, 0],
     [0, 0, 0, 1],
     [0, 0, 0, 0],
     [0, 0, 0, 0]], dtype=int8)

Shifting the diagonal

# code to be run in micropython

import ulab as np

print(np.eye(4, M=6, k=-1, dtype=np.int16))
array([[0, 0, 0, 0],
     [1, 0, 0, 0],
     [0, 1, 0, 0],
     [0, 0, 1, 0],
     [0, 0, 0, 1],
     [0, 0, 0, 0]], dtype=int16)

inv

A square matrix, provided that it is not singular, can be inverted by calling the inv function that takes a single argument. The inversion is based on successive elimination of elements in the lower left triangle, and raises a ValueError exception, if the matrix turns out to be singular (i.e., one of the diagonal entries is zero).

# code to be run in micropython

import ulab as np

m = np.array([[1, 2, 3, 4], [4, 5, 6, 4], [7, 8.6, 9, 4], [3, 4, 5, 6]])

print(np.inv(m))
array([[-2.166666, 1.499999, -0.8333326, 1.0],
     [1.666666, -3.333331, 1.666666, -4.768516e-08],
     [0.1666672, 2.166666, -0.8333327, -1.0],
     [-0.1666666, -0.3333334, 4.96705e-08, 0.5]], dtype=float)

Computation expenses

Note that the cost of inverting a matrix is approximately twice as many floats (RAM), as the number of entries in the original matrix, and approximately as many operations, as the number of entries. Here are a couple of numbers:

# code to be run in micropython

import ulab as np

@timeit
def invert_matrix(m):
    return np.inv(m)

m = np.array([[1, 2,], [4, 5]])
print('2 by 2 matrix:')
invert_matrix(m)

m = np.array([[1, 2, 3, 4], [4, 5, 6, 4], [7, 8.6, 9, 4], [3, 4, 5, 6]])
print('\n4 by 4 matrix:')
invert_matrix(m)

m = np.array([[1, 2, 3, 4, 5, 6, 7, 8], [0, 5, 6, 4, 5, 6, 4, 5],
              [0, 0, 9, 7, 8, 9, 7, 8], [0, 0, 0, 10, 11, 12, 11, 12],
             [0, 0, 0, 0, 4, 6, 7, 8], [0, 0, 0, 0, 0, 5, 6, 7],
             [0, 0, 0, 0, 0, 0, 7, 6], [0, 0, 0, 0, 0, 0, 0, 2]])
print('\n8 by 8 matrix:')
invert_matrix(m)
2 by 2 matrix:
execution time:  65  us

4 by 4 matrix:
execution time:  105  us

8 by 8 matrix:
execution time:  299  us

The above-mentioned scaling is not obeyed strictly. The reason for the discrepancy is that the function call is still the same for all three cases: the input must be inspected, the output array must be created, and so on.

dot

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.dot.html

WARNING: numpy applies upcasting rules for the multiplication of matrices, while ulab simply returns a float matrix.

Once you can invert a matrix, you might want to know, whether the inversion is correct. You can simply take the original matrix and its inverse, and multiply them by calling the dot function, which takes the two matrices as its arguments. If the matrix dimensions do not match, the function raises a ValueError. The result of the multiplication is expected to be the unit matrix, which is demonstrated below.

# code to be run in micropython

import ulab as np

m = np.array([[1, 2, 3], [4, 5, 6], [7, 10, 9]], dtype=np.uint8)
n = np.inv(m)
print("m:\n", m)
print("\nm^-1:\n", n)
# this should be the unit matrix
print("\nm*m^-1:\n", np.dot(m, n))
m:
 array([[1, 2, 3],
     [4, 5, 6],
     [7, 10, 9]], dtype=uint8)

m^-1:
 array([[-1.25, 1.0, -0.25],
     [0.5, -1.0, 0.5],
     [0.4166667, 0.3333334, -0.25]], dtype=float)

m*m^-1:
 array([[1.0, 2.384186e-07, -1.490116e-07],
     [-2.980232e-07, 1.000001, -4.172325e-07],
     [-3.278255e-07, 1.311302e-06, 0.9999992]], dtype=float)

Note that for matrix multiplication you don’t necessarily need square matrices, it is enough, if their dimensions are compatible (i.e., the the left-hand-side matrix has as many columns, as does the right-hand-side matrix rows):

# code to be run in micropython

import ulab as np

m = np.array([[1, 2, 3, 4], [5, 6, 7, 8]], dtype=np.uint8)
n = np.array([[1, 2], [3, 4], [5, 6], [7, 8]], dtype=np.uint8)
print(m)
print(n)
print(np.dot(m, n))
array([[1, 2, 3, 4],
     [5, 6, 7, 8]], dtype=uint8)
array([[1, 2],
     [3, 4],
     [5, 6],
     [7, 8]], dtype=uint8)
array([[7.0, 10.0],
     [23.0, 34.0]], dtype=float)

det

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.det.html

The det function takes a square matrix as its single argument, and calculates the determinant. The calculation is based on successive elimination of the matrix elements, and the return value is a float, even if the input array was of integer type.

# code to be run in micropython

import ulab as np

a = np.array([[1, 2], [3, 4]], dtype=np.uint8)
print(np.det(a))
-2.0

Benchmark

Since the routine for calculating the determinant is pretty much the same as for finding the inverse of a matrix, the execution times are similar:

# code to be run in micropython

@timeit
def matrix_det(m):
    return np.inv(m)

m = np.array([[1, 2, 3, 4, 5, 6, 7, 8], [0, 5, 6, 4, 5, 6, 4, 5],
              [0, 0, 9, 7, 8, 9, 7, 8], [0, 0, 0, 10, 11, 12, 11, 12],
             [0, 0, 0, 0, 4, 6, 7, 8], [0, 0, 0, 0, 0, 5, 6, 7],
             [0, 0, 0, 0, 0, 0, 7, 6], [0, 0, 0, 0, 0, 0, 0, 2]])

matrix_det(m)
execution time:  294  us

eig

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.eig.html

The eig function calculates the eigenvalues and the eigenvectors of a real, symmetric square matrix. If the matrix is not symmetric, a ValueError will be raised. The function takes a single argument, and returns a tuple with the eigenvalues, and eigenvectors. With the help of the eigenvectors, amongst other things, you can implement sophisticated stabilisation routines for robots.

# code to be run in micropython

import ulab as np

a = np.array([[1, 2, 1, 4], [2, 5, 3, 5], [1, 3, 6, 1], [4, 5, 1, 7]], dtype=np.uint8)
x, y = np.eig(a)
print('eigenvectors of a:\n', x)
print('\neigenvalues of a:\n', y)
eigenvectors of a:
 array([-1.165288, 0.8029362, 5.585626, 13.77673], dtype=float)

eigenvalues of a:
 array([[0.8151754, -0.4499267, -0.1643907, 0.3256237],
     [0.2211193, 0.7847154, 0.08373602, 0.5729892],
     [-0.1340859, -0.3100657, 0.8742685, 0.3486182],
     [-0.5182822, -0.2926556, -0.4490192, 0.6664218]], dtype=float)

The same matrix diagonalised with numpy yields:

# code to be run in CPython

a = array([[1, 2, 1, 4], [2, 5, 3, 5], [1, 3, 6, 1], [4, 5, 1, 7]], dtype=np.uint8)
x, y = eig(a)
print('eigenvectors of a:\n', x)
print('\neigenvalues of a:\n', y)
eigenvectors of a:
 [13.77672606 -1.16528837  0.80293655  5.58562576]

eigenvalues of a:
 [[ 0.32561419  0.815156    0.44994112 -0.16446602]
 [ 0.57300777  0.22113342 -0.78469926  0.08372081]
 [ 0.34861093 -0.13401142  0.31007764  0.87427868]
 [ 0.66641421 -0.51832581  0.29266348 -0.44897499]]

When comparing results, we should keep two things in mind:

  1. the eigenvalues and eigenvectors are not necessarily sorted in the same way
  2. an eigenvector can be multiplied by an arbitrary non-zero scalar, and it is still an eigenvector with the same eigenvalue. This is why all signs of the eigenvector belonging to 5.58, and 0.80 are flipped in ulab with respect to numpy. This difference, however, is of absolutely no consequence.

Computation expenses

Since the function is based on Givens rotations and runs till convergence is achieved, or till the maximum number of allowed rotations is exhausted, there is no universal estimate for the time required to find the eigenvalues. However, an order of magnitude can, at least, be guessed based on the measurement below:

# code to be run in micropython

import ulab as np

@timeit
def matrix_eig(a):
    return np.eig(a)

a = np.array([[1, 2, 1, 4], [2, 5, 3, 5], [1, 3, 6, 1], [4, 5, 1, 7]], dtype=np.uint8)

matrix_eig(a)
execution time:  111  us

Polynomials

polyval

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.polyval.html

polyval takes two arguments, both arrays or other iterables.

# code to be run in micropython

import ulab as np

p = [1, 1, 1, 0]
x = [0, 1, 2, 3, 4]
print('coefficients: ', p)
print('independent values: ', x)
print('\nvalues of p(x): ', np.polyval(p, x))

# the same works with one-dimensional ndarrays
a = np.array(x)
print('\nndarray (a): ', a)
print('value of p(a): ', np.polyval(p, a))
coefficients:  [1, 1, 1, 0]
independent values:  [0, 1, 2, 3, 4]

values of p(x):  array([0.0, 3.0, 14.0, 39.0, 84.0], dtype=float)

ndarray (a):  array([0.0, 1.0, 2.0, 3.0, 4.0], dtype=float)
value of p(a):  array([0.0, 3.0, 14.0, 39.0, 84.0], dtype=float)

polyfit

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.polyfit.html

polyfit takes two, or three arguments. The last one is the degree of the polynomial that will be fitted, the last but one is an array or iterable with the y (dependent) values, and the first one, an array or iterable with the x (independent) values, can be dropped. If that is the case, x will be generated in the function, assuming uniform sampling.

If the length of x, and y are not the same, the function raises a ValueError.

# code to be run in micropython

import ulab as np

x = np.array([0, 1, 2, 3, 4, 5, 6])
y = np.array([9, 4, 1, 0, 1, 4, 9])
print('independent values:\t', x)
print('dependent values:\t', y)
print('fitted values:\t\t', np.polyfit(x, y, 2))

# the same with missing x
print('\ndependent values:\t', y)
print('fitted values:\t\t', np.polyfit(y, 2))
independent values:  array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0], dtype=float)
dependent values:    array([9.0, 4.0, 1.0, 0.0, 1.0, 4.0, 9.0], dtype=float)
fitted values:               array([1.0, -6.0, 9.000000000000004], dtype=float)

dependent values:    array([9.0, 4.0, 1.0, 0.0, 1.0, 4.0, 9.0], dtype=float)
fitted values:               array([1.0, -6.0, 9.000000000000004], dtype=float)

Execution time

polyfit is based on the inversion of a matrix (there is more on the background in https://en.wikipedia.org/wiki/Polynomial_regression), and it requires the intermediate storage of 2*N*(deg+1) floats, where N is the number of entries in the input array, and deg is the fit’s degree. The additional computation costs of the matrix inversion discussed in inv also apply. The example from above needs around 150 microseconds to return:

# code to be run in micropython

import ulab as np

@timeit
def time_polyfit(x, y, n):
    return np.polyfit(x, y, n)

x = np.array([0, 1, 2, 3, 4, 5, 6])
y = np.array([9, 4, 1, 0, 1, 4, 9])

time_polyfit(x, y, 2)
execution time:  153  us

Fourier transforms

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.fft.ifft.html

fft

Since ulab’s ndarray does not support complex numbers, the invocation of the Fourier transform differs from that in numpy. In numpy, you can simply pass an array or iterable to the function, and it will be treated as a complex array:

# code to be run in CPython

fft.fft([1, 2, 3, 4, 1, 2, 3, 4])
array([20.+0.j,  0.+0.j, -4.+4.j,  0.+0.j, -4.+0.j,  0.+0.j, -4.-4.j,
        0.+0.j])

WARNING: The array that is returned is also complex, i.e., the real and imaginary components are cast together. In ulab, the real and imaginary parts are treated separately: you have to pass two ndarrays to the function, although, the second argument is optional, in which case the imaginary part is assumed to be zero.

WARNING: The function, as opposed to numpy, returns a 2-tuple, whose elements are two ndarrays, holding the real and imaginary parts of the transform separately.

# code to be run in micropython

import ulab as np

x = np.linspace(0, 10, num=1024)
y = np.sin(x)
z = np.zeros(len(x))

a, b = np.fft(x)
print('real part:\t', a)
print('\nimaginary part:\t', b)

c, d = np.fft(x, z)
print('\nreal part:\t', c)
print('\nimaginary part:\t', d)
real part:   array([5119.996, -5.004663, -5.004798, ..., -5.005482, -5.005643, -5.006577], dtype=float)

imaginary part:      array([0.0, 1631.333, 815.659, ..., -543.764, -815.6588, -1631.333], dtype=float)

real part:   array([5119.996, -5.004663, -5.004798, ..., -5.005482, -5.005643, -5.006577], dtype=float)

imaginary part:      array([0.0, 1631.333, 815.659, ..., -543.764, -815.6588, -1631.333], dtype=float)

ifft

The above-mentioned rules apply to the inverse Fourier transform. The inverse is also normalised by N, the number of elements, as is customary in numpy. With the normalisation, we can ascertain that the inverse of the transform is equal to the original array.

# code to be run in micropython

import ulab as np

x = np.linspace(0, 10, num=1024)
y = np.sin(x)

a, b = np.fft(y)

print('original vector:\t', y)

y, z = np.ifft(a, b)
# the real part should be equal to y
print('\nreal part of inverse:\t', y)
# the imaginary part should be equal to zero
print('\nimaginary part of inverse:\t', z)
original vector:     array([0.0, 0.009775016, 0.0195491, ..., -0.5275068, -0.5357859, -0.5440139], dtype=float)

real part of inverse:        array([-2.980232e-08, 0.0097754, 0.0195494, ..., -0.5275064, -0.5357857, -0.5440133], dtype=float)

imaginary part of inverse:   array([-2.980232e-08, -1.451171e-07, 3.693752e-08, ..., 6.44871e-08, 9.34986e-08, 2.18336e-07], dtype=float)

Note that unlike in numpy, the length of the array on which the Fourier transform is carried out must be a power of 2. If this is not the case, the function raises a ValueError exception.

spectrum

In addition to the Fourier transform and its inverse, ulab also sports a function called spectrum, which returns the absolute value of the Fourier transform. This could be used to find the dominant spectral component in a time series. The arguments are treated in the same way as in fft, and ifft.

# code to be run in micropython

import ulab as np

x = np.linspace(0, 10, num=1024)
y = np.sin(x)

a = np.spectrum(y)

print('original vector:\t', y)
print('\nspectrum:\t', a)
original vector:     array([0.0, 0.009775016, 0.0195491, ..., -0.5275068, -0.5357859, -0.5440139], dtype=float)

spectrum:    array([187.8641, 315.3125, 347.8804, ..., 84.4587, 347.8803, 315.3124], dtype=float)

As such, spectrum is really just a shorthand for np.sqrt(a*a + b*b):

# code to be run in micropython

import ulab as np

x = np.linspace(0, 10, num=1024)
y = np.sin(x)

a, b = np.fft(y)

print('\nspectrum calculated the hard way:\t', np.sqrt(a*a + b*b))

a = np.spectrum(y)

print('\nspectrum calculated the lazy way:\t', a)
spectrum calculated the hard way:    array([187.8641, 315.3125, 347.8804, ..., 84.4587, 347.8803, 315.3124], dtype=float)

spectrum calculated the lazy way:    array([187.8641, 315.3125, 347.8804, ..., 84.4587, 347.8803, 315.3124], dtype=float)

Computation and storage costs

RAM

The FFT routine of ulab calculates the transform in place. This means that beyond reserving space for the two ndarrays that will be returned (the computation uses these two as intermediate storage space), only a handful of temporary variables, all floats or 32-bit integers, are required.

Speed of FFTs

A comment on the speed: a 1024-point transform implemented in python would cost around 90 ms, and 13 ms in assembly, if the code runs on the pyboard, v.1.1. You can gain a factor of four by moving to the D series https://github.com/peterhinch/micropython-fourier/blob/master/README.md#8-performance.

# code to be run in micropython

import ulab as np

x = np.linspace(0, 10, num=1024)
y = np.sin(x)

np.fft(y)

@timeit
def np_fft(y):
    return np.fft(y)

a, b = np_fft(y)
execution time:  1985  us

The C implementation runs in less than 2 ms on the pyboard (we have just measured that), and has been reported to run in under 0.8 ms on the D series board. That is an improvement of at least a factor of four.

Calculating FFTs of real signals

Now, if you have real signals, and you are really pressed for time, you can still gain a bit on speed without sacrificing anything at all.

If you take the FFT of a real-valued signal, the real part of the transform will be symmetric, while the imaginary part will be anti-symmetric in frequency.

If, on the other hand, the signal is imaginary-valued, then the real part of the transform will be anti-symmetric, and the imaginary part will be symmetric in frequency. These two statements follow from the definition of the Fourier transform.

By combining the two observations above, if you place the first signal, \(y_1(t)\), into the real part, and the second signal, \(y_2(t)\), into the imaginary part of your input vector, i.e., \(y(t) = y_1(t) + iy_2(t)\), and take the Fourier transform of the combined signal, then the Fourier transforms of the two components can be recovered as

:raw-latex:`\begin{eqnarray} Y_1(k) &=& \frac{1}{2}\left(Y(k) + Y^*(N-k)\right) \\ Y_2(k) &=& -\frac{i}{2}\left(Y(k) - Y^*(N-k)\right) \end{eqnarray}` where \(N\) is the length of \(y_1\), and \(Y_1, Y_2\), and \(Y\), respectively, are the Fourier transforms of \(y_1, y_2\), and \(y = y_1 + iy_2\).

Filter routines

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.convolve.html

convolve

Returns the discrete, linear convolution of two one-dimensional sequences.

Only the full mode is supported, and the mode named parameter is not accepted. Note that all other modes can be had by slicing a full result.

# code to be run in micropython

import ulab as np

x = np.array((1,2,3))
y = np.array((1,10,100,1000))

print(np.convolve(x, y))
array([1.0, 12.0, 123.0, 1230.0, 2300.0, 3000.0], dtype=float)

Extending ulab

New functions can easily be added to ulab in a couple of simple steps. At the C level, the type definition of an ndarray is as follows:

typedef struct _ndarray_obj_t {
    mp_obj_base_t base;
    size_t m, n;
    mp_obj_array_t *array;
    size_t bytes;
} ndarray_obj_t;

Creating a new ndarray

A new ndarray can be created by calling

ndarray_obj_t *new_ndarray = create_new_ndarray(m, n, typecode);

where m, and n are the number of rows and columns, respectively, and typecode is one of the values from

enum NDARRAY_TYPE {
    NDARRAY_UINT8 = 'B',
    NDARRAY_INT8 = 'b',
    NDARRAY_UINT16 = 'H',
    NDARRAY_INT16 = 'h',
    NDARRAY_FLOAT = 'f',
};

or

enum NDARRAY_TYPE {
    NDARRAY_UINT8 = 'B',
    NDARRAY_INT8 = 'b',
    NDARRAY_UINT16 = 'H',
    NDARRAY_INT16 = 'h',
    NDARRAY_FLOAT = 'd',
};

The ambiguity is caused by the fact that not all platforms implement double, and there one has to take floats. But you haven’t actually got to be concerned by this, because at the very beginning of ndarray.h, this is already taken care of: the preprocessor figures out, what the float implementation of the hardware platform is, and defines the NDARRAY_FLOAT typecode accordingly. All you have to keep in mind is that wherever you would use float or double, you have to use mp_float_t. That type is defined in py/mpconfig.h of the micropython code base.

Therefore, a 4-by-5 matrix of type float can be created as

ndarray_obj_t *new_ndarray = create_new_ndarray(4, 5, NDARRAY_FLOAT);

This function also fills in the ndarray structure’s m, n, and bytes members, as well initialises the array member with zeros.

Alternatively, a one-to-one copy of an ndarray can be gotten by calling

mp_obj_t copy_of_input_object = ndarray_copy(object_in);

Note, however, that this function takes an input object of type mp_obj_t, and returns a copy of type mp_obj_t, i.e., something that can be taken from, and can immediately be returned to the interpreter. If you want to work on the data in the copy, you still have to create a pointer to it

ndarray_obj_t *ndarray = MP_OBJ_TO_PTR(copy_of_input_object);

The values stored in array can be modified or retrieved by accessing array->items. Note that array->items is a void pointer, therefore, it must be cast before trying to access the elements. array has at least two useful members. One of them is len, which is the number of elements that the array holds, while the second one is the typecode that we passed to create_new_ndarray earlier.

Thus, staying with our example of a 4-by-5 float matrix, we can loop through all entries as

mp_float_t *items = (mp_float_t *)new_ndarray->array->items;
mp_float_t item;

for(size_t i=0; i < new_ndarray->array->len; i++) {
    item = items[i];
    // do something with item...
}

or, since the data are stored in the pointer in a C-like fashion, as

mp_float_t *items = (mp_float_t *)new_ndarray->array->items;
mp_float_t item;

for(size_t m=0; m < new_ndarray->m; m++) { // loop through the rows
    for(size_t n=0; n < new_ndarray->n; n++) { // loop through the columns
        item = items[m*new_ndarray->n+n]; // get the (m,n) entry
        // do something with item...
    }
}

Accessing data in the ndarray

We have already seen, how the entries of an array can be accessed. If the object in question comes from the user (i.e., via the micropython interface), we can get a pointer to it by calling

ndarray_obj_t *ndarray = MP_OBJ_TO_PTR(object_in);

Once the pointer is at our disposal, we can get a pointer to the underlying numerical array as discussed earlier.

If you need to find out the typecode of the array, you can get it by accessing the typecode member of array, i.e.,

ndarray->array->typecode

should be equal to B, b, H, h, or f. The size of a single item is returned by the function mp_binary_get_size('@', ndarray->array->typecode, NULL). This number is equal to 1, if the typecode is B, or b, 2, if the typecode is H, or h, 4, if the typecode is f, and 8 for d.

Alternatively, the size can be found out by dividing ndarray->bytes with the product of m, and n, i.e.,

ndarray->bytes/(ndarray->m*ndarray*n)

is equal to mp_binary_get_size('@', ndarray->array->typecode, NULL).

Making certain that we have an ndarray

A number of operations make sense for ndarrays only, therefore, before doing any heavy work on the data, it might be reasonable to check, whether the input argument is of the proper type. This you do by evaluating

mp_obj_is_type(object_in, &ulab_ndarray_type)

which should return true.

Boilerplate of sorts

To summarise the contents of the previous three sections, here is a useless function that prints out the size (m, and n) of an array, creates a copy of the input, and fills up the resulting matrix with 13.

mp_obj_t useless_function(mp_obj_t object_in) {
    if(!mp_obj_is_type(object_in, &ulab_ndarray_type)) {
        mp_raise_TypeError("useless_function takes only ndarray arguments");
    }

    mp_obj_t object_out = ndarray_copy(object_int);

    ndarray_obj_t *ndarray = MP_OBJ_TO_PTR(object_out);
    printf("\nsize (m): %ld, size (n): %ld\n", ndarray->m, ndarray->n);
    printf("\nlength (len): %ld, typecode: %d\n", ndarray->array->len, ndarray->array->typecode);
    if(ndarray->array->typecode == NDARRAY_UINT8) {
        // cast the pointer to the items
        uint8_t *items = (uint8_t *)ndarray->array->items;
        // retrieve the length of the array, and loop over the elements
        for(size_t i=0; i < ndarray->array->len; i++) items[i] = 13;
    } else if(ndarray->array->typecode == NDARRAY_INT8) {
        int8_t *items = (int8_t *)ndarray->array->items;
        for(size_t i=0; i < ndarray->array->len; i++) items[i] = 13;
    } else if(ndarray->array->typecode == NDARRAY_UINT16) {
        uint16_t *items = (uint16_t *)ndarray->array->items;
        for(size_t i=0; i < ndarray->array->len; i++) items[i] = 13;
    } else if(ndarray->array->typecode == NDARRAY_INT16) {
        int16_t *items = (int16_t *)ndarray->array->items;
        for(size_t i=0; i < ndarray->array->len; i++) items[i] = 13;
    } else {
        float *items = (mp_float_t *)ndarray->array->items;
        for(size_t i=0; i < ndarray->array->len; i++) items[i] = 13;
    }
    return object_out;
}

If, on the other hand, you want to create an ndarray from scratch, and return that, you could work along the following lines:

mp_obj_t useless_function(mp_obj_t object_in) {
    uint16_t m = mp_obj_get_int(object_in);

    ndarray_obj_t *ndarray = create_new_ndarray(1, m, NDARRAY_UINT8);

    uint8_t *items = (uint8_t *)ndarray->array->items;
    // do something with the array's entries
    // ...

    // and at the very end, return an mp_object_t
    return MP_PTR_TO_OBJ(ndarray);
}

In the boilerplate above, we cast the pointer to array->items to the required type. There are certain operations, however, when you do not need the casting. If you do not want to change the array’s values, only their position within the array, you can get away with copying the memory content, regardless the type. A good example for such a scenario is the transpose function in https://github.com/v923z/micropython-ulab/blob/master/code/linalg.c.

Compiling your module

Once you have implemented the functionality you wanted, you have to include the source code in the make file by adding it to micropython.mk:

USERMODULES_DIR := $(USERMOD_DIR)

# Add all C files to SRC_USERMOD.
SRC_USERMOD += $(USERMODULES_DIR)/ndarray.c
SRC_USERMOD += $(USERMODULES_DIR)/linalg.c
SRC_USERMOD += $(USERMODULES_DIR)/vectorise.c
SRC_USERMOD += $(USERMODULES_DIR)/poly.c
SRC_USERMOD += $(USERMODULES_DIR)/fft.c
SRC_USERMOD += $(USERMODULES_DIR)/numerical.c
SRC_USERMOD += $(USERMODULES_DIR)/ulab.c

SRC_USERMOD += $(USERMODULES_DIR)/your_module.c

CFLAGS_USERMOD += -I$(USERMODULES_DIR)

In addition, you also have to add the function objects to ulab.c, and create a QString for the function name:

...
    MP_DEFINE_CONST_FUN_OBJ_1(useless_function_obj, userless_function);
...
    STATIC const mp_map_elem_t ulab_globals_table[] = {
...
    { MP_OBJ_NEW_QSTR(MP_QSTR_useless), (mp_obj_t)&useless_function_obj },
...
};

The exact form of the function object definition

MP_DEFINE_CONST_FUN_OBJ_1(useless_function_obj, userless_function);

depends naturally on what exactly you implemented, i.e., how many arguments the function takes, whether only positional arguments allowed and so on. For a thorough discussion on how function objects have to be defined, see, e.g., the user module programming manual.

At this point, you should be able to compile the module with your extension by running make on the command line

make USER_C_MODULES=../../../ulab all

for the unix port, and

make BOARD=PYBV11 CROSS_COMPILE=<arm_tools_path>/bin/arm-none-eabi- USER_C_MODULES=../../../ulab all

for the pyboard, provided that the you have defined

#define MODULE_ULAB_ENABLED (1)

somewhere in micropython/port/unix/mpconfigport.h, or micropython/stm32/mpconfigport.h, respectively.