Welcome to the ulab book!

Introduction

Enter ulab

ulab is a numpy-like module for micropython and its derivatives, meant to simplify and speed up common mathematical operations on arrays. ulab implements a small subset of numpy and scipy. The functions were chosen such that they might be useful in the context of a microcontroller. However, the project is a living one, and suggestions for new features are always welcome.

This document discusses how you can use the library, starting from building your own firmware, through questions like what affects the firmware size, what are the trade-offs, and what are the most important differences to numpy and scipy, respectively. The document is organised as follows:

The chapter after this one helps you with firmware customisation.

The third chapter gives a very concise summary of the ulab functions and array methods. This chapter can be used as a quick reference.

The chapters after that are an in-depth review of most functions. Here you can find usage examples, benchmarks, as well as a thorough discussion of such concepts as broadcasting, and views versus copies.

The final chapter of this book can be regarded as the programming manual. The inner working of ulab is dissected here, and you will also find hints as to how to implement your own numpy-compatible functions.

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 (in fact, there are a couple of python-only implementations of numpy functions out there), 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

1. conforms to numpy as much as possible

2. is so frugal with RAM as possible,

3. and yet, fast. Much faster than pure python. Think of speed-ups of 30-50!

The main points of ulab are

• compact, iterable and slicable containers of numerical data in one to four dimensions. These containers support all the relevant unary and binary operators (e.g., len, ==, +, *, etc.)

• vectorised computations on micropython iterables and numerical arrays (in numpy-speak, universal functions)

• computing statistical properties (mean, standard deviation etc.) on arrays

• basic linear algebra routines (matrix inversion, multiplication, reshaping, transposition, determinant, and eigenvalues, Cholesky decomposition and so on)

• polynomial fits to numerical data, and evaluation of polynomials

• fast Fourier transforms

• filtering of data (convolution and second-order filters)

• function minimisation, fitting, and numerical approximation routines

• interfacing between numerical data and peripheral hardware devices

ulab implements close to a hundred functions and array methods. At the time of writing this manual (for version 4.0.0), the library adds approximately 120 kB of extra compiled code to the micropython (pyboard.v.1.17) firmware. However, if you are tight with flash space, you can easily shave tens of kB off the firmware. In fact, if only a small sub-set of functions are needed, you can get away with less than 10 kB of flash space. 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. while the source of this user manual is under https://github.com/v923z/micropython-ulab/tree/master/docs.

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 to be 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 us 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 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 two environments differ at the python level in how the class properties can be accessed. We will point out the differences and possible workarounds at the relevant places in this document.

Customising the firmware

As mentioned above, ulab has considerably grown since its conception, which also means that it might no longer fit on the microcontroller of your choice. There are, however, a couple of ways of customising the firmware, and thereby reducing its size.

All ulab options are listed in a single header file, ulab.h, which contains pre-processor flags for each feature that can be fine-tuned. The first couple of lines of the file look like this

// The pre-processor constants in this file determine how ulab behaves:
//
// - how many dimensions ulab can handle
// - which functions are included in the compiled firmware
// - whether the python syntax is numpy-like, or modular
// - whether arrays can be sliced and iterated over
// - which binary/unary operators are supported
//
// A considerable amount of flash space can be saved by removing (setting
// the corresponding constants to 0) the unnecessary functions and features.

// Values defined here can be overridden by your own config file as
// make -DULAB_CONFIG_FILE="my_ulab_config.h"
#if defined(ULAB_CONFIG_FILE)
#include ULAB_CONFIG_FILE
#endif

// Adds support for complex ndarrays
#ifndef ULAB_SUPPORTS_COMPLEX
#define ULAB_SUPPORTS_COMPLEX               (1)
#endif

// Determines, whether scipy is defined in ulab. The sub-modules and functions
// of scipy have to be defined separately
#define ULAB_HAS_SCIPY                      (1)

// The maximum number of dimensions the firmware should be able to support
// Possible values lie between 1, and 4, inclusive
#define ULAB_MAX_DIMS                       2

// By setting this constant to 1, iteration over array dimensions will be implemented
// as a function (ndarray_rewind_array), instead of writing out the loops in macros
// This reduces firmware size at the expense of speed
#define ULAB_HAS_FUNCTION_ITERATOR          (0)

// If NDARRAY_IS_ITERABLE is 1, the ndarray object defines its own iterator function
// This option saves approx. 250 bytes of flash space
#define NDARRAY_IS_ITERABLE                 (1)

// Slicing can be switched off by setting this variable to 0
#define NDARRAY_IS_SLICEABLE                (1)

// The default threshold for pretty printing. These variables can be overwritten
// at run-time via the set_printoptions() function
#define ULAB_HAS_PRINTOPTIONS               (1)
#define NDARRAY_PRINT_THRESHOLD             10
#define NDARRAY_PRINT_EDGEITEMS             3

// determines, whether the dtype is an object, or simply a character
// the object implementation is numpythonic, but requires more space
#define ULAB_HAS_DTYPE_OBJECT               (0)

// the ndarray binary operators
#define NDARRAY_HAS_BINARY_OPS              (1)

// Firmware size can be reduced at the expense of speed by using function
// pointers in iterations. For each operator, he function pointer saves around
// 2 kB in the two-dimensional case, and around 4 kB in the four-dimensional case.

#define NDARRAY_BINARY_USES_FUN_POINTER     (0)

#define NDARRAY_HAS_BINARY_OP_ADD           (1)
#define NDARRAY_HAS_BINARY_OP_EQUAL         (1)
#define NDARRAY_HAS_BINARY_OP_LESS          (1)
#define NDARRAY_HAS_BINARY_OP_LESS_EQUAL    (1)
#define NDARRAY_HAS_BINARY_OP_MORE          (1)
#define NDARRAY_HAS_BINARY_OP_MORE_EQUAL    (1)
#define NDARRAY_HAS_BINARY_OP_MULTIPLY      (1)
#define NDARRAY_HAS_BINARY_OP_NOT_EQUAL     (1)
#define NDARRAY_HAS_BINARY_OP_POWER         (1)
#define NDARRAY_HAS_BINARY_OP_SUBTRACT      (1)
#define NDARRAY_HAS_BINARY_OP_TRUE_DIVIDE   (1)
...

The meaning of flags with names _HAS_ should be obvious, so we will just explain the other options.

To see how much you can gain by un-setting the functions that you do not need, here are some pointers. In four dimensions, including all functions adds around 120 kB to the micropython firmware. On the other hand, if you are interested in Fourier transforms only, and strip everything else, you get away with less than 5 kB extra.

Compatibility with numpy

The functions implemented in ulab are organised in four sub-modules at the C level, namely, numpy, scipy, utils, and user. This modularity is elevated to python, meaning that in order to use functions that are part of numpy, you have to import numpy as

from ulab import numpy as np

x = np.array([4, 5, 6])
p = np.array([1, 2, 3])
np.polyval(p, x)

There are a couple of exceptions to this rule, namely fft, and linalg, which are sub-modules even in numpy, thus you have to write them out as

from ulab import numpy as np

A = np.array([1, 2, 3, 4]).reshape()
np.linalg.trace(A)

Some of the functions in ulab are re-implementations of scipy functions, and they are to be imported as

from ulab import numpy as np
from ulab import scipy as spy


x = np.array([1, 2, 3])
spy.special.erf(x)

numpy-compatibility has an enormous benefit : namely, by trying to import, we can guarantee that the same, unmodified code runs in CPython, as in micropython. The following snippet is platform-independent, thus, the python code can be tested and debugged on a computer before loading it onto the microcontroller.

try:
    from ulab import numpy as np
    from ulab import scipy as spy
except ImportError:
    import numpy as np
    import scipy as spy

x = np.array([1, 2, 3])
spy.special.erf(x)

The impact of dimensionality

Reducing the number of dimensions

ulab supports tensors of rank four, but this is expensive in terms of flash: with all available functions and options, the library adds around 100 kB to the firmware. However, if such high dimensions are not required, significant reductions in size can be gotten by changing the value of

#define ULAB_MAX_DIMS                   2

Two dimensions cost a bit more than half of four, while you can get away with around 20 kB of flash in one dimension, because all those functions that don’t make sense (e.g., matrix inversion, eigenvalues etc.) are automatically stripped from the firmware.

Using the function iterator

In higher dimensions, the firmware size increases, because each dimension (axis) adds another level of nested loops. An example of this is the macro of the binary operator in three dimensions

#define BINARY_LOOP(results, type_out, type_left, type_right, larray, lstrides, rarray, rstrides, OPERATOR)
    type_out *array = (type_out *)results->array;
    size_t j = 0;
    do {
        size_t k = 0;
        do {
            size_t l = 0;
            do {
                *array++ = *((type_left *)(larray)) OPERATOR *((type_right *)(rarray));
                (larray) += (lstrides)[ULAB_MAX_DIMS - 1];
                (rarray) += (rstrides)[ULAB_MAX_DIMS - 1];
                l++;
            } while(l < (results)->shape[ULAB_MAX_DIMS - 1]);
            (larray) -= (lstrides)[ULAB_MAX_DIMS - 1] * (results)->shape[ULAB_MAX_DIMS-1];
            (larray) += (lstrides)[ULAB_MAX_DIMS - 2];
            (rarray) -= (rstrides)[ULAB_MAX_DIMS - 1] * (results)->shape[ULAB_MAX_DIMS-1];
            (rarray) += (rstrides)[ULAB_MAX_DIMS - 2];
            k++;
        } while(k < (results)->shape[ULAB_MAX_DIMS - 2]);
        (larray) -= (lstrides)[ULAB_MAX_DIMS - 2] * results->shape[ULAB_MAX_DIMS-2];
        (larray) += (lstrides)[ULAB_MAX_DIMS - 3];
        (rarray) -= (rstrides)[ULAB_MAX_DIMS - 2] * results->shape[ULAB_MAX_DIMS-2];
        (rarray) += (rstrides)[ULAB_MAX_DIMS - 3];
        j++;
    } while(j < (results)->shape[ULAB_MAX_DIMS - 3]);

In order to reduce firmware size, it might make sense in higher dimensions to make use of the function iterator by setting the

#define ULAB_HAS_FUNCTION_ITERATOR      (1)

constant to 1. This allows the compiler to call the ndarray_rewind_array function, so that it doesn’t have to unwrap the loops for k, and j. Instead of the macro above, we now have

#define BINARY_LOOP(results, type_out, type_left, type_right, larray, lstrides, rarray, rstrides, OPERATOR)
    type_out *array = (type_out *)(results)->array;
    size_t *lcoords = ndarray_new_coords((results)->ndim);
    size_t *rcoords = ndarray_new_coords((results)->ndim);
    for(size_t i=0; i < (results)->len/(results)->shape[ULAB_MAX_DIMS -1]; i++) {
        size_t l = 0;
        do {
            *array++ = *((type_left *)(larray)) OPERATOR *((type_right *)(rarray));
            (larray) += (lstrides)[ULAB_MAX_DIMS - 1];
            (rarray) += (rstrides)[ULAB_MAX_DIMS - 1];
            l++;
        } while(l < (results)->shape[ULAB_MAX_DIMS - 1]);
        ndarray_rewind_array((results)->ndim, larray, (results)->shape, lstrides, lcoords);
        ndarray_rewind_array((results)->ndim, rarray, (results)->shape, rstrides, rcoords);
    } while(0)

Since the ndarray_rewind_array function is implemented only once, a lot of space can be saved. Obviously, function calls cost time, thus such trade-offs must be evaluated for each application. The gain also depends on which functions and features you include. Operators and functions that involve two arrays are expensive, because at the C level, the number of cases that must be handled scales with the squares of the number of data types. As an example, the innocent-looking expression

from ulab import numpy as np

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

c = a + b

requires 25 loops in C, because the dtypes of both a, and b can assume 5 different values, and the addition has to be resolved for all possible cases. Hint: each binary operator costs between 3 and 4 kB in two dimensions.

The ulab version string

As is customary with python packages, information on the package version can be found be querying the __version__ string.

# code to be run in micropython

import ulab

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

you are running ulab version 2.1.0-2D

The first three numbers indicate the major, minor, and sub-minor versions of ulab (defined by the ULAB_VERSION constant in ulab.c). We usually change the minor version, whenever a new function is added to the code, and the sub-minor version will be incremented, if a bug fix is implemented.

2D tells us that the particular firmware supports tensors of rank 2 (defined by ULAB_MAX_DIMS in ulab.h).

If you find a bug, please, include the version string in your report!

Should you need the numerical value of ULAB_MAX_DIMS, you can get it from the version string in the following way:

# code to be run in micropython

import ulab

version = ulab.__version__
version_dims = version.split('-')[1]
version_num = int(version_dims.replace('D', ''))

print('version string: ', version)
print('version dimensions: ', version_dims)
print('numerical value of dimensions: ', version_num)

version string:  2.1.0-2D
version dimensions:  2D
numerical value of dimensions:  2

ulab with complex arrays

If the firmware supports complex arrays, -c is appended to the version string as can be seen below.

# code to be run in micropython

import ulab

version = ulab.__version__

print('version string: ', version)

version string:  4.0.0-2D-c

Finding out what your firmware supports

ulab implements a number of array operators and functions, but this does not mean that all of these functions and methods are actually compiled into the firmware. You can fine-tune your firmware by setting/unsetting any of the _HAS_ constants in ulab.h.

Functions included in the firmware

The version string will not tell you everything about your firmware, because the supported functions and sub-modules can still arbitrarily be included or excluded. One way of finding out what is compiled into the firmware is calling dir with ulab as its argument.

# code to be run in micropython

from ulab import numpy as np
from ulab import scipy as spy


print('===== constants, functions, and modules of numpy =====\n\n', dir(np))

# since fft and linalg are sub-modules, print them separately
print('\nfunctions included in the fft module:\n', dir(np.fft))
print('\nfunctions included in the linalg module:\n', dir(np.linalg))

print('\n\n===== modules of scipy =====\n\n', dir(spy))
print('\nfunctions included in the optimize module:\n', dir(spy.optimize))
print('\nfunctions included in the signal module:\n', dir(spy.signal))
print('\nfunctions included in the special module:\n', dir(spy.special))

===== constants, functions, and modules of numpy =====

 ['__class__', '__name__', 'bool', 'sort', 'sum', 'acos', 'acosh', 'arange', 'arctan2', 'argmax', 'argmin', 'argsort', 'around', 'array', 'asin', 'asinh', 'atan', 'atanh', 'ceil', 'clip', 'concatenate', 'convolve', 'cos', 'cosh', 'cross', 'degrees', 'diag', 'diff', 'e', 'equal', 'exp', 'expm1', 'eye', 'fft', 'flip', 'float', 'floor', 'frombuffer', 'full', 'get_printoptions', 'inf', 'int16', 'int8', 'interp', 'linalg', 'linspace', 'log', 'log10', 'log2', 'logspace', 'max', 'maximum', 'mean', 'median', 'min', 'minimum', 'nan', 'ndinfo', 'not_equal', 'ones', 'pi', 'polyfit', 'polyval', 'radians', 'roll', 'set_printoptions', 'sin', 'sinh', 'sqrt', 'std', 'tan', 'tanh', 'trapz', 'uint16', 'uint8', 'vectorize', 'zeros']

functions included in the fft module:
 ['__class__', '__name__', 'fft', 'ifft']

functions included in the linalg module:
 ['__class__', '__name__', 'cholesky', 'det', 'dot', 'eig', 'inv', 'norm', 'trace']


===== modules of scipy =====

 ['__class__', '__name__', 'optimize', 'signal', 'special']

functions included in the optimize module:
 ['__class__', '__name__', 'bisect', 'fmin', 'newton']

functions included in the signal module:
 ['__class__', '__name__', 'sosfilt', 'spectrogram']

functions included in the special module:
 ['__class__', '__name__', 'erf', 'erfc', 'gamma', 'gammaln']

Methods included in the firmware

The dir function applied to the module or its sub-modules gives information on what the module and sub-modules include, but is not enough to find out which methods the ndarray class supports. We can list the methods by calling dir with the array object itself:

# code to be run in micropython

from ulab import numpy as np

print(dir(np.array))

['__class__', '__name__', 'copy', 'sort', '__bases__', '__dict__', 'dtype', 'flatten', 'itemsize', 'reshape', 'shape', 'size', 'strides', 'tobytes', 'transpose']

Operators included in the firmware

A list of operators cannot be generated as shown above. If you really need to find out, whether, e.g., the ** operator is supported by the firmware, you have to try it:

# code to be run in micropython

from ulab import numpy as np

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

try:
    print(a ** b)
except Exception as e:
    print('operator is not supported: ', e)

operator is not supported:  unsupported types for __pow__: 'ndarray', 'ndarray'

The exception above would be raised, if the firmware was compiled with the

#define NDARRAY_HAS_BINARY_OP_POWER         (0)

definition.

ndarray, the base class

The ndarray is the underlying container of numerical data. It can be thought of as 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.

The ndarray consists of a short header, and a pointer that holds the data. The pointer always points to a contiguous segment in memory (numpy is more flexible in this regard), and the header tells the interpreter, how the data from this segment is to be read out, and what the bytes mean. Some operations, e.g., reshape, are fast, because they do not operate on the data, they work on the header, and therefore, only a couple of bytes are manipulated, even if there are a million data entries. A more detailed exposition of how operators are implemented can be found in the section titled Programming ulab.

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, do not. You can find out, what type of float your particular platform implements by looking at the output of the .itemsize class property, or looking at the exact dtype, when you print out an array.

In addition to the five above-mentioned numerical types, it is also possible to define Boolean arrays, which can be used in the indexing of data. However, Boolean arrays are really nothing but arrays of type uint8 with an extra flag.

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. We 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 of the code block.

The ndinfo function

A concise summary of a couple of the properties of an ndarray can be printed out by calling the ndinfo function. In addition to finding out what the shape and strides of the array array, we also get the itemsize, as well as the type. An interesting piece of information is the data pointer, which tells us, what the address of the data segment of the ndarray is. We will see the significance of this in the section Slicing and indexing.

Note that this function simply prints some information, but does not return anything. If you need to get a handle of the data contained in the printout, you should call the dedicated shape, strides, or itemsize functions directly.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(5), dtype=np.float)
b = np.array(range(25), dtype=np.uint8).reshape((5, 5))
np.ndinfo(a)
print('\n')
np.ndinfo(b)

class: ndarray
shape: (5,)
strides: (8,)
itemsize: 8
data pointer: 0x7f8f6fa2e240
type: float


class: ndarray
shape: (5, 5)
strides: (5, 1)
itemsize: 1
data pointer: 0x7f8f6fa2e2e0
type: uint8

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. In addition, if ULAB_SUPPORTS_COMPLEX is set to 1 in ulab.h, the dtype can also take on the value of complex.

# code to be run in micropython

from ulab import numpy 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=float64)

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 a bit faster. This statement is especially true, if the dtypes of the source and output arrays are the same, because then the contents can simply be copied without further ado. While type conversion is also possible, it will always be slower than straight copying.

# code to be run in micropython

from ulab import numpy 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=float64)

c:   array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float64)

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

from ulab import numpy 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=float64)

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

from ulab import numpy 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 nine functions that can be used for initialising an array. Starred functions accept complex as the value of the dtype, if the firmware was compiled with complex support.

1. numpy.arange

2. numpy.concatenate

3. numpy.diag*

4. numpy.empty*

5. numpy.eye*

6. numpy.frombuffer

7. numpy.full*

8. numpy.linspace*

9. numpy.logspace

10. numpy.ones*

11. numpy.zeros*

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

from ulab import numpy as np

print(np.arange(10))
print(np.arange(2, 10))
print(np.arange(2, 10, 3))
print(np.arange(2, 10, 3, dtype=np.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=float64)

concatenate

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

The function joins a sequence of arrays, if they are compatible in shape, i.e., if all shapes except the one along the joining axis are equal.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(25), dtype=np.uint8).reshape((5, 5))
b = np.array(range(15), dtype=np.uint8).reshape((3, 5))

c = np.concatenate((a, b), axis=0)
print(c)

array([[0, 1, 2, 3, 4],
       [5, 6, 7, 8, 9],
       [10, 11, 12, 13, 14],
       [15, 16, 17, 18, 19],
       [20, 21, 22, 23, 24],
       [0, 1, 2, 3, 4],
       [5, 6, 7, 8, 9],
       [10, 11, 12, 13, 14]], dtype=uint8)

WARNING: numpy accepts arbitrary dtypes in the sequence of arrays, in ulab the dtypes must be identical. If you want to concatenate different types, you have to convert all arrays to the same type first. Here b is of float type, so it cannot directly be concatenated to a. However, if we cast the dtype of b, the concatenation works:

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(25), dtype=np.uint8).reshape((5, 5))
b = np.array(range(15), dtype=np.float).reshape((5, 3))
d = np.array(b+1, dtype=np.uint8)
print('a: ', a)
print('='*20 + '\nd: ', d)
c = np.concatenate((d, a), axis=1)
print('='*20 + '\nc: ', c)

a:  array([[0, 1, 2, 3, 4],
       [5, 6, 7, 8, 9],
       [10, 11, 12, 13, 14],
       [15, 16, 17, 18, 19],
       [20, 21, 22, 23, 24]], dtype=uint8)
====================
d:  array([[1, 2, 3],
       [4, 5, 6],
       [7, 8, 9],
       [10, 11, 12],
       [13, 14, 15]], dtype=uint8)
====================
c:  array([[1, 2, 3, 0, 1, 2, 3, 4],
       [4, 5, 6, 5, 6, 7, 8, 9],
       [7, 8, 9, 10, 11, 12, 13, 14],
       [10, 11, 12, 15, 16, 17, 18, 19],
       [13, 14, 15, 20, 21, 22, 23, 24]], dtype=uint8)

diag

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

Extract a diagonal, or construct a diagonal array.

The function takes a positional argument, an ndarray, or any micropython iterable, and an optional keyword argument, a shift, with a default value of 0. If the first argument is a two-dimensional array (or a two-dimensional iterable, e.g., a list of lists), the function returns a one-dimensional array containing the diagonal entries. The diagonal can be shifted by an amount given in the second argument. If the shift is larger than the length of the corresponding axis, an empty array is returned.

If the first argument is a one-dimensional array, the function returns a two-dimensional square tensor with its diagonal elements given by the first argument. Again, the diagonal be shifted by an amount given by the keyword argument.

The diag function can accept a complex array, if the firmware was compiled with complex support.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3], dtype=np.uint8)
print(np.diag(a))

print('\ndiagonal shifted by 2')
print(np.diag(a, k=2))

print('\ndiagonal shifted by -2')
print(np.diag(a, k=-2))

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

diagonal shifted by 2
array([[0, 0, 1, 0, 0],
       [0, 0, 0, 2, 0],
       [0, 0, 0, 0, 3],
       [0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0]], dtype=uint8)

diagonal shifted by -2
array([[0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0],
       [1, 0, 0, 0, 0],
       [0, 2, 0, 0, 0],
       [0, 0, 3, 0, 0]], dtype=uint8)

# code to be run in micropython

from ulab import numpy as np

a = np.arange(16).reshape((4, 4))
print(a)
print('\ndiagonal of a:')
print(np.diag(a))

print('\ndiagonal of a:')
print(np.diag(a))

print('\ndiagonal of a, shifted by 2')
print(np.diag(a, k=2))

print('\ndiagonal of a, shifted by 5')
print(np.diag(a, k=5))

array([[0, 1, 2, 3],
       [4, 5, 6, 7],
       [8, 9, 10, 11],
       [12, 13, 14, 15]], dtype=int16)

diagonal of a:
array([0, 5, 10, 15], dtype=int16)

diagonal of a:
array([0, 5, 10, 15], dtype=int16)

diagonal of a, shifted by 2
array([2, 7], dtype=int16)

diagonal of a, shifted by 5
array([], dtype=int16)

empty

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

empty is simply an alias for zeros, i.e., as opposed to numpy, the entries of the tensor will be initialised to zero.

The empty function can accept complex as the value of the dtype, if the firmware was compiled with complex support.

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.

The eye function can accept complex as the value of the dtype, if the firmware was compiled with complex support.

With a single argument

# code to be run in micropython

from ulab import numpy 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=float64)

Specifying the dimensions of the matrix

# code to be run in micropython

from ulab import numpy as np

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

array([[0, 0, 0, 0, 0, 0],
       [1, 0, 0, 0, 0, 0],
       [0, 1, 0, 0, 0, 0],
       [0, 0, 1, 0, 0, 0]], dtype=int16)

# code to be run in micropython

from ulab import numpy as np

print(np.eye(4, M=6, dtype=np.int8))

array([[1, 0, 0, 0, 0, 0],
       [0, 1, 0, 0, 0, 0],
       [0, 0, 1, 0, 0, 0],
       [0, 0, 0, 1, 0, 0]], dtype=int8)

frombuffer

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

The function interprets a contiguous buffer as a one-dimensional array, and thus can be used for piping buffered data directly into an array. This method of analysing, e.g., ADC data is much more efficient than passing the ADC buffer into the array constructor, because frombuffer simply creates the ndarray header and blindly copies the memory segment, without inspecting the underlying data.

The function takes a single positional argument, the buffer, and three keyword arguments. These are the dtype with a default value of float, the offset, with a default of 0, and the count, with a default of -1, meaning that all data are taken in.

# code to be run in micropython

from ulab import numpy as np

buffer = b'\x01\x02\x03\x04\x05\x06\x07\x08'
print('buffer: ', buffer)

a = np.frombuffer(buffer, dtype=np.uint8)
print('a, all data read: ', a)

b = np.frombuffer(buffer, dtype=np.uint8, offset=2)
print('b, all data with an offset: ', b)

c = np.frombuffer(buffer, dtype=np.uint8, offset=2, count=3)
print('c, only 3 items with an offset: ', c)

buffer:  b'x01x02x03x04x05x06x07x08'
a, all data read:  array([1, 2, 3, 4, 5, 6, 7, 8], dtype=uint8)
b, all data with an offset:  array([3, 4, 5, 6, 7, 8], dtype=uint8)
c, only 3 items with an offset:  array([3, 4, 5], dtype=uint8)

full

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

The function returns an array of arbitrary dimension, whose elements are all equal to the second positional argument. The first argument is a tuple describing the shape of the tensor. The dtype keyword argument with a default value of float can also be supplied.

The full function can accept a complex scalar, or complex as the value of dtype, if the firmware was compiled with complex support.

# code to be run in micropython

from ulab import numpy as np

# create an array with the default type
print(np.full((2, 4), 3))

print('\n' + '='*20 + '\n')
# the array type is uint8 now
print(np.full((2, 4), 3, dtype=np.uint8))

array([[3.0, 3.0, 3.0, 3.0],
       [3.0, 3.0, 3.0, 3.0]], dtype=float64)

====================

array([[3, 3, 3, 3],
       [3, 3, 3, 3]], dtype=uint8)

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.)

The linspace function can accept complex as the value of the dtype, if the firmware was compiled with complex support. The output dtype is automatically complex, if either of the endpoints is a complex scalar.

# code to be run in micropython

from ulab import numpy 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, 0.2040816326530612, 0.4081632653061225, ..., 9.591836734693871, 9.795918367346932, 9.999999999999993], dtype=float64)
num=5:                       array([0.0, 2.5, 5.0, 7.5, 10.0], dtype=float64)
num=5:                       array([0.0, 2.0, 4.0, 6.0, 8.0], dtype=float64)
num=5:                       array([0, 0, 1, 2, 2, 3, 4], dtype=uint8)

logspace

linspace’ equivalent for logarithmically spaced data is logspace. This function produces a sequence of numbers, in which the quotient of consecutive numbers is constant. This is a geometric sequence.

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.logspace.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, exactly as in linspace, when dtype is of integer type, the sequence is not necessarily evenly spaced in log space.

In addition to the keyword arguments found in linspace, logspace also accepts the base argument. The default value is 10.

# code to be run in micropython

from ulab import numpy as np

# generate a sequence with defaults
print('default sequence:\t', np.logspace(0, 3))

# num=5
print('num=5:\t\t\t', np.logspace(1, 10, num=5))

# num=5, endpoint=False
print('num=5:\t\t\t', np.logspace(1, 10, num=5, endpoint=False))

# num=5, endpoint=False
print('num=5:\t\t\t', np.logspace(1, 10, num=5, endpoint=False, base=2))

default sequence:    array([1.0, 1.151395399326447, 1.325711365590109, ..., 754.3120063354646, 868.5113737513561, 1000.000000000004], dtype=float64)
num=5:                       array([10.0, 1778.279410038923, 316227.766016838, 56234132.5190349, 10000000000.0], dtype=float64)
num=5:                       array([10.0, 630.9573444801933, 39810.71705534974, 2511886.431509581, 158489319.2461114], dtype=float64)
num=5:                       array([2.0, 6.964404506368993, 24.25146506416637, 84.44850628946524, 294.066778879241], dtype=float64)

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 tuple specifying the shape.

The ones/zeros functions can accept complex as the value of the dtype, if the firmware was compiled with complex support.

# code to be run in micropython

from ulab import numpy 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=float64)

When specifying the shape, make sure that the length of the tuple is not larger than the maximum dimension of your firmware.

# code to be run in micropython

from ulab import numpy as np
import ulab

print('maximum number of dimensions: ', ulab.__version__)

print(np.zeros((2, 2, 2)))

maximum number of dimensions:  2.1.0-2D

Traceback (most recent call last):
  File "/dev/shm/micropython.py", line 7, in <module>
TypeError: too many dimensions

Customising array printouts

ndarrays are pretty-printed, i.e., if the number of entries along the last axis 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

from ulab import numpy 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=float64)

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

from ulab import numpy 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=float64)

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=float64)

a truncated with 2 edgeitems:        array([0.0, 1.0, ..., 18.0, 19.0], dtype=float64)

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

from ulab import numpy as np

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

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

Methods and properties of ndarrays

Arrays have several properties that can queried, and some methods that can be called. With the exception of the flatten and transpose operators, properties return an object that describe some feature of the array, while the methods return a new array-like object. The imag, and real properties are included in the firmware only, when it was compiled with complex support.

1. .byteswap

2. .copy

3. .dtype

4. .flat

5. .flatten

6. .imag*

7. .itemsize

8. .real*

9. .reshape

10. .shape

11. .size

12. .T

13. .tobytes

14. .tolist

15. .transpose

16. .sort

.byteswap

numpy https://numpy.org/doc/stable/reference/generated/numpy.char.chararray.byteswap.html

The method takes a single keyword argument, inplace, with values True or False, and swaps the bytes in the array. If inplace = False, a new ndarray is returned, otherwise the original values are overwritten.

The frombuffer function is a convenient way of receiving data from peripheral devices that work with buffers. However, it is not guaranteed that the byte order (in other words, the endianness) of the peripheral device matches that of the microcontroller. The .byteswap method makes it possible to change the endianness of the incoming data stream.

Obviously, byteswapping makes sense only for those cases, when a datum occupies more than one byte, i.e., for the uint16, int16, and float dtypes. When dtype is either uint8, or int8, the method simply returns a view or copy of self, depending upon the value of inplace.

# code to be run in micropython

from ulab import numpy as np

buffer = b'\x01\x02\x03\x04\x05\x06\x07\x08'
print('buffer: ', buffer)

a = np.frombuffer(buffer, dtype=np.uint16)
print('a: ', a)
b = a.byteswap()
print('b: ', b)

buffer:  b'x01x02x03x04x05x06x07x08'
a:  array([513, 1027, 1541, 2055], dtype=uint16)
b:  array([258, 772, 1286, 1800], dtype=uint16)

.copy

The .copy method creates a new deep copy of an array, i.e., the entries of the source array are copied into the target array.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3, 4], dtype=np.int8)
b = a.copy()
print('a: ', a)
print('='*20)
print('b: ', b)

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

.dtype

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

The .dtype property is the dtype of an array. This can then be used for initialising another array with the matching type. ulab implements two versions of dtype; one that is numpy-like, i.e., one, which returns a dtype object, and one that is significantly cheaper in terms of flash space, but does not define a dtype object, and holds a single character (number) instead.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3, 4], dtype=np.int8)
b = np.array([5, 6, 7], dtype=a.dtype)
print('a: ', a)
print('dtype of a: ', a.dtype)
print('\nb: ', b)

a:  array([1, 2, 3, 4], dtype=int8)
dtype of a:  dtype('int8')

b:  array([5, 6, 7], dtype=int8)

If the ulab.h header file sets the pre-processor constant ULAB_HAS_DTYPE_OBJECT to 0 as

#define ULAB_HAS_DTYPE_OBJECT               (0)

then the output of the previous snippet will be

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3, 4], dtype=np.int8)
b = np.array([5, 6, 7], dtype=a.dtype)
print('a: ', a)
print('dtype of a: ', a.dtype)
print('\nb: ', b)

a:  array([1, 2, 3, 4], dtype=int8)
dtype of a:  98

b:  array([5, 6, 7], dtype=int8)

Here 98 is nothing but the ASCII value of the character b, which is the type code for signed 8-bit integers. The object definition adds around 600 bytes to the firmware.

.flat

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

.flat returns the array’s flat iterator. For one-dimensional objects the flat iterator is equivalent to the standart iterator, while for higher dimensional tensors, it amounts to first flattening the array, and then iterating over it. Note, however, that the flat iterator does not consume RAM beyond what is required for holding the position of the iterator itself, while flattening produces a new copy.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3, 4], dtype=np.int8)
for _a in a:
    print(_a)

a = np.array([[1, 2, 3, 4], [5, 6, 7, 8]], dtype=np.int8)
print('a:\n', a)

for _a in a:
    print(_a)

for _a in a.flat:
    print(_a)

1
2
3
4
a:
 array([[1, 2, 3, 4],
       [5, 6, 7, 8]], dtype=int8)
array([1, 2, 3, 4], dtype=int8)
array([5, 6, 7, 8], dtype=int8)
1
2
3
4
5
6
7
8

.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 along the last axis in the tensor), or in fortran style (i.e., moving along the first axis in the tensor).

# code to be run in micropython

from ulab import numpy 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)

.imag

numpy: https://numpy.org/doc/stable/reference/generated/numpy.ndarray.imag.html

The .imag property is defined only, if the firmware was compiled with complex support, and returns a copy with the imaginary part of an array. If the array is real, then the output is straight zeros with the dtype of the input. If the input is complex, the output dtype is always float, irrespective of the values.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3], dtype=np.uint16)
print("a:\t", a)
print("a.imag:\t", a.imag)

b = np.array([1, 2+1j, 3-1j], dtype=np.complex)
print("\nb:\t", b)
print("b.imag:\t", b.imag)

a:   array([1, 2, 3], dtype=uint16)
a.imag:      array([0, 0, 0], dtype=uint16)

b:   array([1.0+0.0j, 2.0+1.0j, 3.0-1.0j], dtype=complex)
b.imag:      array([0.0, 1.0, -1.0], dtype=float64)

.itemsize

numpy: https://numpy.org/doc/stable/reference/generated/numpy.ndarray.itemsize.html

The .itemsize property is an integer with the size of elements in the array.

# code to be run in micropython

from ulab import numpy 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=float64)
itemsize of b: 8

.real

numpy: https://numpy.org/doc/stable/reference/generated/numpy.ndarray.real.html

The .real property is defined only, if the firmware was compiled with complex support, and returns a copy with the real part of an array.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3], dtype=np.uint16)
print("a:\t", a)
print("a.real:\t", a.real)

b = np.array([1, 2+1j, 3-1j], dtype=np.complex)
print("\nb:\t", b)
print("b.real:\t", b.real)

a:   array([1, 2, 3], dtype=uint16)
a.real:      array([1, 2, 3], dtype=uint16)

b:   array([1.0+0.0j, 2.0+1.0j, 3.0-1.0j], dtype=complex)
b.real:      array([1.0, 2.0, 3.0], dtype=float64)

.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

from ulab import numpy 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)

# code to be run in CPython

Note that `ndarray.reshape()` can also be called by assigning to `ndarray.shape`.

.shape

numpy: https://numpy.org/doc/stable/reference/generated/numpy.ndarray.shape.html

The .shape property is a tuple whose elements are the length of the array along each axis.

# code to be run in micropython

from ulab import numpy 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: (4,)

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

By assigning a tuple to the .shape property, the array can be reshaped:

# code to be run in micropython

from ulab import numpy as np

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

a.shape = (3, 3)
print('\na:\n', a)

a:
 array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0], dtype=float64)

a:
 array([[1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0],
       [7.0, 8.0, 9.0]], dtype=float64)

.size

numpy: https://numpy.org/doc/stable/reference/generated/numpy.ndarray.size.html

The .size property is an integer specifying the number of elements in the array.

# code to be run in micropython

from ulab import numpy 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

.T

The .T property of the ndarray is equivalent to .transpose.

.tobytes

numpy: https://numpy.org/doc/stable/reference/generated/numpy.ndarray.tobytes.html

The .tobytes method can be used for acquiring a handle of the underlying data pointer of an array, and it returns a new bytearray that can be fed into any method that can accep a bytearray, e.g., ADC data can be buffered into this bytearray, or the bytearray can be fed into a DAC. Since the bytearray is really nothing but the bare data container of the array, any manipulation on the bytearray automatically modifies the array itself.

Note that the method raises a ValueError exception, if the array is not dense (i.e., it has already been sliced).

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(8), dtype=np.uint8)
print('a: ', a)
b = a.tobytes()
print('b: ', b)

# modify b
b[0] = 13

print('='*20)
print('b: ', b)
print('a: ', a)

a:  array([0, 1, 2, 3, 4, 5, 6, 7], dtype=uint8)
b:  bytearray(b'x00x01x02x03x04x05x06x07')
====================
b:  bytearray(b'rx01x02x03x04x05x06x07')
a:  array([13, 1, 2, 3, 4, 5, 6, 7], dtype=uint8)

.tolist

numpy: https://numpy.org/doc/stable/reference/generated/numpy.ndarray.tolist.html

The .tolist method can be used for converting the numerical array into a (nested) python lists.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(4), dtype=np.uint8)
print('a: ', a)
b = a.tolist()
print('b: ', b)

c = a.reshape((2, 2))
print('='*20)
print('c: ', c)
d = c.tolist()
print('d: ', d)

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

.transpose

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

Returns the transposed array. Only defined, if the number of maximum dimensions is larger than 1.

# code to be run in micropython

from ulab import numpy 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)

The transpose of the array can also be gotten through the T property:

# code to be run in micropython

from ulab import numpy as np

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=np.uint8)
print('a:\n', a)
print('\ntranspose of a:\n', a.T)

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

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

.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

from ulab import numpy 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(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(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, the array, and returns either the length of the first axis.

# code to be run in micropython

from ulab import numpy 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:  (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:  2
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

from ulab import numpy 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

from ulab import numpy 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

from ulab import numpy 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

from ulab import numpy 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. Broadcasting is available, meaning that the two operands do not even have to have the same shape. If the lengths along the respective axes are equal, or one of them is 1, or the axis is missing, the element-wise operation can still be carried out. A thorough explanation of broadcasting can be found under https://numpy.org/doc/stable/user/basics.broadcasting.html.

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

from ulab import numpy as np

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

array([False, False, True], dtype=bool)

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

# code to be run in micropython

from ulab import numpy as np

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

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

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:

1. Operations on two ndarrays of the same dtype preserve their dtype, even when the results overflow.

2. if either of the operands is a float, the result is automatically a float

3. When 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 into an array of dtype uint8, while -1000 will be converted into int16. An mp_obj_float, will always be promoted to dtype float. Similarly, if ulab supports complex arrays, the result of a binary operation involving a complex array is always complex. Other micropython types (e.g., lists, tuples, etc.) raise a TypeError exception.

left hand side	right hand side	ulab result	numpy result
uint8	int8	int16	int16
uint8	int16	int16	int16
uint8	uint16	uint16	uint16
int8	int16	int16	int16
int8	uint16	uint16	int32
uint16	int16	float	int32

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

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

Upcasting can be seen in action in the following snippet:

# code to be run in micropython

from ulab import numpy 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=float64)
a*c:         array([1.0, 4.0, 9.0, 16.0], dtype=float64)

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

from ulab import numpy 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

The python implementation above is not 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

from ulab import numpy as np

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

array([True, True, True, True, False, False, False, False], dtype=bool)

WARNING: at the moment, due to micropython’s implementation details, the ndarray must be on the left hand side of the relational operators.

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'

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 reduced-dimensional view is created and returned.

# code to be run in micropython

from ulab import numpy 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

Views vs. copies

numpy has a very important concept called views, which is a powerful extension of python’s own notion of slicing. 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.

In order to see what slicing does, let us take the string a = '012345679'! We can extract every second character by creating the slice ::2, which is equivalent to 0:len(a):2, i.e., increments the character pointer by 2 starting from 0, and traversing the string up to the very end.

# code to be run in CPython

string = '0123456789'
string[::2]

'02468'

Now, we can do the same with numerical arrays.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(10), dtype=np.uint8)
print('a:\t', a)

print('a[::2]:\t', a[::2])

a:   array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], dtype=uint8)
a[::2]:      array([0, 2, 4, 6, 8], dtype=uint8)

This looks similar to string above, but there is a very important difference that is not so obvious. Namely, string[::2] produces a partial copy of string, while a[::2] only produces a view of a. What this means is that a, and a[::2] share their data, and the only difference between the two is, how the data are read out. In other words, internally, a[::2] has the same data pointer as a. We can easily convince ourselves that this is indeed the case by calling the ndinfo function: the data pointer entry is the same in the two printouts.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(10), dtype=np.uint8)
print('a: ', a, '\n')
np.ndinfo(a)
print('\n' + '='*20)
print('a[::2]: ', a[::2], '\n')
np.ndinfo(a[::2])

a:  array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], dtype=uint8)

class: ndarray
shape: (10,)
strides: (1,)
itemsize: 1
data pointer: 0x7ff6c6193220
type: uint8

====================
a[::2]:  array([0, 2, 4, 6, 8], dtype=uint8)

class: ndarray
shape: (5,)
strides: (2,)
itemsize: 1
data pointer: 0x7ff6c6193220
type: uint8

If you are still a bit confused about the meaning of views, the section Slicing and assigning to slices should clarify the issue.

Indexing

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

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(10), dtype=np.uint8)
print("a: ", a)
print("the first, and last element of a:\n", a[0], a[-1])
print("the second, and last but one element of a:\n", a[1], a[-2])

a:  array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], dtype=uint8)
the first, and last element of a:
 0 9
the second, and last but one element of a:
 1 8

Indexing can be applied to higher-dimensional tensors, too. When the length of the indexing sequences is smaller than the number of dimensions, a new view is returned, otherwise, we get a single number.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(9), dtype=np.uint8).reshape((3, 3))
print("a:\n", a)
print("a[0]:\n", a[0])
print("a[1,1]: ", a[1,1])

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

Indices can also be 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; when the rank of the tensor is higher than 1, the function raises a NotImplementedError exception, though this will be rectified in a future version of ulab.

# code to be run in micropython

from ulab import numpy 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

from ulab import numpy 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=uint16)

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

100*sin(b):  array([-75.68024953079282, -75.68024953079282, -75.68024953079282, 14.11200080598672, 14.11200080598672, 14.11200080598672, 42.01670368266409, 42.01670368266409, 42.01670368266409], dtype=float)

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

Boolean indices can also be used in assignments, if the array is one-dimensional. The following example replaces the data in an array, wherever some condition is fulfilled.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(9), dtype=np.uint8)
b = np.array(range(9)) + 12

print(a[b < 15])

a[b < 15] = 123
print(a)

array([0, 1, 2], dtype=uint8)
array([123, 123, 123, 3, 4, 5, 6, 7, 8], dtype=uint8)

On the right hand side of the assignment we can even have another array.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(9), dtype=np.uint8)
b = np.array(range(9)) + 12

print(a[b < 15], b[b < 15])

a[b < 15] = b[b < 15]
print(a)

array([0, 1, 2], dtype=uint8) array([12.0, 13.0, 14.0], dtype=float)
array([12, 13, 14, 3, 4, 5, 6, 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

# code to be run in micropython

from ulab import numpy 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 row
print('\na[-1]:\n', a[-1])

# the last two rows backwards
print('\na[-1:-3:-1]:\n', a[-1:-3:-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:-3:-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

from ulab import numpy as np

a = np.zeros((3, 3), dtype=np.uint8)
print('a:\n', a)

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

a = np.zeros((3, 3), dtype=np.uint8)

# 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]:
 array([[0, 0, 3],
    [0, 0, 3],
    [0, 0, 3]], dtype=uint8)

Now, you should notice that we re-set the array a after the first assignment. Do you care to see what happens, if we do not do that? Well, here are the results:

# code to be run in micropython

from ulab import numpy as np

a = np.zeros((3, 3), dtype=np.uint8)
b = a[:,:]
# assign 1 to the first row
b[0] = 1

# assigning to the last column
b[:,2] = 3
print('a: ', a)

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

Note that both assignments involved b, and not a, yet, when we print out a, its entries are updated. This proves our earlier statement about the behaviour of views: in the statement b = a[:,:] we simply created a view of a, and not a deep copy of it, meaning that whenever we modify b, we actually modify a, because the underlying data container of a and b are shared between the two object. Having a single data container for two seemingly different objects provides an extremely powerful way of manipulating sub-sets of numerical data.

If you want to work on a copy of your data, you can use the .copy method of the ndarray. The following snippet should drive the point home:

# code to be run in micropython

from ulab import numpy as np

a = np.zeros((3, 3), dtype=np.uint8)
b = a.copy()

# get the address of the underlying data pointer

np.ndinfo(a)
print()
np.ndinfo(b)

# assign 1 to the first row of b, and do not touch a
b[0] = 1

print()
print('a: ', a)
print('='*20)
print('b: ', b)

class: ndarray
shape: (3, 3)
strides: (3, 1)
itemsize: 1
data pointer: 0x7ff737ea3220
type: uint8

class: ndarray
shape: (3, 3)
strides: (3, 1)
itemsize: 1
data pointer: 0x7ff737ea3340
type: uint8

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

The .copy method can also be applied to views: below, a[0] is a view of a, out of which we create a deep copy called b. This is a row vector now. We can then do whatever we want to with b, and that leaves a unchanged.

# code to be run in micropython

from ulab import numpy as np

a = np.zeros((3, 3), dtype=np.uint8)
b = a[0].copy()
print('b: ', b)
print('='*20)
# assign 1 to the first entry of b, and do not touch a
b[0] = 1
print('a: ', a)
print('='*20)
print('b: ', b)

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

The fact that the underlying data of a view is the same as that of the original array has another important consequence, namely, that the creation of a view is cheap. Both in terms of RAM, and execution time. A view is really nothing but a short header with a data array that already exists, and is filled up. Hence, creating the view requires only the creation of its header. This operation is fast, and uses virtually no RAM.

Numpy functions

This section of the manual discusses those functions that were adapted from numpy. Starred functions accept complex arrays as arguments, if the firmware was compiled with complex support.

1. numpy.all*

2. numpy.any*

3. numpy.argmax

4. numpy.argmin

5. numpy.argsort

6. numpy.asarray*

7. numpy.clip

8. numpy.compress*

9. numpy.conjugate*

10. numpy.convolve*

11. numpy.delete

12. numpy.diff

13. numpy.dot

14. numpy.equal

15. numpy.flip*

16. numpy.imag*

17. numpy.interp

18. numpy.isfinite

19. numpy.isinf

20. numpy.load

21. numpy.loadtxt

22. numpy.max

23. numpy.maximum

24. numpy.mean

25. numpy.median

26. numpy.min

27. numpy.minimum

28. numpy.nozero

29. numpy.not_equal

30. numpy.polyfit

31. numpy.polyval

32. numpy.real*

33. numpy.roll

34. numpy.save

35. numpy.savetxt

36. numpy.size

37. numpy.sort

38. numpy.sort_complex*

39. numpy.std

40. numpy.sum

41. numpy.trace

42. numpy.trapz

43. numpy.where

all

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

The function takes one positional, and one keyword argument, the axis, with a default value of None, and tests, whether all array elements along the given axis evaluate to True. If the keyword argument is None, the flattened array is inspected.

Elements of an array evaluate to True, if they are not equal to zero, or the Boolean False. The return value if a Boolean ndarray.

If the firmware was compiled with complex support, the function can accept complex arrays.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(12)).reshape((3, 4))

print('\na:\n', a)

b = np.all(a)
print('\nall of the flattened array:\n', b)

c = np.all(a, axis=0)
print('\nall of a along 0th axis:\n', c)

d = np.all(a, axis=1)
print('\nall of a along 1st axis:\n', d)

a:
 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]], dtype=float64)

all of the flattened array:
 False

all of a along 0th axis:
 array([False, True, True, True], dtype=bool)

all of a along 1st axis:
 array([False, True, True], dtype=bool)

any

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

The function takes one positional, and one keyword argument, the axis, with a default value of None, and tests, whether any array element along the given axis evaluates to True. If the keyword argument is None, the flattened array is inspected.

Elements of an array evaluate to True, if they are not equal to zero, or the Boolean False. The return value if a Boolean ndarray.

If the firmware was compiled with complex support, the function can accept complex arrays.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(12)).reshape((3, 4))

print('\na:\n', a)

b = np.any(a)
print('\nany of the flattened array:\n', b)

c = np.any(a, axis=0)
print('\nany of a along 0th axis:\n', c)

d = np.any(a, axis=1)
print('\nany of a along 1st axis:\n', d)

a:
 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]], dtype=float64)

any of the flattened array:
 True

any of a along 0th axis:
 array([True, True, True, True], dtype=bool)

any of a along 1st axis:
 array([True, True, True], dtype=bool)

argmax

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

See numpy.max.

argmin

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

See numpy.max.

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

from ulab import numpy as np

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

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

c = np.argsort(a, axis=None)
print('\nflattened a sorted:\n', c)

a:
 array([[1.0, 12.0, 3.0, 0.0],
       [5.0, 3.0, 4.0, 1.0],
       [9.0, 11.0, 1.0, 8.0],
       [7.0, 10.0, 0.0, 1.0]], dtype=float64)

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)

Traceback (most recent call last):
  File "/dev/shm/micropython.py", line 12, in <module>
NotImplementedError: argsort is not implemented for flattened arrays

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

from ulab import numpy as np

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

asarray

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

The function takes a single positional argument, and an optional keyword argument, dtype, with a default value of None.

If the positional argument is an ndarray, and its dtypes is identical to the value of the keyword argument, or if the keyword argument is None, then the positional argument is simply returned. If the original dtype, and the value of the keyword argument are different, then a copy is returned, with appropriate dtype conversion.

If the positional argument is an iterable, then the function is simply an alias for array.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(9), dtype=np.uint8)
b = np.asarray(a)
c = np.asarray(a, dtype=np.int8)
print('a:{}'.format(a))
print('b:{}'.format(b))
print('a == b: {}'.format(a is b))

print('\nc:{}'.format(c))
print('a == c: {}'.format(a is c))

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

c:array([0, 1, 2, 3, 4, 5, 6, 7, 8], dtype=int8)
a == c: False

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)) 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

from ulab import numpy as np

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

b = 3 * np.ones(len(a), dtype=np.float)
print('\na:\t\t', a)
print('b:\t\t', b)
print('clipped:\t', np.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=float64)
clipped:     array([3.0, 3.0, 3.0, 3.0, 4.0, 5.0, 6.0, 7.0, 7.0], dtype=float64)

compress

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

The function returns selected slices of an array along given axis. If the axis keyword is None, the flattened array is used.

If the firmware was compiled with complex support, the function can accept complex arguments.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(6)).reshape((2, 3))

print('a:\n', a)
print('\ncompress(a):\n', np.compress([0, 1], a, axis=0))

a:
 array([[0.0, 1.0, 2.0],
       [3.0, 4.0, 5.0]], dtype=float64)

compress(a):
 array([[3.0, 4.0, 5.0]], dtype=float64)

conjugate

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

If the firmware was compiled with complex support, the function calculates the complex conjugate of the input array. If the input array is of real dtype, then the output is simply a copy, preserving the dtype.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3, 4], dtype=np.uint8)
b = np.array([1+1j, 2-2j, 3+3j, 4-4j], dtype=np.complex)

print('a:\t\t', a)
print('conjugate(a):\t', np.conjugate(a))
print()
print('b:\t\t', b)
print('conjugate(b):\t', np.conjugate(b))

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

b:           array([1.0+1.0j, 2.0-2.0j, 3.0+3.0j, 4.0-4.0j], dtype=complex)
conjugate(b):        array([1.0-1.0j, 2.0+2.0j, 3.0-3.0j, 4.0+4.0j], dtype=complex)

convolve

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

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

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.

If the firmware was compiled with complex support, the function can accept complex arrays.

# code to be run in micropython

from ulab import numpy as np

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

print(np.convolve(x, y))

array([1.0, 12.0, 123.0, 1230.0, 2300.0, 3000.0], dtype=float64)

delete

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

The function returns a new array with sub-arrays along an axis deleted. It takes two positional arguments, the array, and the indices, which will be removed, as well as the axis keyword argument with a default value of None. If the axis is None, the will be flattened first.

The second positional argument can be a scalar, or any micropython iterable. Since range can also be passed in place of the indices, slicing can be emulated. If the indices are negative, the elements are counted from the end of the axis.

Note that the function creates a copy of the indices first, because it is not guaranteed that the indices are ordered. Keep this in mind, when working with large arrays.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(25), dtype=np.uint8).reshape((5,5))
print('a:\n', a)
print('\naxis = 0\n', np.delete(a, 2, axis=0))
print('\naxis = 1\n', np.delete(a, -2, axis=1))
print('\naxis = None\n', np.delete(a, [0, 1, 2, 22]))

a:
 array([[0, 1, 2, 3, 4],
       [5, 6, 7, 8, 9],
       [10, 11, 12, 13, 14],
       [15, 16, 17, 18, 19],
       [20, 21, 22, 23, 24]], dtype=uint8)

axis = 0
 array([[0, 1, 2, 3, 4],
       [5, 6, 7, 8, 9],
       [15, 16, 17, 18, 19],
       [20, 21, 22, 23, 24]], dtype=uint8)

axis = 1
 array([[0, 1, 2, 4],
       [5, 6, 7, 9],
       [10, 11, 12, 14],
       [15, 16, 17, 19],
       [20, 21, 22, 24]], dtype=uint8)

axis = None
 array([3, 4, 5, ..., 21, 23, 24], 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

from ulab import numpy as np

a = np.array(range(9), dtype=np.uint8)
a[3] = 10
print('a:\n', a)

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

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

a:
 array([0, 1, 2, 10, 4, 5, 6, 7, 8], dtype=uint8)

first derivative:
 array([1, 1, 8, 250, 1, 1, 1, 1], dtype=uint8)

second derivative:
 array([0, 249, 14, 249, 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=float64)

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=float64)

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=float64)

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

from ulab import numpy as np

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

m:
 array([[1, 2, 3],
       [4, 5, 6],
       [7, 10, 9]], dtype=uint8)

m^-1:
 array([[-1.25, 1.0, -0.25],
       [0.4999999999999998, -1.0, 0.5],
       [0.4166666666666668, 0.3333333333333333, -0.25]], dtype=float64)

m*m^-1:
 array([[1.0, 0.0, 0.0],
       [4.440892098500626e-16, 1.0, 0.0],
       [8.881784197001252e-16, 0.0, 1.0]], dtype=float64)

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

from ulab import numpy as np

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

array([[1, 2, 3, 4],
       [5, 6, 7, 8]], dtype=uint8)
array([[1, 2],
       [3, 4],
       [5, 6],
       [7, 8]], dtype=uint8)
array([[50.0, 60.0],
       [114.0, 140.0]], dtype=float64)

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

from ulab import numpy as np

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

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

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

a:  array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float64)
b:  array([0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], dtype=float64)

a == b:  array([True, False, False, False, False, False, False, False, False], dtype=bool)
a != b:  array([False, True, True, True, True, True, True, True, True], dtype=bool)
a == 2:  array([False, False, True, False, False, False, False, False, False], dtype=bool)

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.

If the firmware was compiled with complex support, the function can accept complex arrays.

# code to be run in micropython

from ulab import numpy as np

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

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

a:   array([1.0, 2.0, 3.0, 4.0, 5.0], dtype=float64)
a flipped:   array([5.0, 4.0, 3.0, 2.0, 1.0], dtype=float64)

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)

imag

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

The imag function returns the imaginary part of an array, or scalar. It cannot accept a generic iterable as its argument. The function is defined only, if the firmware was compiled with complex support.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3], dtype=np.uint16)
print("a:\t\t", a)
print("imag(a):\t", np.imag(a))

b = np.array([1, 2+1j, 3-1j], dtype=np.complex)
print("\nb:\t\t", b)
print("imag(b):\t", np.imag(b))

a:           array([1, 2, 3], dtype=uint16)
imag(a):     array([0, 0, 0], dtype=uint16)

b:           array([1.0+0.0j, 2.0+1.0j, 3.0-1.0j], dtype=complex)
imag(b):     array([0.0, 1.0, -1.0], dtype=float64)

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 data variable, 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

from ulab import numpy as np

x = np.array([1, 2, 3, 4, 5]) - 0.2
xp = np.array([1, 2, 3, 4])
fp = np.array([1, 2, 3, 5])

print(x)
print(np.interp(x, xp, fp))
print(np.interp(x, xp, fp, left=0.0))
print(np.interp(x, xp, fp, right=10.0))

array([0.8, 1.8, 2.8, 3.8, 4.8], dtype=float64)
array([1.0, 1.8, 2.8, 4.6, 5.0], dtype=float64)
array([0.0, 1.8, 2.8, 4.6, 5.0], dtype=float64)
array([1.0, 1.8, 2.8, 4.6, 10.0], dtype=float64)

isfinite

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

Returns a Boolean array of the same shape as the input, or a True/False, if the input is a scalar. In the return value, all elements are True at positions, where the input value was finite. Integer types are automatically finite, therefore, if the input is of integer type, the output will be the True tensor.

# code to be run in micropython

from ulab import numpy as np

print('isfinite(0): ', np.isfinite(0))

a = np.array([1, 2, np.nan])
print('\n' + '='*20)
print('a:\n', a)
print('\nisfinite(a):\n', np.isfinite(a))

b = np.array([1, 2, np.inf])
print('\n' + '='*20)
print('b:\n', b)
print('\nisfinite(b):\n', np.isfinite(b))

c = np.array([1, 2, 3], dtype=np.uint16)
print('\n' + '='*20)
print('c:\n', c)
print('\nisfinite(c):\n', np.isfinite(c))

isfinite(0):  True

====================
a:
 array([1.0, 2.0, nan], dtype=float64)

isfinite(a):
 array([True, True, False], dtype=bool)

====================
b:
 array([1.0, 2.0, inf], dtype=float64)

isfinite(b):
 array([True, True, False], dtype=bool)

====================
c:
 array([1, 2, 3], dtype=uint16)

isfinite(c):
 array([True, True, True], dtype=bool)

isinf

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

Similar to isfinite, but the output is True at positions, where the input is infinite. Integer types return the False tensor.

# code to be run in micropython

from ulab import numpy as np

print('isinf(0): ', np.isinf(0))

a = np.array([1, 2, np.nan])
print('\n' + '='*20)
print('a:\n', a)
print('\nisinf(a):\n', np.isinf(a))

b = np.array([1, 2, np.inf])
print('\n' + '='*20)
print('b:\n', b)
print('\nisinf(b):\n', np.isinf(b))

c = np.array([1, 2, 3], dtype=np.uint16)
print('\n' + '='*20)
print('c:\n', c)
print('\nisinf(c):\n', np.isinf(c))

isinf(0):  False

====================
a:
 array([1.0, 2.0, nan], dtype=float64)

isinf(a):
 array([False, False, False], dtype=bool)

====================
b:
 array([1.0, 2.0, inf], dtype=float64)

isinf(b):
 array([False, False, True], dtype=bool)

====================
c:
 array([1, 2, 3], dtype=uint16)

isinf(c):
 array([False, False, False], dtype=bool)

load

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

The function reads data from a file in numpy’s platform-independent format, and returns the generated array. If the endianness of the data in the file and the microcontroller differ, the bytes are automatically swapped.

# code to be run in micropython

from ulab import numpy as np

a = np.load('a.npy')
print(a)

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],
       [20.0, 21.0, 22.0, 23.0, 24.0]], dtype=float64)

loadtxt

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

The function reads data from a text file, and returns the generated array. It takes a file name as the single positional argument, and the following keyword arguments:

1. comments='#'

2. dtype=float

3. delimiter=','

4. max_rows (with a default of all rows)

5. skip_rows=0

6. usecols (with a default of all columns)

If dtype is supplied and is not float, the data entries will be converted to the appropriate integer type by rounding the values.

# code to be run in micropython

from ulab import numpy as np

print('read all data')
print(np.loadtxt('loadtxt.dat'))

print('\nread maximum 5 rows (first row is a comment line)')
print(np.loadtxt('loadtxt.dat', max_rows=5))

print('\nread maximum 5 rows, convert dtype (first row is a comment line)')
print(np.loadtxt('loadtxt.dat', max_rows=5, dtype=np.uint8))

print('\nskip the first 3 rows, convert dtype (first row is a comment line)')
print(np.loadtxt('loadtxt.dat', skiprows=3, dtype=np.uint8))

read all data
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],
       [20.0, 21.0, 22.0, 23.0],
       [24.0, 25.0, 26.0, 27.0],
       [28.00000000000001, 29.0, 30.0, 31.0],
       [32.0, 33.0, 34.00000000000001, 35.0]], dtype=float64)

read maximum 5 rows (first row is a comment line)
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]], dtype=float64)

read maximum 5 rows, convert dtype (first row is a comment line)
array([[0, 1, 2, 3],
       [4, 5, 6, 7],
       [8, 9, 10, 11],
       [12, 13, 14, 15]], dtype=uint8)

skip the first 3 rows, convert dtype (first row is a comment line)
array([[8, 9, 10, 11],
       [12, 13, 14, 15],
       [16, 17, 18, 19],
       [20, 21, 22, 23],
       [24, 25, 26, 27],
       [28, 29, 30, 31],
       [32, 33, 34, 35]], dtype=uint8)

mean

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

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

from ulab import numpy as np

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print('a: \n', a)
print('mean, flat: ', np.mean(a))
print('mean, horizontal: ', np.mean(a, axis=1))
print('mean, vertical: ', np.mean(a, axis=0))

a:
 array([[1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0],
       [7.0, 8.0, 9.0]], dtype=float64)
mean, flat:  5.0
mean, horizontal:  array([2.0, 5.0, 8.0], dtype=float64)
mean, vertical:  array([4.0, 5.0, 6.0], dtype=float64)

max

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

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

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

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmin.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

from ulab import numpy as np

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

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

a: array([1.0, 2.0, 0.0, 1.0, 10.0], dtype=float64)
min of a: 0.0
argmin of a: 2

b:
 array([[1.0, 2.0, 0.0],
       [1.0, 10.0, -1.0]], dtype=float64)
min of b (flattened): -1.0
min of b (axis=0): array([1.0, 2.0, -1.0], dtype=float64)
min of b (axis=1): array([0.0, -1.0], dtype=float64)

median

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

The function computes the median along the specified axis, and returns the median of the array elements. If the axis keyword argument is None, the arrays is flattened first. The dtype of the results is always float.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(12), dtype=np.int8).reshape((3, 4))
print('a:\n', a)
print('\nmedian of the flattened array: ', np.median(a))
print('\nmedian along the vertical axis: ', np.median(a, axis=0))
print('\nmedian along the horizontal axis: ', np.median(a, axis=1))

a:
 array([[0, 1, 2, 3],
       [4, 5, 6, 7],
       [8, 9, 10, 11]], dtype=int8)

median of the flattened array:  5.5

median along the vertical axis:  array([4.0, 5.0, 6.0, 7.0], dtype=float64)

median along the horizontal axis:  array([1.5, 5.5, 9.5], dtype=float64)

min

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

See numpy.max.

minimum

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

See numpy.maximum

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. 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

from ulab import numpy as np

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

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

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

minimum of a, and b:
array([1.0, 2.0, 3.0, 2.0, 1.0], dtype=float64)

maximum of a, and b:
array([5.0, 4.0, 3.0, 4.0, 5.0], dtype=float64)

maximum of 1, and 5.5:
5.5

nonzero

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

nonzero returns the indices of the elements of an array that are not zero. If the number of dimensions of the array is larger than one, a tuple of arrays is returned, one for each dimension, containing the indices of the non-zero elements in that dimension.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(9)) - 5
print('a:\n', a)
print(np.nonzero(a))

a = a.reshape((3,3))
print('\na:\n', a)
print(np.nonzero(a))

a:
 array([-5.0, -4.0, -3.0, -2.0, -1.0, 0.0, 1.0, 2.0, 3.0], dtype=float64)
(array([0, 1, 2, 3, 4, 6, 7, 8], dtype=uint16),)

a:
 array([[-5.0, -4.0, -3.0],
       [-2.0, -1.0, 0.0],
       [1.0, 2.0, 3.0]], dtype=float64)
(array([0, 0, 0, 1, 1, 2, 2, 2], dtype=uint16), array([0, 1, 2, 0, 1, 0, 1, 2], dtype=uint16))

not_equal

See numpy.equal.

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 as range(len(y)).

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

# code to be run in micropython

from ulab import numpy as np

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

# the same with missing x
print('\ndependent values:\t', y)
print('fitted values:\t\t', np.polyfit(y, 2))

independent values:  array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0], dtype=float64)
dependent values:    array([9.0, 4.0, 1.0, 0.0, 1.0, 4.0, 9.0], dtype=float64)
fitted values:               array([1.0, -6.0, 9.000000000000004], dtype=float64)

dependent values:    array([9.0, 4.0, 1.0, 0.0, 1.0, 4.0, 9.0], dtype=float64)
fitted values:               array([1.0, -6.0, 9.000000000000004], dtype=float64)

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 linalg.inv also apply. The example from above needs around 150 microseconds to return:

# code to be run in micropython

from ulab import numpy as np

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

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

time_polyfit(x, y, 2)

execution time:  153  us

polyval

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

polyval takes two arguments, both arrays or generic micropython iterables returning scalars.

# code to be run in micropython

from ulab import numpy as np

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

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

coefficients:  [1, 1, 1, 0]
independent values:  [0, 1, 2, 3, 4]

values of p(x):  array([0.0, 3.0, 14.0, 39.0, 84.0], dtype=float64)

ndarray (a):  array([0.0, 1.0, 2.0, 3.0, 4.0], dtype=float64)
value of p(a):  array([0.0, 3.0, 14.0, 39.0, 84.0], dtype=float64)

real

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

The real function returns the real part of an array, or scalar. It cannot accept a generic iterable as its argument. The function is defined only, if the firmware was compiled with complex support.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3], dtype=np.uint16)
print("a:\t\t", a)
print("real(a):\t", np.real(a))

b = np.array([1, 2+1j, 3-1j], dtype=np.complex)
print("\nb:\t\t", b)
print("real(b):\t", np.real(b))

a:           array([1, 2, 3], dtype=uint16)
real(a):     array([1, 2, 3], dtype=uint16)

b:           array([1.0+0.0j, 2.0+1.0j, 3.0-1.0j], dtype=complex)
real(b):     array([1.0, 2.0, 3.0], dtype=float64)

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

from ulab import numpy as np

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

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

# this should be the original vector
a = np.roll(a, -2)
print("a rolled to the right:\t", a)

a:                   array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float64)
a rolled to the left:        array([7.0, 8.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0], dtype=float64)
a rolled to the right:       array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float64)

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

from ulab import numpy as np

a = np.array(range(12)).reshape((3, 4))
print("a:\n", a)
a = np.roll(a, 2, axis=0)
print("\na rolled up:\n", a)

a = np.array(range(12)).reshape((3, 4))
print("a:\n", a)
a = np.roll(a, -1, axis=1)
print("\na rolled to the left:\n", a)

a = np.array(range(12)).reshape((3, 4))
print("a:\n", a)
a = np.roll(a, 1, axis=None)
print("\na rolled with None:\n", a)

a:
 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]], dtype=float64)

a rolled up:
 array([[4.0, 5.0, 6.0, 7.0],
       [8.0, 9.0, 10.0, 11.0],
       [0.0, 1.0, 2.0, 3.0]], dtype=float64)
a:
 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]], dtype=float64)

a rolled to the left:
 array([[1.0, 2.0, 3.0, 0.0],
       [5.0, 6.0, 7.0, 4.0],
       [9.0, 10.0, 11.0, 8.0]], dtype=float64)
a:
 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]], dtype=float64)

a rolled with None:
 array([[11.0, 0.0, 1.0, 2.0],
       [3.0, 4.0, 5.0, 6.0],
       [7.0, 8.0, 9.0, 10.0]], dtype=float64)

save

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

With the help of this function, numerical array can be saved in numpy’s platform-independent format.

The function takes two positional arguments, the name of the output file, and the array.

# code to be run in CPython

a = np.array(range(25)).reshape((5, 5))
np.save('a.npy', a)

savetxt

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

With the help of this function, numerical array can be saved in a text file. The function takes two positional arguments, the name of the output file, and the array, and also implements the comments='#' delimiter=' ', the header='', and footer='' keyword arguments. The input is treated as of type float, i.e., the output is always in the floating point representation.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(12), dtype=np.uint8).reshape((3, 4))
np.savetxt('savetxt.dat', a)

with open('savetxt.dat', 'r') as fin:
    print(fin.read())

np.savetxt('savetxt.dat', a,
           comments='!',
           delimiter=';',
           header='col1;col2;col3;col4',
           footer='saved data')

with open('savetxt.dat', 'r') as fin:
    print(fin.read())

0.000000000000000 1.000000000000000 2.000000000000000 3.000000000000000
4.000000000000000 5.000000000000000 6.000000000000000 7.000000000000000
8.000000000000000 9.000000000000000 10.000000000000000 11.000000000000000

!col1;col2;col3;col4
0.000000000000000;1.000000000000000;2.000000000000000;3.000000000000000
4.000000000000000;5.000000000000000;6.000000000000000;7.000000000000000
8.000000000000000;9.000000000000000;10.000000000000000;11.000000000000000
!saved data

size

The function takes a single positional argument, and an optional keyword argument, axis, with a default value of None, and returns the size of an array along that axis. If axis is None, the total length of the array (the product of the elements of its shape) is returned.

# code to be run in micropython

from ulab import numpy as np

a = np.ones((2, 3))

print(a)
print('size(a, axis=0): ', np.size(a, axis=0))
print('size(a, axis=1): ', np.size(a, axis=1))
print('size(a, axis=None): ', np.size(a, axis=None))

array([[1.0, 1.0, 1.0],
       [1.0, 1.0, 1.0]], dtype=float64)
size(a, axis=0):  2
size(a, axis=1):  3
size(a, axis=None):  6

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

from ulab import numpy as np

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

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

c = np.sort(a, axis=None)
print('\nflattened a sorted:\n', c)

a:
 array([[1.0, 12.0, 3.0, 0.0],
       [5.0, 3.0, 4.0, 1.0],
       [9.0, 11.0, 1.0, 8.0],
       [7.0, 10.0, 0.0, 1.0]], dtype=float64)

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=float64)

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=float64)

flattened a sorted:
 array([0.0, 0.0, 1.0, ..., 10.0, 11.0, 12.0], dtype=float64)

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

from ulab import numpy as np

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

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

sort_complex

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

If the firmware was compiled with complex support, the functions sorts the input array first according to its real part, and then the imaginary part. The input must be a one-dimensional array. The output is always of dtype complex, even if the input was real integer.

# code to be run in micropython

from ulab import numpy as np

a = np.array([5, 4, 3, 2, 1], dtype=np.int16)
print('a:\t\t\t', a)
print('sort_complex(a):\t', np.sort_complex(a))
print()

b = np.array([5, 4+3j, 4-2j, 0, 1j], dtype=np.complex)
print('b:\t\t\t', b)
print('sort_complex(b):\t', np.sort_complex(b))

a:                   array([5, 4, 3, 2, 1], dtype=int16)
sort_complex(a):     array([1.0+0.0j, 2.0+0.0j, 3.0+0.0j, 4.0+0.0j, 5.0+0.0j], dtype=complex)

b:                   array([5.0+0.0j, 4.0+3.0j, 4.0-2.0j, 0.0+0.0j, 0.0+1.0j], dtype=complex)
sort_complex(b):     array([0.0+0.0j, 0.0+1.0j, 4.0-2.0j, 4.0+3.0j, 5.0+0.0j], dtype=complex)

std

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

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

from ulab import numpy as np

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print('a: \n', a)
print('sum, flat array: ', np.std(a))
print('std, vertical: ', np.std(a, axis=0))
print('std, horizonal: ', np.std(a, axis=1))

a:
 array([[1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0],
       [7.0, 8.0, 9.0]], dtype=float64)
sum, flat array:  2.581988897471611
std, vertical:  array([2.449489742783178, 2.449489742783178, 2.449489742783178], dtype=float64)
std, horizonal:  array([0.8164965809277261, 0.8164965809277261, 0.8164965809277261], dtype=float64)

sum

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

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

from ulab import numpy as np

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

print('sum, flat array: ', np.sum(a))
print('sum, horizontal: ', np.sum(a, axis=1))
print('std, vertical: ', np.sum(a, axis=0))

a:
 array([[1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0],
       [7.0, 8.0, 9.0]], dtype=float64)
sum, flat array:  45.0
sum, horizontal:  array([6.0, 15.0, 24.0], dtype=float64)
std, vertical:  array([12.0, 15.0, 18.0], dtype=float64)

trace

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

from ulab import numpy as np

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

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

print('='*20 + '\nb: ', b)
print('\ntrace of b: ', np.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=float64)

trace of b:  54.0

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

from ulab import numpy as np

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

print('x: ',  x)
print('y: ',  y)
print('============================')
print('integral of y: ', np.trapz(y))
print('integral of y at x: ', np.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=float64)
y:  array([0.0, 1.0, 4.0, 9.0, 16.0, 25.0, 36.0, 49.0, 64.0, 81.0], dtype=float64)
============================
integral of y:  244.5
integral of y at x:  244.5

where

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

The function takes three positional arguments, condition, x, and y, and returns a new ndarray, whose values are taken from either x, or y, depending on the truthness of condition. The three arguments are broadcast together, and the function raises a ValueError exception, if broadcasting is not possible.

The function is implemented for ndarrays only: other iterable types can be passed after casting them to an ndarray by calling the array constructor.

If the dtypes of x, and y differ, the output is upcast as discussed earlier.

Note that the condition is expanded into an Boolean ndarray. This means that the storage required to hold the condition should be taken into account, whenever the function is called.

The following example returns an ndarray of length 4, with 1 at positions, where condition is smaller than 3, and with -1 otherwise.

# code to be run in micropython


from ulab import numpy as np

condition = np.array([1, 2, 3, 4], dtype=np.uint8)
print(np.where(condition < 3, 1, -1))

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

The next snippet shows, how values from two arrays can be fed into the output:

# code to be run in micropython


from ulab import numpy as np

condition = np.array([1, 2, 3, 4], dtype=np.uint8)
x = np.array([11, 22, 33, 44], dtype=np.uint8)
y = np.array([1, 2, 3, 4], dtype=np.uint8)
print(np.where(condition < 3, x, y))

array([11, 22, 3, 4], dtype=uint8)

Universal functions

Standard mathematical functions 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 only exceptions to this rule are the exp, and sqrt functions, which, if ULAB_SUPPORTS_COMPLEX is set to 1 in ulab.h, can return complex arrays, depending on the argument. All 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 (starred functions can operate on, or can return complex arrays):

acos, acosh, arctan2, around, asin, asinh, atan, arctan2, atanh, ceil, cos, degrees, exp*, expm1, floor, log, log10, log2, radians, 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, only the exponential of the matrix entries.

# code to be run in micropython

from ulab import numpy as np

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

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

# with 1D arrays
print('\n=============\nb:\n', b)
print('exp(b):\n', np.exp(b))

# as well as with matrices
c = np.array(range(9)).reshape((3, 3))
print('\n=============\nc:\n', c)
print('exp(c):\n', np.exp(c))

a:   range(0, 9)
exp(a):      array([1.0, 2.718281828459045, 7.38905609893065, 20.08553692318767, 54.59815003314424, 148.4131591025766, 403.4287934927351, 1096.633158428459, 2980.957987041728], dtype=float64)

=============
b:
 array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float64)
exp(b):
 array([1.0, 2.718281828459045, 7.38905609893065, 20.08553692318767, 54.59815003314424, 148.4131591025766, 403.4287934927351, 1096.633158428459, 2980.957987041728], dtype=float64)

=============
c:
 array([[0.0, 1.0, 2.0],
       [3.0, 4.0, 5.0],
       [6.0, 7.0, 8.0]], dtype=float64)
exp(c):
 array([[1.0, 2.718281828459045, 7.38905609893065],
       [20.08553692318767, 54.59815003314424, 148.4131591025766],
       [403.4287934927351, 1096.633158428459, 2980.957987041728]], dtype=float64)

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

from ulab import numpy as np
import math

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

@timeit
def timed_vector(iterable):
    return np.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

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 broadcasting as discussed in the section on ndarrays. Scalars (micropython integers or floats) are also allowed.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2.2, 33.33, 444.444])
print('a:\n', a)
print('\narctan2(a, 1.0)\n', np.arctan2(a, 1.0))
print('\narctan2(1.0, a)\n', np.arctan2(1.0, a))
print('\narctan2(a, a): \n', np.arctan2(a, a))

a:
 array([1.0, 2.2, 33.33, 444.444], dtype=float64)

arctan2(a, 1.0)
 array([0.7853981633974483, 1.14416883366802, 1.5408023243361, 1.568546328341769], dtype=float64)

arctan2(1.0, a)
 array([0.7853981633974483, 0.426627493126876, 0.02999400245879636, 0.002249998453127392], dtype=float64)

arctan2(a, a):
 array([0.7853981633974483, 0.7853981633974483, 0.7853981633974483, 0.7853981633974483], dtype=float64)

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

from ulab import numpy as np

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

a:           array([1.0, 2.2, 33.33, 444.444], dtype=float64)

decimals = 0         array([1.0, 2.0, 33.0, 444.0], dtype=float64)

decimals = 1         array([1.0, 2.2, 33.3, 444.4], dtype=float64)

decimals = -1        array([0.0, 0.0, 30.0, 440.0], dtype=float64)

exp

If ULAB_SUPPORTS_COMPLEX is set to 1 in ulab.h, the exponential function can also take complex arrays as its argument, in which case the return value is also complex.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3])
print('a:\t\t', a)
print('exp(a):\t\t', np.exp(a))
print()

b = np.array([1+1j, 2+2j, 3+3j], dtype=np.complex)
print('b:\t\t', b)
print('exp(b):\t\t', np.exp(b))

a:           array([1.0, 2.0, 3.0], dtype=float64)
exp(a):              array([2.718281828459045, 7.38905609893065, 20.08553692318767], dtype=float64)

b:           array([1.0+1.0j, 2.0+2.0j, 3.0+3.0j], dtype=complex)
exp(b):              array([1.468693939915885+2.287355287178842j, -3.074932320639359+6.71884969742825j, -19.88453084414699+2.834471132487004j], dtype=complex)

sqrt

If ULAB_SUPPORTS_COMPLEX is set to 1 in ulab.h, the exponential function can also take complex arrays as its argument, in which case the return value is also complex. If the input is real, but the results might be complex, the user is supposed to specify the output dtype in the function call. Otherwise, the square roots of negative numbers will result in NaN.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, -1])
print('a:\t\t', a)
print('sqrt(a):\t\t', np.sqrt(a))
print('sqrt(a):\t\t', np.sqrt(a, dtype=np.complex))

a:           array([1.0, -1.0], dtype=float64)
sqrt(a):             array([1.0, nan], dtype=float64)
sqrt(a):             array([1.0+0.0j, 0.0+1.0j], dtype=complex)

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

from ulab import numpy as np

def f(x):
    return x*x

vf = np.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=float64)
f on an ndarray:     array([1.0, 4.0, 9.0, 16.0], dtype=float64)
f on a list:         array([4.0, 9.0, 16.0], dtype=float64)

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

from ulab import numpy as np

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

vf1 = np.vectorize(f, otypes=np.uint8)
vf2 = np.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=float64)

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

from ulab import numpy as np

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

vf = np.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

from ulab import numpy as np

def f(x):
    return x*x

vf = np.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.

numpy.fft

Functions related to Fourier transforms can be called by prepending them with numpy.fft.. The module defines the following two functions:

1. numpy.fft.fft

2. numpy.fft.ifft

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 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

from ulab import numpy as np

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

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

c, d = np.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)

ulab with complex support

If the ULAB_SUPPORTS_COMPLEX, and ULAB_FFT_IS_NUMPY_COMPATIBLE pre-processor constants are set to 1 in ulab.h as

// Adds support for complex ndarrays
#ifndef ULAB_SUPPORTS_COMPLEX
#define ULAB_SUPPORTS_COMPLEX               (1)
#endif

#ifndef ULAB_FFT_IS_NUMPY_COMPATIBLE
#define ULAB_FFT_IS_NUMPY_COMPATIBLE    (1)
#endif

then the FFT routine will behave in a numpy-compatible way: the single input array can either be real, in which case the imaginary part is assumed to be zero, or complex. The output is also complex.

While numpy-compatibility might be a desired feature, it has one side effect, namely, the FFT routine consumes approx. 50% more RAM. The reason for this lies in the implementation details.

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

from ulab import numpy as np

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

a, b = np.fft.fft(y)

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

y, z = np.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.

ulab with complex support

The fft.ifft function can also be made numpy-compatible by setting the ULAB_SUPPORTS_COMPLEX, and ULAB_FFT_IS_NUMPY_COMPATIBLE pre-processor constants to 1.

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

from ulab import numpy as np

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

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

numpy.linalg

Functions in the linalg module can be called by prepending them by numpy.linalg.. The module defines the following seven functions:

1. numpy.linalg.cholesky

2. numpy.linalg.det

3. numpy.linalg.eig

4. numpy.linalg.inv

5. numpy.linalg.norm

6. numpy.linalg.qr

cholesky

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

The function of the Cholesky decomposition takes a positive definite, symmetric square matrix as its single argument, and returns the 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

from ulab import numpy as np

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

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

from ulab import numpy as np

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

from ulab import numpy as np

@timeit
def matrix_det(m):
    return np.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

from ulab import numpy as np

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

eigenvectors of a:
 array([[0.8151560042509081, -0.4499411232970823, -0.1644660242574522, 0.3256141906686505],
       [0.2211334179893007, 0.7846992598235538, 0.08372081379922657, 0.5730077734355189],
       [-0.1340114162071679, -0.3100776411558949, 0.8742786816656, 0.3486109343758527],
       [-0.5183258053659028, -0.292663481927148, -0.4489749870391468, 0.6664142156731531]], dtype=float)

eigenvalues of a:
 array([-1.165288365404889, 0.8029365530314914, 5.585625756072663, 13.77672605630074], 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', y)
print('\neigenvalues of a:\n', x)

eigenvectors 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]]

eigenvalues of a:
 [13.77672606 -1.16528837  0.80293655  5.58562576]

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

1. the eigenvalues and eigenvectors are not necessarily sorted in the same way

2. an eigenvector can be multiplied by an arbitrary non-zero scalar, and it is still an eigenvector with the same eigenvalue. This is why all signs of the eigenvector belonging to 5.58, and 0.80 are flipped in ulab with respect to numpy. This difference, however, is of absolutely no consequence.

Computation expenses

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

# code to be run in micropython

from ulab import numpy as np

@timeit
def matrix_eig(a):
    return np.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

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

from ulab import numpy as np

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

print(np.linalg.inv(m))

array([[-2.166666666666667, 1.500000000000001, -0.8333333333333337, 1.0],
       [1.666666666666667, -3.333333333333335, 1.666666666666668, -0.0],
       [0.1666666666666666, 2.166666666666668, -0.8333333333333337, -1.0],
       [-0.1666666666666667, -0.3333333333333333, 0.0, 0.5]], dtype=float64)

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

from ulab import numpy as np

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

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

from ulab import numpy as np

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

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

norm of a: 7.416198487095663
norm of b: 16.88194301613414

qr

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

The function computes the QR decomposition of a matrix m of dimensions (M, N), i.e., it returns two such matrices, q’, and r, that m = qr, where q is orthonormal, and r is upper triangular. In addition to the input matrix, which is the first positional argument, the function accepts the mode keyword argument with a default value of reduced. If mode is reduced, q, and r are returned in the reduced representation. Otherwise, the outputs will have dimensions (M, M), and (M, N), respectively.

# code to be run in micropython

from ulab import numpy as np

A = np.arange(6).reshape((3, 2))
print('A: \n', A)

print('complete decomposition')
q, r = np.linalg.qr(A, mode='complete')
print('q: \n', q)
print()
print('r: \n', r)

print('\n\nreduced decomposition')
q, r = np.linalg.qr(A, mode='reduced')
print('q: \n', q)
print()
print('r: \n', r)

A:
 array([[0, 1],
       [2, 3],
       [4, 5]], dtype=int16)
complete decomposition
q:
 array([[0.0, -0.9128709291752768, 0.408248290463863],
       [-0.447213595499958, -0.3651483716701107, -0.8164965809277261],
       [-0.8944271909999159, 0.1825741858350553, 0.408248290463863]], dtype=float64)

r:
 array([[-4.47213595499958, -5.813776741499454],
       [0.0, -1.095445115010332],
       [0.0, 0.0]], dtype=float64)


reduced decomposition
q:
 array([[0.0, -0.9128709291752768],
       [-0.447213595499958, -0.3651483716701107],
       [-0.8944271909999159, 0.1825741858350553]], dtype=float64)

r:
 array([[-4.47213595499958, -5.813776741499454],
       [0.0, -1.095445115010332]], dtype=float64)

scipy.linalg

scipy’s linalg module contains two functions, solve_triangular, and cho_solve. The functions can be called by prepending them by scipy.linalg..

1. scipy.linalg.solve_cho

2. scipy.linalg.solve_triangular

cho_solve

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

Solve the linear equations

:raw-latex:`\begin{equation} \mathbf{A}\cdot\mathbf{x} = \mathbf{b} \end{equation}`

given the Cholesky factorization of \(\mathbf{A}\). As opposed to scipy, the function simply takes the Cholesky-factorised matrix, \(\mathbf{A}\), and \(\mathbf{b}\) as inputs.

# code to be run in micropython

from ulab import numpy as np
from ulab import scipy as spy

A = np.array([[3, 0, 0, 0], [2, 1, 0, 0], [1, 0, 1, 0], [1, 2, 1, 8]])
b = np.array([4, 2, 4, 2])

print(spy.linalg.cho_solve(A, b))

array([-0.01388888888888906, -0.6458333333333331, 2.677083333333333, -0.01041666666666667], dtype=float64)

solve_triangular

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

Solve the linear equation

:raw-latex:`\begin{equation} \mathbf{a}\cdot\mathbf{x} = \mathbf{b} \end{equation}`

with the assumption that \(\mathbf{a}\) is a triangular matrix. The two position arguments are \(\mathbf{a}\), and \(\mathbf{b}\), and the optional keyword argument is lower with a default value of False. lower determines, whether data are taken from the lower, or upper triangle of \(\mathbf{a}\).

Note that \(\mathbf{a}\) itself does not have to be a triangular matrix: if it is not, then the values are simply taken to be 0 in the upper or lower triangle, as dictated by lower. However, \(\mathbf{a}\cdot\mathbf{x}\) will yield \(\mathbf{b}\) only, when \(\mathbf{a}\) is triangular. You should keep this in mind, when trying to establish the validity of the solution by back substitution.

# code to be run in micropython

from ulab import numpy as np
from ulab import scipy as spy

a = np.array([[3, 0, 0, 0], [2, 1, 0, 0], [1, 0, 1, 0], [1, 2, 1, 8]])
b = np.array([4, 2, 4, 2])

print('a:\n')
print(a)
print('\nb: ', b)

x = spy.linalg.solve_triangular(a, b, lower=True)

print('='*20)
print('x: ', x)
print('\ndot(a, x): ', np.dot(a, x))

a:

array([[3.0, 0.0, 0.0, 0.0],
       [2.0, 1.0, 0.0, 0.0],
       [1.0, 0.0, 1.0, 0.0],
       [1.0, 2.0, 1.0, 8.0]], dtype=float64)

b:  array([4.0, 2.0, 4.0, 2.0], dtype=float64)
====================
x:  array([1.333333333333333, -0.6666666666666665, 2.666666666666667, -0.08333333333333337], dtype=float64)

dot(a, x):  array([4.0, 2.0, 4.0, 2.0], dtype=float64)

With get the same solution, \(\mathbf{x}\), with the following matrix, but the dot product of \(\mathbf{a}\), and \(\mathbf{x}\) is no longer \(\mathbf{b}\):

# code to be run in micropython

from ulab import numpy as np
from ulab import scipy as spy

a = np.array([[3, 2, 1, 0], [2, 1, 0, 1], [1, 0, 1, 4], [1, 2, 1, 8]])
b = np.array([4, 2, 4, 2])

print('a:\n')
print(a)
print('\nb: ', b)

x = spy.linalg.solve_triangular(a, b, lower=True)

print('='*20)
print('x: ', x)
print('\ndot(a, x): ', np.dot(a, x))

a:

array([[3.0, 2.0, 1.0, 0.0],
       [2.0, 1.0, 0.0, 1.0],
       [1.0, 0.0, 1.0, 4.0],
       [1.0, 2.0, 1.0, 8.0]], dtype=float64)

b:  array([4.0, 2.0, 4.0, 2.0], dtype=float64)
====================
x:  array([1.333333333333333, -0.6666666666666665, 2.666666666666667, -0.08333333333333337], dtype=float64)

dot(a, x):  array([5.333333333333334, 1.916666666666666, 3.666666666666667, 2.0], dtype=float64)

scipy.optimize

Functions in the optimize module can be called by prepending them by scipy.optimize.. The module defines the following three functions:

1. scipy.optimize.bisect

2. scipy.optimize.fmin

3. scipy.optimize.newton

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.

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

from ulab import scipy as spy

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

print(spy.optimize.bisect(f, 0, 4))

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

print('with 0.1 accuracy: ',  spy.optimize.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

from ulab import scipy as spy

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_scipy(f, a, b):
    return spy.optimize.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_scipy(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

from ulab import scipy as spy

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

print(spy.optimize.fmin(f, 3.0))
print(spy.optimize.fmin(f, 3.0, xatol=0.1))

0.9996093749999952
1.199999999999996

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

from ulab import scipy as spy

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

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

1.260135727246117

scipy.signal

This module defines the single function:

1. scipy.signal.sosfilt

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 sampled 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 must 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

from ulab import numpy as np
from ulab import scipy as spy

x = np.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 = spy.signal.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

from ulab import numpy as np
from ulab import scipy as spy

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

y, zf = spy.signal.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)

scipy.special

scipy’s special module defines several functions that behave as do the standard mathematical functions of the numpy, i.e., they can be called 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).

At present, ulab’s special module contains the following functions:

erf, erfc, gamma, and gammaln, and they can be called by prepending them by scipy.special..

# code to be run in micropython

from ulab import numpy as np
from ulab import scipy as spy

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

print('a: ', a)
print(spy.special.erf(a))

print('\nb: ', b)
print(spy.special.erfc(b))

a:  range(0, 9)
array([0.0, 0.8427007929497149, 0.9953222650189527, 0.9999779095030014, 0.9999999845827421, 1.0, 1.0, 1.0, 1.0], dtype=float64)

b:  array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float64)
array([1.0, 0.1572992070502851, 0.004677734981047265, 2.209049699858544e-05, 1.541725790028002e-08, 1.537459794428035e-12, 2.151973671249892e-17, 4.183825607779414e-23, 1.122429717298293e-29], dtype=float64)

ulab utilities

There might be cases, when the format of your data does not conform to ulab, i.e., there is no obvious way to map the data to any of the five supported dtypes. A trivial example is an ADC or microphone signal with 32-bit resolution. For such cases, ulab defines the utils module, which, at the moment, has four functions that are not numpy compatible, but which should ease interfacing ndarrays to peripheral devices.

The utils module can be enabled by setting the ULAB_HAS_UTILS_MODULE constant to 1 in ulab.h:

#ifndef ULAB_HAS_UTILS_MODULE
#define ULAB_HAS_UTILS_MODULE               (1)
#endif

This still does not compile any functions into the firmware. You can add a function by setting the corresponding pre-processor constant to 1. E.g.,

#ifndef ULAB_UTILS_HAS_FROM_INT16_BUFFER
#define ULAB_UTILS_HAS_FROM_INT16_BUFFER    (1)
#endif

from_int32_buffer, from_uint32_buffer

With the help of utils.from_int32_buffer, and utils.from_uint32_buffer, it is possible to convert 32-bit integer buffers to ndarrays of float type. These functions have a syntax similar to numpy.frombuffer; they support the count=-1, and offset=0 keyword arguments. However, in addition, they also accept out=None, and byteswap=False.

Here is an example without keyword arguments

# code to be run in micropython

from ulab import numpy as np
from ulab import utils

a = bytearray([1, 1, 0, 0, 0, 0, 0, 255])
print('a: ', a)
print()
print('unsigned integers: ', utils.from_uint32_buffer(a))

b = bytearray([1, 1, 0, 0, 0, 0, 0, 255])
print('\nb:  ', b)
print()
print('signed integers: ', utils.from_int32_buffer(b))

a:  bytearray(b'x01x01x00x00x00x00x00xff')

unsigned integers:  array([257.0, 4278190080.000001], dtype=float64)

b:   bytearray(b'x01x01x00x00x00x00x00xff')

signed integers:  array([257.0, -16777216.0], dtype=float64)

The meaning of count, and offset is similar to that in numpy.frombuffer. count is the number of floats that will be converted, while offset would discard the first offset number of bytes from the buffer before the conversion.

In the example above, repeated calls to either of the functions returns a new ndarray. You can save RAM by supplying the out keyword argument with a pre-defined ndarray of sufficient size, in which case the results will be inserted into the ndarray. If the dtype of out is not float, a TypeError exception will be raised.

# code to be run in micropython

from ulab import numpy as np
from ulab import utils

a = np.array([1, 2], dtype=np.float)
b = bytearray([1, 0, 1, 0, 0, 1, 0, 1])
print('b: ', b)
utils.from_uint32_buffer(b, out=a)
print('a: ', a)

b:  bytearray(b'x01x00x01x00x00x01x00x01')
a:  array([65537.0, 16777472.0], dtype=float64)

Finally, since there is no guarantee that the endianness of a particular peripheral device supplying the buffer is the same as that of the microcontroller, from_(u)intbuffer allows a conversion via the byteswap keyword argument.

# code to be run in micropython

from ulab import numpy as np
from ulab import utils

a = bytearray([1, 0, 0, 0, 0, 0, 0, 1])
print('a: ', a)
print('buffer without byteswapping: ', utils.from_uint32_buffer(a))
print('buffer with byteswapping: ', utils.from_uint32_buffer(a, byteswap=True))

a:  bytearray(b'x01x00x00x00x00x00x00x01')
buffer without byteswapping:  array([1.0, 16777216.0], dtype=float64)
buffer with byteswapping:  array([16777216.0, 1.0], dtype=float64)

from_int16_buffer, from_uint16_buffer

These two functions are identical to utils.from_int32_buffer, and utils.from_uint32_buffer, with the exception that they convert 16-bit integers to floating point ndarrays.

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, also known as the power spectrum. 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. This means that, if the firmware was compiled with complex support, the input can also be a complex array.

# code to be run in micropython

from ulab import numpy as np
from ulab import utils as utils

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

a = utils.spectrogram(y)

print('original vector:\n', y)
print('\nspectrum:\n', a)

original vector:
 array([0.0, 0.009775015390171337, 0.01954909674625918, ..., -0.5275140569487312, -0.5357931822978732, -0.5440211108893697], dtype=float64)

spectrum:
 array([187.8635087634579, 315.3112063607119, 347.8814873399374, ..., 84.45888934298905, 347.8814873399374, 315.3112063607118], dtype=float64)

As such, spectrogram is really just a shorthand for np.sqrt(a*a + b*b), however, it saves significant amounts of RAM: the expression a*a + b*b has to allocate memory for a*a, b*b, and finally, their sum. In contrast, spectrogram calculates the spectrum internally, and stores it in the memory segment that was reserved for the real part of the Fourier transform.

# code to be run in micropython

from ulab import numpy as np
from ulab import utils as utils

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

a, b = np.fft.fft(y)

print('\nspectrum calculated the hard way:\n', np.sqrt(a*a + b*b))

a = utils.spectrogram(y)

print('\nspectrum calculated the lazy way:\n', a)

spectrum calculated the hard way:
 array([187.8635087634579, 315.3112063607119, 347.8814873399374, ..., 84.45888934298905, 347.8814873399374, 315.3112063607118], dtype=float64)

spectrum calculated the lazy way:
 array([187.8635087634579, 315.3112063607119, 347.8814873399374, ..., 84.45888934298905, 347.8814873399374, 315.3112063607118], dtype=float64)

Tricks

This section of the book discusses a couple of tricks that can be exploited to either speed up computations, or save on RAM. However, there is probably no silver bullet, and you have to evaluate your code in terms of execution speed (if the execution is time critical), or RAM used. You should also keep in mind that, if a particular code snippet is optimised on some hardware, there is no guarantee that on another piece of hardware, you will get similar improvements. Hardware implementations are vastly different. Some microcontrollers do not even have an FPU, so you should not be surprised that you get significantly different benchmarks. Just to underline this statement, you can study the collection of benchmarks.

Use an ndarray, if you can

Many functions in ulab are implemented in a universal fashion, meaning that both generic micropython iterables, and ndarrays can be passed as an argument. E.g., both

from ulab import numpy as np

np.sum([1, 2, 3, 4, 5])

and

from ulab import numpy as np

a = np.array([1, 2, 3, 4, 5])
np.sum(a)

will return the micropython variable 15 as the result. Still, np.sum(a) is evaluated significantly faster, because in np.sum([1, 2, 3, 4, 5]), the interpreter has to fetch 5 micropython variables, convert them to float, and sum the values, while the C type of a is known, thus the interpreter can invoke a single for loop for the evaluation of the sum. In the for loop, there are no function calls, the iteration simply walks through the pointer holding the values of a, and adds the values to an accumulator. If the array a is already available, then you can gain a factor of 3 in speed by calling sum on the array, instead of using the list. Compared to the python implementation of the same functionality, the speed-up is around 40 (again, this might depend on the hardware).

On the other hand, if the array is not available, then there is not much point in converting the list to an ndarray and passing that to the function. In fact, you should expect a slow-down: the constructor has to iterate over the list elements, and has to convert them to a numerical type. On top of that, it also has to reserve RAM for the ndarray.

Use a reasonable dtype

Just as in numpy, the default dtype is float. But this does not mean that that is the most suitable one in all scenarios. If data are streamed from an 8-bit ADC, and you only want to know the maximum, or the sum, then it is quite reasonable to use uint8 for the dtype. Storing the same data in float array would cost 4 or 8 times as much RAM, with absolutely no gain. Do not rely on the default value of the constructor’s keyword argument, and choose one that fits!

Beware the axis!

Whenever ulab iterates over multi-dimensional arrays, the outermost loop is the first axis, then the second axis, and so on. E.g., when the sum of

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

is being calculated, first the data pointer walks along [1, 2, 3, 4] (innermost loop, last axis), then is moved back to the position, where 5 is stored (this is the nesting loop), and traverses [5, 6, 7, 8], and so on. Moving the pointer back to 5 is more expensive, than moving it along an axis, because the position of 5 has to be calculated, whereas moving from 5 to 6 is simply an addition to the address. Thus, while the matrix

b = array([[1, 5, 9],
           [2, 6, 10],
           [3, 7, 11],
           [4, 8, 12]], dtype=uint8)

holds the same data as a, the summation over the entries in b is slower, because the pointer has to be re-wound three times, as opposed to twice in a. For small matrices the savings are not significant, but you would definitely notice the difference, if you had

a = array(range(2000)).reshape((2, 1000))
b = array(range(2000)).reshape((1000, 2))

The moral is that, in order to improve on the execution speed, whenever possible, you should try to make the last axis the longest. As a side note, numpy can re-arrange its loops, and puts the longest axis in the innermost loop. This is why the longest axis is sometimes referred to as the fast axis. In ulab, the order of the axes is fixed.

Reduce the number of artifacts

Before showing a real-life example, let us suppose that we want to interpolate uniformly sampled data, and the absolute magnitude is not really important, we only care about the ratios between neighbouring value. One way of achieving this is calling the interp functions. However, we could just as well work with slices.

# code to be run in CPython

a = array([0, 10, 2, 20, 4], dtype=np.uint8)
b = np.zeros(9, dtype=np.uint8)

b[::2] = 2 * a
b[1::2] = a[:-1] + a[1:]

b //= 2
b

array([ 0,  5, 10,  6,  2, 11, 20, 12,  4], dtype=uint8)

b now has values from a at every even position, and interpolates the values on every odd position. If only the relative magnitudes are important, then we can even save the division by 2, and we end up with

# code to be run in CPython

a = array([0, 10, 2, 20, 4], dtype=np.uint8)
b = np.zeros(9, dtype=np.uint8)

b[::2] = 2 * a
b[1::2] = a[:-1] + a[1:]

b

array([ 0, 10, 20, 12,  4, 22, 40, 24,  8], dtype=uint8)

Importantly, we managed to keep the results in the smaller dtype, uint8. Now, while the two assignments above are terse and pythonic, the code is not the most efficient: the right hand sides are compound statements, generating intermediate results. To store them, RAM has to be allocated. This takes time, and leads to memory fragmentation. Better is to write out the assignments in 4 instructions:

# code to be run in CPython

b = np.zeros(9, dtype=np.uint8)

b[::2] = a
b[::2] += a
b[1::2] = a[:-1]
b[1::2] += a[1:]

b

array([ 0, 10, 20, 12,  4, 22, 40, 24,  8], dtype=uint8)

The results are the same, but no extra RAM is allocated, except for the views a[:-1], and a[1:], but those had to be created even in the origin implementation.

Upscaling images

And now the example: there are low-resolution thermal cameras out there. Low resolution might mean 8 by 8 pixels. Such a small number of pixels is just not reasonable to plot, no matter how small the display is. If you want to make the camera image a bit more pleasing, you can upscale (stretch) it in both dimensions. This can be done exactly as we up-scaled the linear array:

# code to be run in CPython

b = np.zeros((15, 15), dtype=np.uint8)

b[1::2,::2] = a[:-1,:]
b[1::2,::2] += a[1:, :]
b[1::2,::2] //= 2
b[::,1::2] = a[::,:-1:2]
b[::,1::2] += a[::,2::2]
b[::,1::2] //= 2

Up-scaling by larger numbers can be done in a similar fashion, you simply have more assignments.

There are cases, when one cannot do away with the intermediate results. Two prominent cases are the where function, and indexing by means of a Boolean array. E.g., in

# code to be run in CPython

a = array([1, 2, 3, 4, 5])
b = a[a < 4]
b

array([1, 2, 3])

the expression a < 4 produces the Boolean array,

# code to be run in CPython

a < 4

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

If you repeatedly have such conditions in a loop, you might have to peridically call the garbage collector to remove the Boolean arrays that are used only once.

# code to be run in CPython

Programming ulab

Earlier we have seen, how ulab’s functions and methods can be accessed in micropython. This last section of the book explains, how these functions are implemented. By the end of this chapter, not only would you be able to extend ulab, and write your own numpy-compatible functions, but through a deeper understanding of the inner workings of the functions, you would also be able to see what the trade-offs are at the python level.

Code organisation

As mentioned earlier, the python functions are organised into sub-modules at the C level. The C sub-modules can be found in ./ulab/code/.

The ndarray object

General comments

ndarrays are efficient containers of numerical data of the same type (i.e., signed/unsigned chars, signed/unsigned integers or mp_float_ts, which, depending on the platform, are either C floats, or C doubles). Beyond storing the actual data in the void pointer *array, the type definition has eight additional members (on top of the base type). Namely, the dtype, which tells us, how the bytes are to be interpreted. Moreover, the itemsize, which stores the size of a single entry in the array, boolean, an unsigned integer, which determines, whether the arrays is to be treated as a set of Booleans, or as numerical data, ndim, the number of dimensions (uint8_t), len, the length of the array (the number of entries), the shape (*size_t), the strides (*int32_t). The length is simply the product of the numbers in shape.

The type definition is as follows:

typedef struct _ndarray_obj_t {
    mp_obj_base_t base;
    uint8_t dtype;
    uint8_t itemsize;
    uint8_t boolean;
    uint8_t ndim;
    size_t len;
    size_t shape[ULAB_MAX_DIMS];
    int32_t strides[ULAB_MAX_DIMS];
    void *array;
} ndarray_obj_t;

Memory layout

The values of an ndarray are stored in a contiguous segment in the RAM. The ndarray can be dense, meaning that all numbers in the linear memory segment belong to a linar combination of coordinates, and it can also be sparse, i.e., some elements of the linear storage space will be skipped, when the elements of the tensor are traversed.

In the RAM, the position of the item \(M(n_1, n_2, ..., n_{k-1}, n_k)\) in a dense tensor of rank \(k\) is given by the linear combination

:raw-latex:`\begin{equation} P(n_1, n_2, ..., n_{k-1}, n_k) = n_1 s_1 + n_2 s_2 + ... + n_{k-1}s_{k-1} + n_ks_k = \sum_{i=1}^{k}n_is_i \end{equation}` where \(s_i\) are the strides of the tensor, defined as

:raw-latex:`\begin{equation} s_i = \prod_{j=i+1}^k l_j \end{equation}`

where \(l_j\) is length of the tensor along the \(j\)th axis. When the tensor is sparse (e.g., when the tensor is sliced), the strides along a particular axis will be multiplied by a non-zero integer. If this integer is different to \(\pm 1\), the linear combination above cannot access all elements in the RAM, i.e., some numbers will be skipped. Note that \(|s_1| > |s_2| > ... > |s_{k-1}| > |s_k|\), even if the tensor is sparse. The statement is trivial for dense tensors, and it follows from the definition of \(s_i\). For sparse tensors, a slice cannot have a step larger than the shape along that axis. But for dense tensors, \(s_i/s_{i+1} = l_i\).

When creating a view, we simply re-calculate the strides, and re-set the *array pointer.

Iterating over elements of a tensor

The shape and strides members of the array tell us how we have to move our pointer, when we want to read out the numbers. For technical reasons that will become clear later, the numbers in shape and in strides are aligned to the right, and begin on the right hand side, i.e., if the number of possible dimensions is ULAB_MAX_DIMS, then shape[ULAB_MAX_DIMS-1] is the length of the last axis, shape[ULAB_MAX_DIMS-2] is the length of the last but one axis, and so on. If the number of actual dimensions, ndim < ULAB_MAX_DIMS, the first ULAB_MAX_DIMS - ndim entries in shape and strides will be equal to zero, but they could, in fact, be assigned any value, because these will never be accessed in an operation.

With this definition of the strides, the linear combination in \(P(n_1, n_2, ..., n_{k-1}, n_k)\) is a one-to-one mapping from the space of tensor coordinates, \((n_1, n_2, ..., n_{k-1}, n_k)\), and the coordinate in the linear array, \(n_1s_1 + n_2s_2 + ... + n_{k-1}s_{k-1} + n_ks_k\), i.e., no two distinct sets of coordinates will result in the same position in the linear array.

Since the strides are given in terms of bytes, when we iterate over an array, the void data pointer is usually cast to uint8_t, and the values are converted using the proper data type stored in ndarray->dtype. However, there might be cases, when it makes perfect sense to cast *array to a different type, in which case the strides have to be re-scaled by the value of ndarray->itemsize.

Iterating using the unwrapped loops

The following macro definition is taken from vector.h, and demonstrates, how we can iterate over a single array in four dimensions.

#define ITERATE_VECTOR(type, array, source, sarray) do {
    size_t i=0;
    do {
        size_t j = 0;
        do {
            size_t k = 0;
            do {
                size_t l = 0;
                do {
                    *(array)++ = f(*((type *)(sarray)));
                    (sarray) += (source)->strides[ULAB_MAX_DIMS - 1];
                    l++;
                } while(l < (source)->shape[ULAB_MAX_DIMS-1]);
                (sarray) -= (source)->strides[ULAB_MAX_DIMS - 1] * (source)->shape[ULAB_MAX_DIMS-1];
                (sarray) += (source)->strides[ULAB_MAX_DIMS - 2];
                k++;
            } while(k < (source)->shape[ULAB_MAX_DIMS-2]);
            (sarray) -= (source)->strides[ULAB_MAX_DIMS - 2] * (source)->shape[ULAB_MAX_DIMS-2];
            (sarray) += (source)->strides[ULAB_MAX_DIMS - 3];
            j++;
        } while(j < (source)->shape[ULAB_MAX_DIMS-3]);
        (sarray) -= (source)->strides[ULAB_MAX_DIMS - 3] * (source)->shape[ULAB_MAX_DIMS-3];
        (sarray) += (source)->strides[ULAB_MAX_DIMS - 4];
        i++;
    } while(i < (source)->shape[ULAB_MAX_DIMS-4]);
} while(0)

We start with the innermost loop, the one recursing l. array is already of type mp_float_t, while the source array, sarray, has been cast to uint8_t in the calling function. The numbers contained in sarray have to be read out in the proper type dictated by ndarray->dtype. This is what happens in the statement *((type *)(sarray)), and this number is then fed into the function f. Vectorised mathematical functions produce dense arrays, and for this reason, we can simply advance the array pointer.

The advancing of the sarray pointer is a bit more involving: first, in the innermost loop, we simply move forward by the amount given by the last stride, which is (source)->strides[ULAB_MAX_DIMS - 1], because the shape and the strides are aligned to the right. We move the pointer as many times as given by (source)->shape[ULAB_MAX_DIMS-1], which is the length of the very last axis. Hence the the structure of the loop

size_t l = 0;
do {
    ...
    l++;
} while(l < (source)->shape[ULAB_MAX_DIMS-1]);

Once we have exhausted the last axis, we have to re-wind the pointer, and advance it by an amount given by the last but one stride. Keep in mind that in the the innermost loop we moved our pointer (source)->shape[ULAB_MAX_DIMS-1] times by (source)->strides[ULAB_MAX_DIMS - 1], i.e., we re-wind it by moving it backwards by (source)->strides[ULAB_MAX_DIMS - 1] * (source)->shape[ULAB_MAX_DIMS-1]. In the next step, we move forward by (source)->strides[ULAB_MAX_DIMS - 2], which is the last but one stride.

(sarray) -= (source)->strides[ULAB_MAX_DIMS - 1] * (source)->shape[ULAB_MAX_DIMS-1];
(sarray) += (source)->strides[ULAB_MAX_DIMS - 2];

This pattern must be repeated for each axis of the array, and this is how we arrive at the four nested loops listed above.

Re-winding arrays by means of a function

In addition to un-wrapping the iteration loops by means of macros, there is another way of traversing all elements of a tensor: we note that, since \(|s_1| > |s_2| > ... > |s_{k-1}| > |s_k|\), \(P(n1, n2, ..., n_{k-1}, n_k)\) changes most slowly in the last coordinate. Hence, if we start from the very beginning, (\(n_i = 0\) for all \(i\)), and walk along the linear RAM segment, we increment the value of \(n_k\) as long as \(n_k < l_k\). Once \(n_k = l_k\), we have to reset \(n_k\) to 0, and increment \(n_{k-1}\) by one. After each such round, \(n_{k-1}\) will be incremented by one, as long as \(n_{k-1} < l_{k-1}\). Once \(n_{k-1} = l_{k-1}\), we reset both \(n_k\), and \(n_{k-1}\) to 0, and increment \(n_{k-2}\) by one.

Rewinding the arrays in this way is implemented in the function ndarray_rewind_array in ndarray.c.

void ndarray_rewind_array(uint8_t ndim, uint8_t *array, size_t *shape, int32_t *strides, size_t *coords) {
    // resets the data pointer of a single array, whenever an axis is full
    // since we always iterate over the very last axis, we have to keep track of
    // the last ndim-2 axes only
    array -= shape[ULAB_MAX_DIMS - 1] * strides[ULAB_MAX_DIMS - 1];
    array += strides[ULAB_MAX_DIMS - 2];
    for(uint8_t i=1; i < ndim-1; i++) {
        coords[ULAB_MAX_DIMS - 1 - i] += 1;
        if(coords[ULAB_MAX_DIMS - 1 - i] == shape[ULAB_MAX_DIMS - 1 - i]) { // we are at a dimension boundary
            array -= shape[ULAB_MAX_DIMS - 1 - i] * strides[ULAB_MAX_DIMS - 1 - i];
            array += strides[ULAB_MAX_DIMS - 2 - i];
            coords[ULAB_MAX_DIMS - 1 - i] = 0;
            coords[ULAB_MAX_DIMS - 2 - i] += 1;
        } else { // coordinates can change only, if the last coordinate changes
            return;
        }
    }
}

and the function would be called as in the snippet below. Note that the innermost loop is factored out, so that we can save the if(...) statement for the last axis.

size_t *coords = ndarray_new_coords(results->ndim);
for(size_t i=0; i < results->len/results->shape[ULAB_MAX_DIMS -1]; i++) {
    size_t l = 0;
    do {
        ...
        l++;
    } while(l < results->shape[ULAB_MAX_DIMS - 1]);
    ndarray_rewind_array(results->ndim, array, results->shape, strides, coords);
} while(0)

The advantage of this method is that the implementation is independent of the number of dimensions: the iteration requires more or less the same flash space for 2 dimensions as for 22. However, the price we have to pay for this convenience is the extra function call.

Iterating over two ndarrays simultaneously: broadcasting

Whenever we invoke a binary operator, call a function with two arguments of ndarray type, or assign something to an ndarray, we have to iterate over two views at the same time. The task is trivial, if the two ndarrays in question have the same shape (but not necessarily the same set of strides), because in this case, we can still iterate in the same loop. All that happens is that we move two data pointers in sync.

The problem becomes a bit more involving, when the shapes of the two ndarrays are not identical. For such cases, numpy defines so-called broadcasting, which boils down to two rules.

1. The shapes in the tensor with lower rank has to be prepended with axes of size 1 till the two ranks become equal.

2. Along all axes the two tensors should have the same size, or one of the sizes must be 1.

If, after applying the first rule the second is not satisfied, the two ndarrays cannot be broadcast together.

Now, let us suppose that we have two compatible ndarrays, i.e., after applying the first rule, the second is satisfied. How do we iterate over the elements in the tensors?

We should recall, what exactly we do, when iterating over a single array: normally, we move the data pointer by the last stride, except, when we arrive at a dimension boundary (when the last axis is exhausted). At that point, we move the pointer by an amount dictated by the strides. And this is the key: dictated by the strides. Now, if we have two arrays that are originally not compatible, we define new strides for them, and use these in the iteration. With that, we are back to the case, where we had two compatible arrays.

Now, let us look at the second broadcasting rule: if the two arrays have the same size, we take both ndarrays’ strides along that axis. If, on the other hand, one of the ndarrays is of length 1 along one of its axes, we set the corresponding strides to 0. This will ensure that that data pointer is not moved, when we iterate over both ndarrays at the same time.

Thus, in order to implement broadcasting, we first have to check, whether the two above-mentioned rules can be satisfied, and if so, we have to find the two new sets strides.

The ndarray_can_broadcast function from ndarray.c takes two ndarrays, and returns true, if the two arrays can be broadcast together. At the same time, it also calculates new strides for the two arrays, so that they can be iterated over at the same time.

bool ndarray_can_broadcast(ndarray_obj_t *lhs, ndarray_obj_t *rhs, uint8_t *ndim, size_t *shape, int32_t *lstrides, int32_t *rstrides) {
    // returns True or False, depending on, whether the two arrays can be broadcast together
    // numpy's broadcasting rules are as follows:
    //
    // 1. the two shapes are either equal
    // 2. one of the shapes is 1
    memset(lstrides, 0, sizeof(size_t)*ULAB_MAX_DIMS);
    memset(rstrides, 0, sizeof(size_t)*ULAB_MAX_DIMS);
    lstrides[ULAB_MAX_DIMS - 1] = lhs->strides[ULAB_MAX_DIMS - 1];
    rstrides[ULAB_MAX_DIMS - 1] = rhs->strides[ULAB_MAX_DIMS - 1];
    for(uint8_t i=ULAB_MAX_DIMS; i > 0; i--) {
        if((lhs->shape[i-1] == rhs->shape[i-1]) || (lhs->shape[i-1] == 0) || (lhs->shape[i-1] == 1) ||
        (rhs->shape[i-1] == 0) || (rhs->shape[i-1] == 1)) {
            shape[i-1] = MAX(lhs->shape[i-1], rhs->shape[i-1]);
            if(shape[i-1] > 0) (*ndim)++;
            if(lhs->shape[i-1] < 2) {
                lstrides[i-1] = 0;
            } else {
                lstrides[i-1] = lhs->strides[i-1];
            }
            if(rhs->shape[i-1] < 2) {
                rstrides[i-1] = 0;
            } else {
                rstrides[i-1] = rhs->strides[i-1];
            }
        } else {
            return false;
        }
    }
    return true;
}

A good example of how the function would be called can be found in vector.c, in the vector_arctan2 function:

mp_obj_t vector_arctan2(mp_obj_t y, mp_obj_t x) {
    ...
    uint8_t ndim = 0;
    size_t *shape = m_new(size_t, ULAB_MAX_DIMS);
    int32_t *xstrides = m_new(int32_t, ULAB_MAX_DIMS);
    int32_t *ystrides = m_new(int32_t, ULAB_MAX_DIMS);
    if(!ndarray_can_broadcast(ndarray_x, ndarray_y, &ndim, shape, xstrides, ystrides)) {
        mp_raise_ValueError(translate("operands could not be broadcast together"));
        m_del(size_t, shape, ULAB_MAX_DIMS);
        m_del(int32_t, xstrides, ULAB_MAX_DIMS);
        m_del(int32_t, ystrides, ULAB_MAX_DIMS);
    }

    uint8_t *xarray = (uint8_t *)ndarray_x->array;
    uint8_t *yarray = (uint8_t *)ndarray_y->array;

    ndarray_obj_t *results = ndarray_new_dense_ndarray(ndim, shape, NDARRAY_FLOAT);
    mp_float_t *rarray = (mp_float_t *)results->array;
    ...

After the new strides have been calculated, the iteration loop is identical to what we discussed in the previous section.

Contracting an ndarray

There are many operations that reduce the number of dimensions of an ndarray by 1, i.e., that remove an axis from the tensor. The drill is the same as before, with the exception that first we have to remove the strides and shape that corresponds to the axis along which we intend to contract. The numerical_reduce_axes function from numerical.c does that.

static void numerical_reduce_axes(ndarray_obj_t *ndarray, int8_t axis, size_t *shape, int32_t *strides) {
    // removes the values corresponding to a single axis from the shape and strides array
    uint8_t index = ULAB_MAX_DIMS - ndarray->ndim + axis;
    if((ndarray->ndim == 1) && (axis == 0)) {
        index = 0;
        shape[ULAB_MAX_DIMS - 1] = 0;
        return;
    }
    for(uint8_t i = ULAB_MAX_DIMS - 1; i > 0; i--) {
        if(i > index) {
            shape[i] = ndarray->shape[i];
            strides[i] = ndarray->strides[i];
        } else {
            shape[i] = ndarray->shape[i-1];
            strides[i] = ndarray->strides[i-1];
        }
    }
}

Once the reduced strides and shape are known, we place the axis in question in the innermost loop, and wrap it with the loops, whose coordinates are in the strides, and shape arrays. The RUN_STD macro from numerical.h is a good example. The macro is expanded in the numerical_sum_mean_std_ndarray function.

static mp_obj_t numerical_sum_mean_std_ndarray(ndarray_obj_t *ndarray, mp_obj_t axis, uint8_t optype, size_t ddof) {
    uint8_t *array = (uint8_t *)ndarray->array;
    size_t *shape = m_new(size_t, ULAB_MAX_DIMS);
    memset(shape, 0, sizeof(size_t)*ULAB_MAX_DIMS);
    int32_t *strides = m_new(int32_t, ULAB_MAX_DIMS);
    memset(strides, 0, sizeof(uint32_t)*ULAB_MAX_DIMS);

    int8_t ax = mp_obj_get_int(axis);
    if(ax < 0) ax += ndarray->ndim;
    if((ax < 0) || (ax > ndarray->ndim - 1)) {
        mp_raise_ValueError(translate("index out of range"));
    }
    numerical_reduce_axes(ndarray, ax, shape, strides);
    uint8_t index = ULAB_MAX_DIMS - ndarray->ndim + ax;
    ndarray_obj_t *results = NULL;
    uint8_t *rarray = NULL;
    ...

Here is the macro for the three-dimensional case:

#define RUN_STD(ndarray, type, array, results, r, shape, strides, index, div) do {
    size_t k = 0;
    do {
        size_t l = 0;
        do {
            RUN_STD1((ndarray), type, (array), (results), (r), (index), (div));
            (array) -= (ndarray)->strides[(index)] * (ndarray)->shape[(index)];
            (array) += (strides)[ULAB_MAX_DIMS - 1];
            l++;
        } while(l < (shape)[ULAB_MAX_DIMS - 1]);
        (array) -= (strides)[ULAB_MAX_DIMS - 2] * (shape)[ULAB_MAX_DIMS-2];
        (array) += (strides)[ULAB_MAX_DIMS - 3];
        k++;
    } while(k < (shape)[ULAB_MAX_DIMS - 2]);
} while(0)

In RUN_STD, we simply move our pointers; the calculation itself happens in the RUN_STD1 macro below. (Note that this is the implementation of the numerically stable Welford algorithm.)

#define RUN_STD1(ndarray, type, array, results, r, index, div)
({
    mp_float_t M, m, S = 0.0, s = 0.0;
    M = m = *(mp_float_t *)((type *)(array));
    for(size_t i=1; i < (ndarray)->shape[(index)]; i++) {
        (array) += (ndarray)->strides[(index)];
        mp_float_t value = *(mp_float_t *)((type *)(array));
        m = M + (value - M) / (mp_float_t)i;
        s = S + (value - M) * (value - m);
        M = m;
        S = s;
    }
    (array) += (ndarray)->strides[(index)];
    *(r)++ = MICROPY_FLOAT_C_FUN(sqrt)((ndarray)->shape[(index)] * s / (div));
})

Upcasting

When in an operation the dtypes of two arrays are different, the result’s dtype will be decided by the following upcasting rules:

1. Operations with two ndarrays of the same dtype preserve their dtype, even when the results overflow.

2. if either of the operands is a float, the result automatically becomes a float

3. otherwise

   • uint8 + int8 => int16,

   • uint8 + int16 => int16

   • uint8 + uint16 => uint16

   • int8 + int16 => int16

   • int8 + uint16 => uint16 (in numpy, the result is a int32)

   • uint16 + int16 => float (in numpy, the result is a int32)

4. When one operand of a binary operation is a generic scalar micropython variable, i.e., mp_obj_int, or mp_obj_float, it will be converted to a linear array of length 1, and with the smallest dtype that can accommodate the variable in question. After that the broadcasting rules apply, as described in the section Iterating over two ndarrays simultaneously: broadcasting

Upcasting is resolved in place, wherever it is required. Notable examples can be found in ndarray_operators.c

Slicing and indexing

An ndarray can be indexed with three types of objects: integer scalars, slices, and another ndarray, whose elements are either integer scalars, or Booleans. Since slice and integer indices can be thought of as modifications of the strides, these indices return a view of the ndarray. This statement does not hold for ndarray indices, and therefore, the return a copy of the array.

Extending ulab

The user module is disabled by default, as can be seen from the last couple of lines of ulab.h

// user-defined module
#ifndef ULAB_USER_MODULE
#define ULAB_USER_MODULE                (0)
#endif

The module contains a very simple function, user_dummy, and this function is bound to the module itself. In other words, even if the module is enabled, one has to import:

import ulab
from ulab import user

user.dummy_function(2.5)

which should just return 5.0. Even if numpy-compatibility is required (i.e., if most functions are bound at the top level to ulab directly), having to import the module has a great advantage. Namely, only the user.h and user.c files have to be modified, thus it should be relatively straightforward to update your local copy from github.

Now, let us see, how we can add a more meaningful function.

Creating a new ndarray

In the General comments sections we have seen the type definition of an ndarray. This structure can be generated by means of a couple of functions listed in ndarray.c.

ndarray_new_ndarray

The ndarray_new_ndarray functions is called by all other array-generating functions. It takes the number of dimensions, ndim, a uint8_t, the shape, a pointer to size_t, the strides, a pointer to int32_t, and dtype, another uint8_t as its arguments, and returns a new array with all entries initialised to 0.

Assuming that ULAB_MAX_DIMS > 2, a new dense array of dimension 3, of shape (3, 4, 5), of strides (1000, 200, 10), and dtype uint16_t can be generated by the following instructions

size_t *shape = m_new(size_t, ULAB_MAX_DIMS);
shape[ULAB_MAX_DIMS - 1] = 5;
shape[ULAB_MAX_DIMS - 2] = 4;
shape[ULAB_MAX_DIMS - 3] = 3;

int32_t *strides = m_new(int32_t, ULAB_MAX_DIMS);
strides[ULAB_MAX_DIMS - 1] = 10;
strides[ULAB_MAX_DIMS - 2] = 200;
strides[ULAB_MAX_DIMS - 3] = 1000;

ndarray_obj_t *new_ndarray = ndarray_new_ndarray(3, shape, strides, NDARRAY_UINT16);

ndarray_new_dense_ndarray

The functions simply calculates the strides from the shape, and calls ndarray_new_ndarray. Assuming that ULAB_MAX_DIMS > 2, a new dense array of dimension 3, of shape (3, 4, 5), and dtype mp_float_t can be generated by the following instructions

size_t *shape = m_new(size_t, ULAB_MAX_DIMS);
shape[ULAB_MAX_DIMS - 1] = 5;
shape[ULAB_MAX_DIMS - 2] = 4;
shape[ULAB_MAX_DIMS - 3] = 3;

ndarray_obj_t *new_ndarray = ndarray_new_dense_ndarray(3, shape, NDARRAY_FLOAT);

ndarray_new_linear_array

Since the dimensions of a linear array are known (1), the ndarray_new_linear_array takes the length, a size_t, and the dtype, an uint8_t. Internally, ndarray_new_linear_array generates the shape array, and calls ndarray_new_dense_array with ndim = 1.

A linear array of length 100, and dtype uint8 could be created by the function call

ndarray_obj_t *new_ndarray = ndarray_new_linear_array(100, NDARRAY_UINT8)

ndarray_new_ndarray_from_tuple

This function takes a tuple, which should hold the lengths of the axes (in other words, the shape), and the dtype, and calls internally ndarray_new_dense_array. A new ndarray can be generated by calling

ndarray_obj_t *new_ndarray = ndarray_new_ndarray_from_tuple(shape, NDARRAY_FLOAT);

where shape is a tuple.

ndarray_new_view

This function crates a view, and takes the source, an ndarray, the number of dimensions, an uint8_t, the shape, a pointer to size_t, the strides, a pointer to int32_t, and the offset, an int32_t as arguments. The offset is the number of bytes by which the void array pointer is shifted. E.g., the python statement

a = np.array([0, 1, 2, 3, 4, 5], dtype=uint8)
b = a[1::2]

produces the array

array([1, 3, 5], dtype=uint8)

which holds its data at position x0 + 1, if a’s pointer is at x0. In this particular case, the offset is 1.

The array b from the example above could be generated as

size_t *shape = m_new(size_t, ULAB_MAX_DIMS);
shape[ULAB_MAX_DIMS - 1] = 3;

int32_t *strides = m_new(int32_t, ULAB_MAX_DIMS);
strides[ULAB_MAX_DIMS - 1] = 2;

int32_t offset = 1;
uint8_t ndim = 1;

ndarray_obj_t *new_ndarray = ndarray_new_view(ndarray_a, ndim, shape, strides, offset);

ndarray_copy_array

The ndarray_copy_array function can be used for copying the contents of an array. Note that the target array has to be created beforehand. E.g., a one-to-one copy can be gotten by

ndarray_obj_t *new_ndarray = ndarray_new_ndarray(source->ndim, source->shape, source->strides, source->dtype);
ndarray_copy_array(source, new_ndarray);

Note that the function cannot be used for forcing type conversion, i.e., the input and output types must be identical, because the function simply calls the memcpy function. On the other hand, the input and output strides do not necessarily have to be equal.

ndarray_copy_view

The ndarray_obj_t *new_ndarray = ... instruction can be saved by calling the ndarray_copy_view function with the single source argument.

Accessing data in the ndarray

Having seen, how arrays can be generated and copied, it is time to look at how the data in an ndarray can be accessed and modified.

For starters, let us suppose that the object in question comes from the user (i.e., via the micropython interface), First, we have to acquire a pointer to the ndarray by calling

ndarray_obj_t *ndarray = MP_OBJ_TO_PTR(object_in);

If it is not clear, whether the object is an ndarray (e.g., if we want to write a function that can take ndarrays, and other iterables as its argument), we find this out by evaluating

mp_obj_is_type(object_in, &ulab_ndarray_type)

which should return true. Once the pointer is at our disposal, we can get a pointer to the underlying numerical array as discussed earlier, i.e.,

uint8_t *array = (uint8_t *)ndarray->array;

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

ndarray->dtype

should be equal to B, b, H, h, or f. The size of a single item is stored in the itemsize member. This number should be equal to 1, if the dtype is B, or b, 2, if the dtype is H, or h, 4, if the dtype is f, and 8 for d.

Boilerplate

In the next section, we will construct a function that generates the element-wise square of a dense array, otherwise, raises a TypeError exception. Dense arrays can easily be iterated over, since we do not have to care about the shape and the strides. If the array is sparse, the section Iterating over elements of a tensor should contain hints as to how the iteration can be implemented.

The function is listed under user.c. The user module is bound to ulab in ulab.c in the lines

#if ULAB_USER_MODULE
    { MP_ROM_QSTR(MP_QSTR_user), MP_ROM_PTR(&ulab_user_module) },
#endif

which assumes that at the very end of ulab.h the

// user-defined module
#ifndef ULAB_USER_MODULE
#define ULAB_USER_MODULE                (1)
#endif

constant has been set to 1. After compilation, you can call a particular user function in python by importing the module first, i.e.,

from ulab import numpy as np
from ulab import user

user.some_function(...)

This separation of user-defined functions from the rest of the code ensures that the integrity of the main module and all its functions are always preserved. Even in case of a catastrophic failure, you can exclude the user module, and start over.

And now the function:

static mp_obj_t user_square(mp_obj_t arg) {
    // the function takes a single dense ndarray, and calculates the
    // element-wise square of its entries

    // raise a TypeError exception, if the input is not an ndarray
    if(!mp_obj_is_type(arg, &ulab_ndarray_type)) {
        mp_raise_TypeError(translate("input must be an ndarray"));
    }
    ndarray_obj_t *ndarray = MP_OBJ_TO_PTR(arg);

    // make sure that the input is a dense array
    if(!ndarray_is_dense(ndarray)) {
        mp_raise_TypeError(translate("input must be a dense ndarray"));
    }

    // if the input is a dense array, create `results` with the same number of
    // dimensions, shape, and dtype
    ndarray_obj_t *results = ndarray_new_dense_ndarray(ndarray->ndim, ndarray->shape, ndarray->dtype);

    // since in a dense array the iteration over the elements is trivial, we
    // can cast the data arrays ndarray->array and results->array to the actual type
    if(ndarray->dtype == NDARRAY_UINT8) {
        uint8_t *array = (uint8_t *)ndarray->array;
        uint8_t *rarray = (uint8_t *)results->array;
        for(size_t i=0; i < ndarray->len; i++, array++) {
            *rarray++ = (*array) * (*array);
        }
    } else if(ndarray->dtype == NDARRAY_INT8) {
        int8_t *array = (int8_t *)ndarray->array;
        int8_t *rarray = (int8_t *)results->array;
        for(size_t i=0; i < ndarray->len; i++, array++) {
            *rarray++ = (*array) * (*array);
        }
    } else if(ndarray->dtype == NDARRAY_UINT16) {
        uint16_t *array = (uint16_t *)ndarray->array;
        uint16_t *rarray = (uint16_t *)results->array;
        for(size_t i=0; i < ndarray->len; i++, array++) {
            *rarray++ = (*array) * (*array);
        }
    } else if(ndarray->dtype == NDARRAY_INT16) {
        int16_t *array = (int16_t *)ndarray->array;
        int16_t *rarray = (int16_t *)results->array;
        for(size_t i=0; i < ndarray->len; i++, array++) {
            *rarray++ = (*array) * (*array);
        }
    } else { // if we end up here, the dtype is NDARRAY_FLOAT
        mp_float_t *array = (mp_float_t *)ndarray->array;
        mp_float_t *rarray = (mp_float_t *)results->array;
        for(size_t i=0; i < ndarray->len; i++, array++) {
            *rarray++ = (*array) * (*array);
        }
    }
    // at the end, return a micropython object
    return MP_OBJ_FROM_PTR(results);
}

To summarise, the steps for implementing a function are

1. If necessary, inspect the type of the input object, which is always a mp_obj_t object

2. If the input is an ndarray_obj_t, acquire a pointer to it by calling ndarray_obj_t *ndarray = MP_OBJ_TO_PTR(arg);

3. Create a new array, or modify the existing one; get a pointer to the data by calling uint8_t *array = (uint8_t *)ndarray->array;, or something equivalent

4. Once the new data have been calculated, return a micropython object by calling MP_OBJ_FROM_PTR(...).

The listing above contains the implementation of the function, but as such, it cannot be called from python: it still has to be bound to the name space. This we do by first defining a function object in

MP_DEFINE_CONST_FUN_OBJ_1(user_square_obj, user_square);

micropython defines a number of MP_DEFINE_CONST_FUN_OBJ_N macros in obj.h. N is always the number of arguments the function takes. We had a function definition static mp_obj_t user_square(mp_obj_t arg), i.e., we dealt with a single argument.

Finally, we have to bind this function object in the globals table of the user module:

STATIC const mp_rom_map_elem_t ulab_user_globals_table[] = {
    { MP_OBJ_NEW_QSTR(MP_QSTR___name__), MP_OBJ_NEW_QSTR(MP_QSTR_user) },
    { MP_OBJ_NEW_QSTR(MP_QSTR_square), (mp_obj_t)&user_square_obj },
};

Thus, the three steps required for the definition of a user-defined function are

1. The low-level implementation of the function itself

2. The definition of a function object by calling MP_DEFINE_CONST_FUN_OBJ_N()

3. Binding this function object to the namespace in the ulab_user_globals_table[]

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