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 a signal coming from the ADC of the pyboard, and all 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 in ulab are implemented in a way that

conforms to numpy as much as possible
is so frugal with RAM as possible,
and yet, fast. Much faster than pure python. Think of a number between 30 and 50!

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 0.54.0), the library adds approximately 40 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.

Resources and legal matters¶

The source code of the module can be found under https://github.com/v923z/micropython-ulab/tree/master/code. The source of this user manual is under https://github.com/v923z/micropython-ulab/tree/master/docs, while the technical details of the implementation are discussed at great length in https://github.com/v923z/micropython-ulab/tree/master/docs/ulab.ipynb. If you want an even thorougher explanation on why the various constructs of the implementation work, and work in that particular way, you can read more on the subject under https://micropython-usermod.readthedocs.io/en/latest/, where I demonstrate, what you have to do, if you want to make a C object behave in a pythonic way.

The MIT licence applies to all material.

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 a ulab 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.

Differences between micropython-ulab and circuitpython-ulab¶

ulab has originally been developed for micropython, but has since been integrated into a number of its flavours. Most of these flavours are simply forks of micropython itself, with some additional functionality. One of the notable exceptions is circuitpython, which has slightly diverged at the core level, and this has some minor consequences. Some of these concern the C implementation details only, which all have been sorted out with the generous and enthusiastic support of Jeff Epler from Adafruit Industries.

There are, however, a couple of instances, where the usage in the two environments is slightly different at the python level. These are how the packages can be imported, and how the class properties can be accessed. In both cases, the circuitpython implementation results in numpy-conform code. numpy-compatibility in micropython will be implemented as soon as micropython itself has the required tools. Till then we have to live with a workaround, which I will point out at the relevant places.

Customising `ulab`¶

ulab implements a great number of functions, which are organised in sub-modules. E.g., functions related to Fourier transforms are located in the ulab.fft sub-module, so you would import fft as

import ulab
from ulab import fft

by which point you can get the FFT of your data by calling fft.fft(...).

The idea of such grouping of functions and methods is to provide a means for granularity: It is quite possible that you do not need all functions in a particular application. If you want to save some flash space, you can easily exclude arbitrary sub-modules from the firmware. The ulab.h header file contains a pre-processor flag for each sub-module. The default setting is 1 for each of them. Setting them to 0 removes the module from the compiled firmware.

The first couple of lines of the file look like this

// vectorise (all functions) takes approx. 6 kB of flash space
#define ULAB_VECTORISE_MODULE (1)

// linalg adds around 6 kB
#define ULAB_LINALG_MODULE (1)

// poly requires approx. 2.5 kB
#define ULAB_POLY_MODULE (1)

In order to simplify navigation in the header, each flag begins with ULAB_, and continues with the name of the sub-module. This name is also the .c file, where the sub-module is implemented. So, e.g., the linear algebra routines can be found in linalg.c, and the corresponding compiler flag is ULAB_LINALG_MODULE. Each section displays a hint as to how much space you can save by un-setting the flag.

At first, having to import everything in this way might appear to be overly complicated, but there is a very good reason behind all this: you can find out at the time of importing, whether a function or sub-module is part of your ulab firmware, or not. The alternative, namely, that you do not have to import anything beyond ulab, could prove catastrophic: you would learn only at run time (at the moment of calling the function in your code) that a particular function is not in the firmware, and that is most probably too late.

Except for fft, the standard sub-modules, vector, linalg, numerical, and poly are all numpy-compatible. User-defined functions that accept ndarrays as their argument should be implemented in the user sub-module, or its sub-modules. Hints as to how to do that can be found in the section Extending ulab.

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 0.50.2

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

Basic ndarray operations¶

Comparison operators*

Universal functions (also support function calls on general iterables, and vectorisation of user-defined python functions.)

Methods of ndarrays¶

Matrix methods¶

Array initialisation functions¶

Statistical and other properties of arrays¶

Linear algebra functions¶

Manipulation of polynomials¶

polyval

polyfit

FFT routines¶

fft**

ifft**

spectrogram**

Filter functions¶

convolve

sosfilt

Comparison of arrays¶

Interpolation, root finding, function minimasation¶

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, which operations can be done on it, and which functions can accept it as an argument. One important property of an ndarray is that it is also a proper micropython iterable.

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 .itemsize class property.

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 the 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 are 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

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 is faster, because simply a new copy is created, without inspection, iteration etc. It is also possible to coerce type conversion of the output (with type conversion, the iteration cannot be avoided, therefore, this case will always be slower than straight copying):

# 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)
d = np.array(b, dtype=np.uint8)

print("a:\t", a)
print("\nb:\t", b)
print("\nc:\t", c)
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([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)

d:   array([1, 2, 3, 4, 5, 6, 7, 8], dtype=uint8)

Note that the default type of the ndarray is float. Hence, if the array is initialised from another array, type conversion will always take place, except, when the output type is specifically supplied. I.e.,

# code to be run in micropython

import ulab as np

a = np.array(range(5), dtype=np.uint8)
b = np.array(a)
print("a:\t", a)
print("\nb:\t", b)

a:   array([0, 1, 2, 3, 4], dtype=uint8)

b:   array([0.0, 1.0, 2.0, 3.0, 4.0], dtype=float)

will iterate over the elements in a, since in the assignment b = np.array(a) no output type was given, therefore, float was assumed. On the other hand,

# code to be run in micropython

import ulab as np

a = np.array(range(5), dtype=np.uint8)
b = np.array(a, dtype=np.uint8)
print("a:\t", a)
print("\nb:\t", b)

a:   array([0, 1, 2, 3, 4], dtype=uint8)

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

will simply copy the content of a into b without any iteration, and will, therefore, be faster. Keep this in mind, whenever the output type, or performance is important.

Array initialisation functions¶

There are four functions that can be used for initialising an array. These are bound to ulab itself at the top level, i.e., no module has to be imported for the function invocations.

arange¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.arange.html

The function returns a one-dimensional array with evenly spaced values. Takes 3 positional arguments (two are optional), and the dtype keyword argument.

# code to be run in micropython

import ulab

print(ulab.arange(10))
print(ulab.arange(2, 10))
print(ulab.arange(2, 10, 3))
print(ulab.arange(2, 10, 3, dtype=ulab.float))

array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], dtype=int16)
array([2, 3, 4, 5, 6, 7, 8, 9], dtype=int16)
array([2, 5, 8], dtype=int16)
array([2.0, 5.0, 8.0], dtype=float)

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 CPython

### Shifting the diagonal

%%micropython -unix 1

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)

# 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)

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)

Customising array printouts¶

ndarrays are pretty-printed, i.e., if the length is larger than 10 (default value), 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)

set_printoptions¶

The default values can be overwritten by means of the set_printoptions function numpy.set_printoptions, which accepts two keywords arguments, the threshold, and the edgeitems. The first of these arguments determines the length of the longest array that will be printed in full, while the second is the number of items that will be printed on the left and right hand side of the ellipsis, if the array is longer than threshold.

# code to be run in micropython

import ulab as np

a = np.array(range(20))
print("a printed with defaults:\t", a)

np.set_printoptions(threshold=200)
print("\na printed in full:\t\t", a)

np.set_printoptions(threshold=10, edgeitems=2)
print("\na truncated with 2 edgeitems:\t", a)

a printed with defaults:     array([0.0, 1.0, 2.0, ..., 17.0, 18.0, 19.0], dtype=float)

a printed in full:           array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], dtype=float)

a truncated with 2 edgeitems:        array([0.0, 1.0, ..., 18.0, 19.0], dtype=float)

get_printoptions¶

The set value of the threshold and edgeitems can be retrieved by calling the get_printoptions function with no arguments. The function returns a dictionary with two keys.

# code to be run in micropython

import ulab as np

np.set_printoptions(threshold=100, edgeitems=20)
print(np.get_printoptions())

{'threshold': 100, 'edgeitems': 20}

Methods of ndarrays¶

.shape¶

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

WARNING: In circuitpython, you can call the method as a property, i.e.,

# 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)

WARNING: On the other hand, since properties are not implemented in micropython, there you would call the method as a function, i.e.,

# 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.

WARNING: In circuitpython, the numpy nomenclature applies, i.e.,

# 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

WARNING: In micropython, size is a method, i.e.,

# 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.

WARNING: In circuitpython:

# 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

WARNING: In micropython:

# 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

# 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 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¶

ulab implements the +, -, *, /, **, <, >, <=, >=, ==, != binary operators that work element-wise. Partial broadcasting is available, meaning that the operands either must have the same shape, or one of them must be a scalar.

The operators raise a ValueError exception, if partial broadcasting is not possible. The only exceptions are the == and != operators that will return False in this case.

WARNING: note that relational operators (<, >, <=, >=, ==, !=) should have the ndarray on their left hand side, when compared to scalars. This means that the following works

# code to be run in micropython

import ulab
a = ulab.array([1, 2, 3])
print(a > 2)

[False, False, True]

while the equivalent statement, 2 < a, will raise a TypeError exception:

# code to be run in micropython

import ulab
a = ulab.array([1, 2, 3])
print(2 < a)

Traceback (most recent call last):
  File "/dev/shm/micropython.py", line 4, in <module>
TypeError: unsupported types for __lt__: 'int', 'ndarray'

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]])

WARNING: circuitpython users should use the equal, and not_equal operators instead of ==, and !=. See the section on array comparison for details.

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:

Operations with two ndarrays of the same dtype preserve their dtype, even when the results overflow.
if either of the operands is a float, the result is automatically a float
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!

When one of the operands is a scalar, it will internally be turned into a single-element ndarray with the smallest possible dtype. Thus, e.g., if the scalar is 123, it will be converted to an array of dtype uint8.

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)

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

import utime

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: at the moment, due to implementation details, the ndarray must be on the left hand side of the relational operators. This will change in a future version of ulab.

That is, while a < 5 and 5 > a have the same meaning, the following code will not work:

# 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(5 > a)

Traceback (most recent call last):
  File "/dev/shm/micropython.py", line 5, in <module>
TypeError: unsupported types for __gt__: 'int', 'ndarray'

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 are defined in the vector sub-module, and can be calculated on any scalar, 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, arctan2, around, 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. The functions can be invoked by importing the vector sub-module first.

# code to be run in micropython

import ulab as np
from ulab import vector

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

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

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

Doing the same with list comprehension requires 30 times more time than with the ndarray, which would become even more, if we converted the resulting list to an ndarray.

# code to be run in micropython

import ulab as np
from ulab import vector
import math

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

@timeit
def timed_vector(iterable):
    return vector.exp(iterable)

@timeit
def timed_list(iterable):
    return [math.exp(i) for i in iterable]

print('iterating over ndarray in ulab')
timed_vector(b)

print('\niterating over list in ulab')
timed_vector(a)

print('\niterating over list in python')
timed_list(a)

iterating over ndarray in ulab
execution time:  441  us

iterating over list in ulab
execution time:  1266  us

iterating over list in python
execution time:  11379  us

Vectorising generic python functions¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.vectorize.html

The examples above use factory functions. In fact, they are nothing but the vectorised versions of the standard mathematical functions. User-defined python functions can also be vectorised by help of vectorize. This function takes a positional argument, namely, the python function that you want to vectorise, and a non-mandatory keyword argument, otypes, which determines the dtype of the output array. The otypes must be None (default), or any of the dtypes defined in ulab. With None, the output is automatically turned into a float array.

The return value of vectorize is a micropython object that can be called as a standard function, but which now accepts either a scalar, an ndarray, or a generic micropython iterable as its sole argument. Note that the function that is to be vectorised must have a single argument.

# code to be run in micropython

import ulab as np
from ulab import vector

def f(x):
    return x*x

vf = vector.vectorize(f)

# calling with a scalar
print('{:20}'.format('f on a scalar: '), vf(44.0))

# calling with an ndarray
a = np.array([1, 2, 3, 4])
print('{:20}'.format('f on an ndarray: '), vf(a))

# calling with a list
print('{:20}'.format('f on a list: '), vf([2, 3, 4]))

f on a scalar:       array([1936.0], dtype=float)
f on an ndarray:     array([1.0, 4.0, 9.0, 16.0], dtype=float)
f on a list:         array([4.0, 9.0, 16.0], dtype=float)

As mentioned, the dtype of the resulting ndarray can be specified via the otypes keyword. The value is bound to the function object that vectorize returns, therefore, if the same function is to be vectorised with different output types, then for each type a new function object must be created.

# code to be run in micropython

import ulab as np
from ulab import vector

l = [1, 2, 3, 4]
def f(x):
    return x*x

vf1 = vector.vectorize(f, otypes=np.uint8)
vf2 = vector.vectorize(f, otypes=np.float)

print('{:20}'.format('output is uint8: '), vf1(l))
print('{:20}'.format('output is float: '), vf2(l))

output is uint8:     array([1, 4, 9, 16], dtype=uint8)
output is float:     array([1.0, 4.0, 9.0, 16.0], dtype=float)

The otypes keyword argument cannot be used for type coercion: if the function evaluates to a float, but otypes would dictate an integer type, an exception will be raised:

# code to be run in micropython

import ulab as np
from ulab import vector

int_list = [1, 2, 3, 4]
float_list = [1.0, 2.0, 3.0, 4.0]
def f(x):
    return x*x

vf = vector.vectorize(f, otypes=np.uint8)

print('{:20}'.format('integer list: '), vf(int_list))
# this will raise a TypeError exception
print(vf(float_list))

integer list:        array([1, 4, 9, 16], dtype=uint8)

Traceback (most recent call last):
  File "/dev/shm/micropython.py", line 14, in <module>
TypeError: can't convert float to int

Benchmarks¶

It should be pointed out that the vectorize function produces the pseudo-vectorised version of the python function that is fed into it, i.e., on the C level, the same python function is called, with the all-encompassing mp_obj_t type arguments, and all that happens is that the for loop in [f(i) for i in iterable] runs purely in C. Since type checking and type conversion in f() is expensive, the speed-up is not so spectacular as when iterating over an ndarray with a factory function: a gain of approximately 30% can be expected, when a native python type (e.g., list) is returned by the function, and this becomes around 50% (a factor of 2), if conversion to an ndarray is also counted.

The following code snippet calculates the square of a 1000 numbers with the vectorised function (which returns an ndarray), with list comprehension, and with list comprehension followed by conversion to an ndarray. For comparison, the execution time is measured also for the case, when the square is calculated entirely in ulab.

# code to be run in micropython

import ulab as np
from ulab import vector

def f(x):
    return x*x

vf = vector.vectorize(f)

@timeit
def timed_vectorised_square(iterable):
    return vf(iterable)

@timeit
def timed_python_square(iterable):
    return [f(i) for i in iterable]

@timeit
def timed_ndarray_square(iterable):
    return np.array([f(i) for i in iterable])

@timeit
def timed_ulab_square(ndarray):
    return ndarray**2

print('vectorised function')
squares = timed_vectorised_square(range(1000))

print('\nlist comprehension')
squares = timed_python_square(range(1000))

print('\nlist comprehension + ndarray conversion')
squares = timed_ndarray_square(range(1000))

print('\nsquaring an ndarray entirely in ulab')
a = np.array(range(1000))
squares = timed_ulab_square(a)

vectorised function
execution time:  7237  us

list comprehension
execution time:  10248  us

list comprehension + ndarray conversion
execution time:  12562  us

squaring an ndarray entirely in ulab
execution time:  560  us

From the comparisons above, it is obvious that python functions should only be vectorised, when the same effect cannot be gotten in ulab only. However, although the time savings are not significant, there is still a good reason for caring about vectorised functions. Namely, user-defined python functions become universal, i.e., they can accept generic iterables as well as ndarrays as their arguments. A vectorised function is still a one-liner, resulting in transparent and elegant code.

A final comment on this subject: the f(x) that we defined is a generic python function. This means that it is not required that it just crunches some numbers. It has to return a number object, but it can still access the hardware in the meantime. So, e.g.,

led = pyb.LED(2)

def f(x):
    if x < 100:
        led.toggle()
    return x*x

is perfectly valid code.

around¶

numpy: https://docs.scipy.org/doc/numpy-1.17.0/reference/generated/numpy.around.html

numpy’s around function can also be found in the vector sub-module. The function implements the decimals keyword argument with default value 0. The first argument must be an ndarray. If this is not the case, the function raises a TypeError exception. Note that numpy accepts general iterables. The out keyword argument known from numpy is not accepted. The function always returns an ndarray of type mp_float_t.

# code to be run in micropython

import ulab as np
from ulab import vector

a = np.array([1, 2.2, 33.33, 444.444])
print('a:\t\t', a)
print('\ndecimals = 0\t', vector.around(a, decimals=0))
print('\ndecimals = 1\t', vector.around(a, decimals=1))
print('\ndecimals = -1\t', vector.around(a, decimals=-1))

a:           array([1.0, 2.2, 33.33, 444.444], dtype=float)

decimals = 0         array([1.0, 2.0, 33.0, 444.0], dtype=float)

decimals = 1         array([1.0, 2.2, 33.3, 444.4], dtype=float)

decimals = -1        array([0.0, 0.0, 30.0, 440.0], dtype=float)

arctan2¶

numpy: https://docs.scipy.org/doc/numpy-1.17.0/reference/generated/numpy.arctan2.html

The two-argument inverse tangent function is also part of the vector sub-module. The function implements only partial broadcasting, i.e., its two arguments either have the same shape, or at least one of them must be a single-element array. Scalars (micropython integers or floats) are also allowed.

# code to be run in micropython

import ulab as np
from ulab import vector

a = np.array([1, 2.2, 33.33, 444.444])
print('a:\t\t', a)
print('\narctan2(a, 1.0)\t', vector.arctan2(a, 1.0))
print('\narctan2(1.0, a)\t', vector.arctan2(1.0, a))
print('\narctan2(a, a): \t', vector.arctan2(a, a))

a:           array([1.0, 2.2, 33.33, 444.444], dtype=float)

arctan2(a, 1.0)      array([0.7853981633974483, 1.14416883366802, 1.5408023243361, 1.568546328341769], dtype=float)

arctan2(1.0, a)      array([0.7853981633974483, 0.426627493126876, 0.02999400245879636, 0.002249998453127392], dtype=float)

arctan2(a, a):       array([0.7853981633974483, 0.7853981633974483, 0.7853981633974483, 0.7853981633974483], dtype=float)

Numerical¶

Function in the numerical sub-module can be called by importing the sub-module first.

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
from ulab import numerical

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

b = np.array([[1, 2, 0], [1, 10, -1]])
print('\nb:\n', b)
print('min of b (flattened):', numerical.min(b))
print('min of b (axis=0):', numerical.min(b, axis=0))
print('min of b (axis=1):', numerical.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
from ulab import numerical

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

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

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

print('std, vertical: ', numerical.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
from ulab import numerical

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

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

# this should be the original vector
numerical.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
from ulab import numerical

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

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

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

numerical.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
from ulab import numerical
from ulab import vector

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 = vector.exp([1, 2, 3, 4, 5])
weight = weight/numerical.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(numerical.mean(samples[-5:]*weight))
    print(samples[-5:])
    # the data are shifted by one position to the left
    numerical.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
from ulab import numerical

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", numerical.flip(a, axis=1))
print("\na flipped vertically\n", numerical.flip(a, axis=0))
print("\na flipped horizontally+vertically\n", numerical.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
from ulab import numerical

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

print('\nfirst derivative:\n', numerical.diff(a, n=1))
print('\nsecond derivative:\n', numerical.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', numerical.diff(c, axis=0))
print('\nfirst derivative, second axis:\n', numerical.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
from ulab import numerical

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 = numerical.sort(a, axis=0)
print('\na sorted along vertical axis:\n', b)

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

c = numerical.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
from ulab import vector
from ulab import numerical

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

b = vector.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. If that happens to be the case, the function will bail out with a ValueError.

# code to be run in micropython

import ulab as np
from ulab import numerical

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 = numerical.argsort(a, axis=0)
print('\na sorted along vertical axis:\n', b)

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

c = numerical.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
from ulab import numerical

a = np.array([0, 5, 1, 3, 2, 4], dtype=np.uint8)
print('\na:\n', a)
b = numerical.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¶

Functions in the linalg module can be called by importing the sub-module first.

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:

argument is 0: the number of elements of the array
argument is 1: the number of rows
argument is 2: the number of columns

# code to be run in micropython

import ulab as np
from ulab import linalg

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

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

inv¶

numpy: https://docs.scipy.org/doc/numpy-1.17.0/reference/generated/numpy.linalg.inv.html

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
from ulab import linalg

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

print(linalg.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
from ulab import linalg

@timeit
def invert_matrix(m):
    return linalg.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
from ulab import linalg

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

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(linalg.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
from ulab import linalg

a = np.array([[1, 2], [3, 4]], dtype=np.uint8)
print(linalg.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 linalg.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
from ulab import linalg

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

the eigenvalues and eigenvectors are not necessarily sorted in the same way
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
from ulab import linalg

@timeit
def matrix_eig(a):
    return linalg.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

Cholesky decomposition¶

numpy: https://docs.scipy.org/doc/numpy-1.17.0/reference/generated/numpy.linalg.cholesky.html

cholesky takes a positive definite, symmetric square matrix as its single argument, and returns square root matrix in the lower triangular form. If the input argument does not fulfill the positivity or symmetry condition, a ValueError is raised.

# code to be run in micropython

import ulab
from ulab import linalg

a = ulab.array([[25, 15, -5], [15, 18,  0], [-5,  0, 11]])
print('a: ', a)
print('\n' + '='*20 + '\nCholesky decomposition\n', linalg.cholesky(a))

a:  array([[25.0, 15.0, -5.0],
     [15.0, 18.0, 0.0],
     [-5.0, 0.0, 11.0]], dtype=float)

====================
Cholesky decomposition
 array([[5.0, 0.0, 0.0],
     [3.0, 3.0, 0.0],
     [-1.0, 1.0, 3.0]], dtype=float)

norm¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.linalg.norm.html

The function takes a vector or matrix without options, and returns its 2-norm, i.e., the square root of the sum of the square of the elements.

# code to be run in micropython

import ulab
from ulab import linalg

a = ulab.array([1, 2, 3, 4, 5])
b = ulab.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

print('norm of a:', linalg.norm(a))
print('norm of b:', linalg.norm(b))

norm of a: 7.416198487095663
norm of b: 16.88194301613414

trace¶

numpy: https://docs.scipy.org/doc/numpy-1.17.0/reference/generated/numpy.linalg.trace.html

The trace function returns the sum of the diagonal elements of a square matrix. If the input argument is not a square matrix, an exception will be raised.

The scalar so returned will inherit the type of the input array, i.e., integer arrays have integer trace, and floating point arrays a floating point trace.

# code to be run in micropython

import ulab
from ulab import linalg

a = ulab.array([[25, 15, -5], [15, 18,  0], [-5,  0, 11]], dtype=ulab.int8)
print('a: ', a)
print('\ntrace of a: ', linalg.trace(a))

b = ulab.array([[25, 15, -5], [15, 18,  0], [-5,  0, 11]], dtype=ulab.float)

print('='*20 + '\nb: ', b)
print('\ntrace of b: ', linalg.trace(b))

a:  array([[25, 15, -5],
     [15, 18, 0],
     [-5, 0, 11]], dtype=int8)

trace of a:  54
====================
b:  array([[25.0, 15.0, -5.0],
     [15.0, 18.0, 0.0],
     [-5.0, 0.0, 11.0]], dtype=float)

trace of b:  54.0

Polynomials¶

Functions in the polynomial sub-module can be invoked by importing the module first.

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
from ulab import poly

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

# the same works with one-dimensional ndarrays
a = np.array(x)
print('\nndarray (a): ', a)
print('value of p(a): ', poly.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
from ulab import poly

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', poly.polyfit(x, y, 2))

# the same with missing x
print('\ndependent values:\t', y)
print('fitted values:\t\t', poly.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
from ulab import poly

@timeit
def time_polyfit(x, y, n):
    return poly.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¶

Functions related to Fourier transforms can be called by importing the fft sub-module first.

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
from ulab import vector
from ulab import fft

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

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

c, d = fft.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
from ulab import vector
from ulab import fft

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

a, b = fft.fft(y)

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

y, z = fft.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.

spectrogram¶

In addition to the Fourier transform and its inverse, ulab also sports a function called spectrogram, 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
from ulab import vector
from ulab import fft

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

a = fft.spectrogram(y)

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

original vector:     array([0.0, 0.009775015390171337, 0.01954909674625918, ..., -0.5275140569487312, -0.5357931822978732, -0.5440211108893639], dtype=float)

spectrum:    array([187.8635087634579, 315.3112063607119, 347.8814873399374, ..., 84.45888934298905, 347.8814873399374, 315.3112063607118], dtype=float)

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

# code to be run in micropython

import ulab as np
from ulab import fft
from ulab import vector

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

a, b = fft.fft(y)

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

a = fft.spectrogram(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
from ulab import vector
from ulab import fft

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

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

Filter routines¶

Functions in the filter module can be called by importing the sub-module first.

convolve¶

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

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
from ulab import filter

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

print(filter.convolve(x, y))

array([1.0, 12.0, 123.0, 1230.0, 2300.0, 3000.0], dtype=float)

sosfilt¶

scipy: https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.sosfilt.html

Filter data along one dimension using cascaded second-order sections.

The function takes two positional arguments, sos, the filter segments of length 6, and the one-dimensional, uniformly sample data set to be filtered. Returns the filtered data, or the filtered data and the final filter delays, if the zi keyword arguments is supplied. The keyword argument be a float ndarray of shape (n_sections, 2). If zi is not passed to the function, the initial values are assumed to be 0.

# code to be run in micropython

import ulab
from ulab import filter as filter

x = ulab.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
sos = [[1, 2, 3, 1, 5, 6], [1, 2, 3, 1, 5, 6]]
y = filter.sosfilt(sos, x)
print('y: ', y)

y:  array([0.0, 1.0, -4.0, 24.0, -104.0, 440.0, -1728.0, 6532.000000000001, -23848.0, 84864.0], dtype=float)

# code to be run in micropython

import ulab
from ulab import filter as filter

x = ulab.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
sos = [[1, 2, 3, 1, 5, 6], [1, 2, 3, 1, 5, 6]]
# initial conditions of the filter
zi = ulab.array([[1, 2], [3, 4]])

y, zf = filter.sosfilt(sos, x, zi=zi)
print('y: ', y)
print('\n' + '='*40 + '\nzf: ', zf)

y:  array([4.0, -16.0, 63.00000000000001, -227.0, 802.9999999999999, -2751.0, 9271.000000000001, -30775.0, 101067.0, -328991.0000000001], dtype=float)

========================================
zf:  array([[37242.0, 74835.0],
     [1026187.0, 1936542.0]], dtype=float)

Comparison of arrays¶

Functions in the compare module can be called by importing the sub-module first.

equal, not_equal¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.equal.html

numpy: https://numpy.org/doc/stable/reference/generated/numpy.not_equal.html

In micropython, equality of arrays or scalars can be established by utilising the ==, !=, <, >, <=, or => binary operators. In circuitpython, == and != will produce unexpected results. In order to avoid this discrepancy, and to maintain compatibility with numpy, ulab implements the equal and not_equal operators that return the same results, irrespective of the python implementation.

These two functions take two ndarrays, or scalars as their arguments. No keyword arguments are implemented.

# code to be run in micropython

import ulab as np

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

print('a: ', a)
print('b: ', b)
print('\na == b: ', np.compare.equal(a, b))
print('a != b: ', np.compare.not_equal(a, b))

# comparison with scalars
print('a == 2: ', np.compare.equal(a, 2))

a:  array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
b:  array([0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], dtype=float)

a == b:  [True, False, False, False, False, False, False, False, False]
a != b:  [False, True, True, True, True, True, True, True, True]
a == 2:  [False, False, True, False, False, False, False, False, False]

minimum¶

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

Returns the minimum of two arrays, or two scalars, or an array, and a scalar. Partial broadcasting is implemented. If the arrays are of different dtype, the output is upcast as in Binary operators. If both inputs are scalars, a scalar is returned. Only positional arguments are implemented.

maximum¶

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

Returns the maximum of two arrays, or two scalars, or an array, and a scalar. Partial broadcasting is implemented. If the arrays are of different dtype, the output is upcast as in Binary operators. If both inputs are scalars, a scalar is returned. Only positional arguments are implemented.

# code to be run in micropython

import ulab

a = ulab.array([1, 2, 3, 4, 5], dtype=ulab.uint8)
b = ulab.array([5, 4, 3, 2, 1], dtype=ulab.float)
print('minimum of a, and b:')
print(ulab.compare.minimum(a, b))

print('\nmaximum of a, and b:')
print(ulab.compare.maximum(a, b))

print('\nmaximum of 1, and 5.5:')
print(ulab.compare.maximum(1, 5.5))

minimum of a, and b:
array([1.0, 2.0, 3.0, 2.0, 1.0], dtype=float)

maximum of a, and b:
array([5.0, 4.0, 3.0, 4.0, 5.0], dtype=float)

maximum of 1, and 5.5:
5.5

clip¶

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

Clips an array, i.e., values that are outside of an interval are clipped to the interval edges. The function is equivalent to maximum(a_min, minimum(a, a_max)). or two scalars, hence partial broadcasting takes place exactly as in minimum. If the arrays are of different dtype, the output is upcast as in Binary operators.

# code to be run in micropython

import ulab

a = ulab.array(range(9), dtype=ulab.uint8)
print('a:\t\t', a)
print('clipped:\t', ulab.compare.clip(a, 3, 7))

b = 3 * ulab.ones(len(a), dtype=ulab.float)
print('\na:\t\t', a)
print('b:\t\t', b)
print('clipped:\t', ulab.compare.clip(a, b, 7))

a:           array([0, 1, 2, 3, 4, 5, 6, 7, 8], dtype=uint8)
clipped:     array([3, 3, 3, 3, 4, 5, 6, 7, 7], dtype=uint8)

a:           array([0, 1, 2, 3, 4, 5, 6, 7, 8], dtype=uint8)
b:           array([3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0], dtype=float)
clipped:     array([3.0, 3.0, 3.0, 3.0, 4.0, 5.0, 6.0, 7.0, 7.0], dtype=float)

Interpolation, root finding, and function minimisation¶

The approx sub-module defines functions for interpolating numerical data, and finding the roots and the minimum of arbitrary functions defined in python. Note that routines that work with user-defined functions still have to call the underlying python code, and therefore, gains in speed are not as significant as with other vectorised operations. As a rule of thumb, a factor of two can be expected, when compared to an optimised python implementation.

interp¶

numpy: https://docs.scipy.org/doc/numpy/numpy.interp

The interp function returns the linearly interpolated values of a one-dimensional numerical array. It requires three positional arguments,x, at which the interpolated values are evaluated, xp, the array of the independent variables of the data, and fp, the array of the dependent values of the data. xp must be a monotonically increasing sequence of numbers.

Two keyword arguments, left, and right can also be supplied; these determine the return values, if x < xp[0], and x > xp[-1], respectively. If these arguments are not supplied, left, and right default to fp[0], and fp[-1], respectively.

# code to be run in micropython

import ulab
from ulab import approx

x = ulab.array([1, 2, 3, 4, 5])
xp = ulab.array([1, 2, 3, 4])
fp = ulab.array([1, 2, 3, 5])
x = x - 0.2
print(x)
print(approx.interp(x, xp, fp))
print(approx.interp(x, xp, fp, left=0.0))
print(approx.interp(x, xp, fp, right=10.0))

array([0.8, 1.8, 2.8, 3.8, 4.8], dtype=float)
array([1.0, 1.8, 2.8, 4.6, 5.0], dtype=float)
array([0.0, 1.8, 2.8, 4.6, 5.0], dtype=float)
array([1.0, 1.8, 2.8, 4.6, 10.0], dtype=float)

newton¶

scipy:https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.newton.html

newton finds a zero of a real, user-defined function using the Newton-Raphson (or secant or Halley’s) method. The routine requires two positional arguments, the function, and the initial value. Three keyword arguments can be supplied to control the iteration. These are the absolute and relative tolerances tol, and rtol, respectively, and the number of iterations before stopping, maxiter. The function retuns a single scalar, the position of the root.

# code to be run in micropython

import ulab
from ulab import approx

def f(x):
    return x*x*x - 2.0

print(approx.newton(f, 3., tol=0.001, rtol=0.01))

1.260135727246117

bisect¶

scipy: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.bisect.html

bisect finds the root of a function of one variable using a simple bisection routine. It takes three positional arguments, the function itself, and two starting points. The function must have opposite signs at the starting points. Returned is the position of the root.

Two keyword arguments, xtol, and maxiter can be supplied to control the accuracy, and the number of bisections, respectively.

# code to be run in micropython

import ulab
from ulab import approx

def f(x):
    return x*x - 1

print(approx.bisect(f, 0, 4))

print('only 8 bisections: ',  approx.bisect(f, 0, 4, maxiter=8))

print('with 0.1 accuracy: ',  approx.bisect(f, 0, 4, xtol=0.1))

0.9999997615814209
only 8 bisections:  0.984375
with 0.1 accuracy:  0.9375

Performance¶

Since the bisect routine calls user-defined python functions, the speed gain is only about a factor of two, if compared to a purely python implementation.

# code to be run in micropython

import ulab
from ulab import approx

def f(x):
    return (x-1)*(x-1) - 2.0

def bisect(f, a, b, xtol=2.4e-7, maxiter=100):
    if f(a) * f(b) > 0:
        raise ValueError

    rtb = a if f(a) < 0.0 else b
    dx = b - a if f(a) < 0.0 else a - b
    for i in range(maxiter):
        dx *= 0.5
        x_mid = rtb + dx
        mid_value = f(x_mid)
        if mid_value < 0:
            rtb = x_mid
        if abs(dx) < xtol:
            break

    return rtb

@timeit
def bisect_approx(f, a, b):
    return approx.bisect(f, a, b)

@timeit
def bisect_timed(f, a, b):
    return bisect(f, a, b)

print('bisect running in python')
bisect_timed(f, 3, 2)

print('bisect running in C')
bisect_approx(f, 3, 2)

bisect running in python
execution time:  1270  us
bisect running in C
execution time:  642  us

fmin¶

scipy: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin.html

The fmin function finds the position of the minimum of a user-defined function by using the downhill simplex method. Requires two positional arguments, the function, and the initial value. Three keyword arguments, xatol, fatol, and maxiter stipulate conditions for stopping.

# code to be run in micropython
import ulab
from ulab import approx

def f(x):
    return (x-1)**2 - 1

print(approx.fmin(f, 3.0))
print(approx.fmin(f, 3.0, xatol=0.1))

0.9996093749999952
1.199999999999996

trapz¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.trapz.html

The function takes one or two one-dimensional ndarrays, and integrates the dependent values (y) using the trapezoidal rule. If the independent variable (x) is given, that is taken as the sample points corresponding to y.

# code to be run in micropython

import ulab
from ulab import approx

x = ulab.linspace(0, 9, num=10)
y = x*x

print('x: ',  x)
print('y: ',  y)
print('============================')
print('integral of y: ', approx.trapz(y))
print('integral of y at x: ', approx.trapz(y, x=x))

x:  array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0], dtype=float)
y:  array([0.0, 1.0, 4.0, 9.0, 16.0, 25.0, 36.0, 49.0, 64.0, 81.0], dtype=float)
============================
integral of y:  244.5
integral of y at x:  244.5

Extending ulab¶

As mentioned at the beginning, ulab is a set of sub-modules, out of which one, user is explicitly reserved for user code. You should implement your functions in this sub-module, or sub-modules thereof.

The new functions can easily be added to user 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 pre-processor 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);
}

Once the function implementation is done, you should add the function object to the globals dictionary of the extra sub-module as

...
    MP_DEFINE_CONST_FUN_OBJ_1(useless_function_obj, userless_function);
...
    STATIC const mp_map_elem_t extra_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.

And with that, you are done. You can simply call the compiler as

make BOARD=PYBV11 USER_C_MODULES=../../../ulab all

and the rest is taken care of.

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.

Introduction¶

Enter ulab¶

Purpose¶

Resources and legal matters¶

Friendly request¶

Differences between micropython-ulab and circuitpython-ulab¶

Customising ulab¶

Supported functions and methods¶

Basic ndarray operations¶

Methods of ndarrays¶

Matrix methods¶

Array initialisation functions¶

Statistical and other properties of arrays¶

Linear algebra functions¶

Manipulation of polynomials¶

FFT routines¶

Filter functions¶

Comparison of arrays¶

Interpolation, root finding, function minimasation¶

ndarray, the basic container¶

Initialising an array¶

Initialising by passing iterables¶

Initialising by passing arrays¶

Array initialisation functions¶

arange¶

ones, zeros¶

eye¶

With a single argument¶

Specifying the dimensions of the matrix¶

linspace¶

Customising array printouts¶

set_printoptions¶

get_printoptions¶

Methods of ndarrays¶

.shape¶

.size¶

.itemsize¶

.reshape¶

.flatten¶

.transpose¶

.sort¶

Unary operators¶

len¶

invert¶

abs¶

neg¶

pos¶

Binary operators¶

Upcasting¶

Benchmarks¶

Comparison operators¶

Iterating over arrays¶

Slicing and indexing¶

Indexing¶

Slicing and assigning to slices¶

Universal functions¶

Computation expenses¶

Vectorising generic python functions¶

Benchmarks¶

around¶

arctan2¶

Numerical¶

min, argmin, max, argmax¶

sum, std, mean¶

roll¶

Simple running weighted average¶

flip¶

diff¶

sort¶

argsort¶

Linalg¶

size¶

inv¶

Computation expenses¶

dot¶

det¶

Benchmark¶

eig¶

Computation expenses¶

Cholesky decomposition¶

Customising `ulab`¶