Introduction¶
In the last chapter of the usermod documentation, I mentioned that I have another story, for another day. The day has come, so here is my story.
Enter ulab¶
ulab
is a numpy-like module for micropython
, meant to simplify
and speed up common mathematical operations on arrays. The primary goal
was to implement a small subset of numpy that might be useful in the
context of a microcontroller. This means low-level data processing of
linear (array) and two-dimensional (matrix) data.
Purpose¶
Of course, the first question that one has to answer is, why on Earth one would need a fast math library on a microcontroller. After all, it is not expected that heavy number crunching is going to take place on bare metal. It is not meant to. On a PC, the main reason for writing fast code is the sheer amount of data that one wants to process. On a microcontroller, the data volume is probably small, but it might lead to catastrophic system failure, if these data are not processed in time, because the microcontroller is supposed to interact with the outside world in a timely fashion. In fact, this latter objective was the initiator of this project: I needed the Fourier transform of a signal coming from the ADC of the pyboard, and all available options were simply too slow.
In addition to speed, another issue that one has to keep in mind when working with embedded systems is the amount of available RAM: I believe, everything here could be implemented in pure python with relatively little effort, but the price we would have to pay for that is not only speed, but RAM, too. python code, if is not frozen, and compiled into the firmware, has to be compiled at runtime, which is not exactly a cheap process. On top of that, if numbers are stored in a list or tuple, which would be the high-level container, then they occupy 8 bytes, no matter, whether they are all smaller than 100, or larger than one hundred million. This is obviously a waste of resources in an environment, where resources are scarce.
Finally, there is a reason for using micropython
in the first place.
Namely, that a microcontroller can be programmed in a very elegant, and
pythonic way. But if it is so, why should we not extend this idea to
other tasks and concepts that might come up in this context? If there
was no other reason than this elegance, I would find that convincing
enough.
Based on the above-mentioned considerations, all functions in ulab
are implemented in a way that
- conforms to
numpy
as much as possible - is so frugal with RAM as possible,
- and yet, fast. Much faster than pure python.
The main points of ulab
are
- compact, iterable and slicable containers of numerical data in 1, and
2 dimensions (arrays and matrices). These containers support all the
relevant unary and binary operators (e.g.,
len
, ==, +, *, etc.) - vectorised computations on micropython iterables and numerical
arrays/matrices (in
numpy
-speak, universal functions) - basic linear algebra routines (matrix inversion, multiplication, reshaping, transposition, determinant, and eigenvalues)
- polynomial fits to numerical data
- fast Fourier transforms
At the time of writing this manual (for version 0.33.2), the library adds approximately 30 kB of extra compiled code to the micropython (pyboard.v.11) firmware. However, if you are tight with flash space, you can easily shave off a couple of kB. See the section on customising ulab.
Resources and legal matters¶
The source code of the module can be found under https://github.com/v923z/micropython-ulab/tree/master/code. The source of this user manual is under https://github.com/v923z/micropython-ulab/tree/master/docs, while the technical details of the implementation are discussed at great length in https://github.com/v923z/micropython-ulab/tree/master/docs/ulab.ipynb. If you want an even thorougher explanation on why the various constructs of the implementation work, and work in that particular way, you can read more on the subject under https://micropython-usermod.readthedocs.io/en/latest/, where I demonstrate, what you have to do, if you want to make a C object behave in a pythonic way.
The MIT licence applies to all material.
Friendly request¶
If you use ulab
, and bump into a bug, or think that a particular
function is missing, or its behaviour does not conform to numpy
,
please, raise a ulab
issue on github,
so that the community can profit from your experiences.
Even better, if you find the project useful, and think that it could be
made better, faster, tighter, and shinier, please, consider
contributing, and issue a pull request with the implementation of your
improvements and new features. ulab
can only become successful, if
it offers what the community needs.
These last comments apply to the documentation, too. If, in your opinion, the documentation is obscure, misleading, or not detailed enough, please, let me know, so that we can fix it.
Differences between micropython-ulab and circuitpython-ulab¶
ulab
has originally been developed for micropython
, but has
since been integrated into a number of its flavours. Most of these
flavours are simply forks of micropython
itself, with some
additional functionality. One of the notable exceptions is
circuitpython
, which has slightly diverged at the core level, and
this has some minor consequences. Some of these concern the C
implementation details only, which all have been sorted out with the
generous and enthusiastic support of Jeff Epler from Adafruit
Industries.
There are, however, a couple of instances, where the usage in the two
environments is slightly different at the python level. These are how
the packges can be imported, and how the class properties can be
accessed. In both cases, the circuitpython
implementation results in
numpy
-conform code. numpy
-compatibility in micropython
will
be implemented as soon as micropython
itself has the required tools.
Till then we have to live with a workaround, which I will point out at
the relevant places.
Customising ulab
¶
ulab
implements a great number of functions, which are organised in
sub-modules. E.g., functions related to Fourier transforms are located
in the ulab.fft
sub-module, so you would import fft
as
import ulab
from ulab import fft
by which point you can get the FFT of your data by calling
fft.fft(...)
.
The idea of such grouping of functions and methods is to provide a means for granularity: It is quite possible that you do not need all functions in a particular application. If you want to save some flash space, you can easily exclude arbitrary sub-modules from the firmware. The ulab.h header file contains a pre-processor flag for each sub-module. The default setting is 1 for each of them. Setting them to 0 removes the module from the compiled firmware.
The first couple of lines of the file look like this
// vectorise (all functions) takes approx. 3 kB of flash space
#define ULAB_VECTORISE_MODULE (1)
// linalg adds around 6 kB
#define ULAB_LINALG_MODULE (1)
// poly is approx. 2.5 kB
#define ULAB_POLY_MODULE (1)
In order to simplify navigation in the header, each flag begins with
ULAB_
, and continues with the name of the sub-module. This name is
also the .c
file, where the sub-module is implemented. So, e.g., the
linear algebra routines can be found in linalg.c
, and the
corresponding compiler flag is ULAB_LINALG_MODULE
. Each section
displays a hint as to how much space you can save by un-setting the
flag.
At first, having to import everything in this way might appear to be
overly complicated, but there is a very good reason behind all this: you
can find out at the time of importing, whether a function or sub-module
is part of your ulab
firmware, or not. The alternative, namely, that
you do not have to import anything beyond ulab
, could prove
catastrophic: you would learn only at run time (at the moment of calling
the function in your code) that a particular function is not in the
firmware, and that is most probably too late.
Except for fft
, the standard sub-modules, vector
, linalg
,
numerical
, and poly
all numpy
-compatible. User-defined
functions that accept ndarray
s as their argument should be
implemented in the extra
sub-module, or its sub-modules. Hints as to
how to do that can be found in the section Extending
ulab.
Supported functions and methods¶
ulab
supports a number of array operators, which are listed here. I
tried to follow the specifications of the numpy
interface as closely
as possible, though, it was not always practical to implement verbatim
behaviour. The differences, if any, are in each case small (e.g., a
function cannot take all possible keyword arguments), and should not
hinder everyday use. In the list below, a single asterisk denotes slight
deviations from numpy
’s nomenclature, and a double asterisk
denotes those cases, where a bit more caution should be exercised,
though this usually means functions that are not supported by numpy
.
The detailed discussion of the various functions always contains a link
to the corresponding numpy
documentation. However, before going down
the rabbit hole, the module also defines a constant, the version, which
can always be queried as
# code to be run in micropython
import ulab as np
print('you are running ulab version', np.__version__)
you are running ulab version 0.24
If you find a bug, please, include this number in your report!
Basic ndarray operations¶
Universal functions (also support function calls on general iterables)
ndarray, the basic container¶
The ndarray
is the underlying container of numerical data. It is
derived from micropython’s own array
object, but has a great number
of extra features starting with how it can be initialised, which
operations can be done on it, and which functions can accept it as an
argument. One important property of an ndarray
is that it is also a
proper micropython
iterable.
Since the ndarray
is a binary container, it is also compact, meaning
that it takes only a couple of bytes of extra RAM in addition to what is
required for storing the numbers themselves. ndarray
s 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 double
s,
but some, e.g., the pyboard.v.11, don’t. You can find out, what type of
float your particular platform implements by looking at the output of
the .itemsize class property.
On the following pages, we will see how one can work with
ndarray
s. Those familiar with numpy
should find that the
nomenclature and naming conventions of numpy
are adhered to as
closely as possible. I will point out the few differences, where
necessary.
For the sake of comparison, in addition to the ulab
code snippets,
sometimes the equivalent numpy
code is also presented. You can find
out, where the snippet is supposed to run by looking at its first line,
the header.
Hint: you can easily port existing numpy
code, if you
import ulab as np
.
Initialising an array¶
A new array can be created by passing either a standard micropython
iterable, or another ndarray
into the constructor.
Initialising by passing iterables¶
If the iterable is one-dimensional, i.e., one whose elements are
numbers, then a row vector will be created and returned. If the iterable
is two-dimensional, i.e., one whose elements are again iterables, a
matrix will be created. If the lengths of the iterables is not
consistent, a ValueError
will be raised. Iterables of different
types can be mixed in the initialisation function.
If the dtype
keyword with the possible
uint8/int8/uint16/int16/float
values is supplied, the new
ndarray
will have that type, otherwise, it assumes float
as
default.
# code to be run in micropython
import ulab as np
a = [1, 2, 3, 4, 5, 6, 7, 8]
b = np.array(a)
print("a:\t", a)
print("b:\t", b)
# a two-dimensional array with mixed-type initialisers
c = np.array([range(5), range(20, 25, 1), [44, 55, 66, 77, 88]], dtype=np.uint8)
print("\nc:\t", c)
# and now we throw an exception
d = np.array([range(5), range(10), [44, 55, 66, 77, 88]], dtype=np.uint8)
print("\nd:\t", d)
a: [1, 2, 3, 4, 5, 6, 7, 8]
b: array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
c: array([[0, 1, 2, 3, 4],
[20, 21, 22, 23, 24],
[44, 55, 66, 77, 88]], dtype=uint8)
Traceback (most recent call last):
File "/dev/shm/micropython.py", line 15, in <module>
ValueError: iterables are not of the same length
ndarray
s are pretty-printed, i.e., if the length is larger than
10, then only the first and last three entries will be printed. Also
note that, as opposed to numpy
, the printout always contains the
dtype
.
# code to be run in micropython
import ulab as np
a = np.array(range(200))
print("a:\t", a)
a: array([0.0, 1.0, 2.0, ..., 197.0, 198.0, 199.0], dtype=float)
Initialising by passing arrays¶
An ndarray
can be initialised by supplying another array. This
statement is almost trivial, since ndarray
s are iterables
themselves, though it should be pointed out that initialising through
arrays should be faster, because simply a new copy is created, without
inspection, iteration etc.
# code to be run in micropython
import ulab as np
a = [1, 2, 3, 4, 5, 6, 7, 8]
b = np.array(a)
c = np.array(b)
print("a:\t", a)
print("b:\t", b)
print("\nc:\t", c)
a: [1, 2, 3, 4, 5, 6, 7, 8]
b: array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
c: array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
Methods of ndarrays¶
.shape¶
The .shape
method (property) returns a 2-tuple with the number of
rows, and columns.
WARNING: In circuitpython
, you can call the method as a
property, i.e.,
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3, 4], dtype=np.int8)
print("a:\n", a)
print("shape of a:", a.shape)
b= np.array([[1, 2], [3, 4]], dtype=np.int8)
print("\nb:\n", b)
print("shape of b:", b.shape)
a:
array([1, 2, 3, 4], dtype=int8)
shape of a: (1, 4)
b:
array([[1, 2],
[3, 4]], dtype=int8)
shape of b: (2, 2)
WARNING: On the other hand, since properties are not implemented in
micropython
, there you would call the method as a function, i.e.,
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3, 4], dtype=np.int8)
print("a:\n", a)
print("shape of a:", a.shape)
b= np.array([[1, 2], [3, 4]], dtype=np.int8)
print("\nb:\n", b)
print("shape of b:", b.shape())
a:
array([1, 2, 3, 4], dtype=int8)
shape of a: (1, 4)
b:
array([[1, 2],
[3, 4]], dtype=int8)
shape of b: (2, 2)
.size¶
The .size
method (property) returns an integer with the number of
elements in the array.
WARNING: In circuitpython
, the numpy
nomenclature applies,
i.e.,
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3], dtype=np.int8)
print("a:\n", a)
print("size of a:", a.size)
b= np.array([[1, 2], [3, 4]], dtype=np.int8)
print("\nb:\n", b)
print("size of b:", b.size)
a:
array([1, 2, 3], dtype=int8)
size of a: 3
b:
array([[1, 2],
[3, 4]], dtype=int8)
size of b: 4
WARNING: In micropython
, size
is a method, i.e.,
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3], dtype=np.int8)
print("a:\n", a)
print("size of a:", a.size)
b= np.array([[1, 2], [3, 4]], dtype=np.int8)
print("\nb:\n", b)
print("size of b:", b.size())
a:
array([1, 2, 3], dtype=int8)
size of a: 3
b:
array([[1, 2],
[3, 4]], dtype=int8)
size of b: 4
.itemsize¶
The .itemsize
method (property) returns an integer with the siz
enumber of elements in the array.
WARNING: In circuitpython
:
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3], dtype=np.int8)
print("a:\n", a)
print("itemsize of a:", a.itemsize)
b= np.array([[1, 2], [3, 4]], dtype=np.float)
print("\nb:\n", b)
print("itemsize of b:", b.itemsize)
a:
array([1, 2, 3], dtype=int8)
itemsize of a: 1
b:
array([[1.0, 2.0],
[3.0, 4.0]], dtype=float)
itemsize of b: 8
WARNING: In micropython
:
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3], dtype=np.int8)
print("a:\n", a)
print("itemsize of a:", a.itemsize)
b= np.array([[1, 2], [3, 4]], dtype=np.float)
print("\nb:\n", b)
print("itemsize of b:", b.itemsize())
a:
array([1, 2, 3], dtype=int8)
itemsize of a: 1
b:
array([[1.0, 2.0],
[3.0, 4.0]], dtype=float)
itemsize of b: 8
.reshape¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.reshape.html
reshape
re-writes the shape properties of an ndarray
, but the
array will not be modified in any other way. The function takes a single
2-tuple with two integers as its argument. The 2-tuple should specify
the desired number of rows and columns. If the new shape is not
consistent with the old, a ValueError
exception will be raised.
# code to be run in micropython
import ulab as np
a = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]], dtype=np.uint8)
print('a (4 by 4):', a)
print('a (2 by 8):', a.reshape((2, 8)))
print('a (1 by 16):', a.reshape((1, 16)))
a (4 by 4): array([[1, 2, 3, 4],
[5, 6, 7, 8],
[9, 10, 11, 12],
[13, 14, 15, 16]], dtype=uint8)
a (2 by 8): array([[1, 2, 3, 4, 5, 6, 7, 8],
[9, 10, 11, 12, 13, 14, 15, 16]], dtype=uint8)
a (1 by 16): array([1, 2, 3, ..., 14, 15, 16], dtype=uint8)
.flatten¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.ndarray.flatten.htm
.flatten
returns the flattened array. The array can be flattened in
C
style (i.e., moving horizontally in the matrix), or in fortran
style (i.e., moving vertically in the matrix). The C
-style
flattening is the default, and it is also fast, because this is just a
verbatim copy of the contents.
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3, 4], dtype=np.int8)
print("a: \t\t", a)
print("a flattened: \t", a.flatten())
b = np.array([[1, 2, 3], [4, 5, 6]], dtype=np.int8)
print("\nb:", b)
print("b flattened (C): \t", b.flatten())
print("b flattened (F): \t", b.flatten(order='F'))
a: array([1, 2, 3, 4], dtype=int8)
a flattened: array([1, 2, 3, 4], dtype=int8)
b: array([[1, 2, 3],
[4, 5, 6]], dtype=int8)
b flattened (C): array([1, 2, 3, 4, 5, 6], dtype=int8)
b flattened (F): array([1, 4, 2, 5, 3, 6], dtype=int8)
.transpose¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.transpose.html
Note that only square matrices can be transposed in place, and in general, an internal copy of the matrix is required. If RAM is a concern, plan accordingly!
# code to be run in micropython
import ulab as np
a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12]], dtype=np.uint8)
print('a:\n', a)
print('shape of a:', a.shape())
a.transpose()
print('\ntranspose of a:\n', a)
print('shape of a:', a.shape())
a:
array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9],
[10, 11, 12]], dtype=uint8)
shape of a: (4, 3)
transpose of a:
array([[1, 4, 7, 10],
[2, 5, 8, 11],
[3, 6, 9, 12]], dtype=uint8)
shape of a: (3, 4)
.sort¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.sort.html
In-place sorting of an ndarray
. For a more detailed exposition, see
sort.
# code to be run in micropython
import ulab as np
a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.uint8)
print('\na:\n', a)
a.sort(axis=0)
print('\na sorted along vertical axis:\n', a)
a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.uint8)
a.sort(a, axis=1)
print('\na sorted along horizontal axis:\n', a)
a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.uint8)
a.sort(a, axis=None)
print('\nflattened a sorted:\n', a)
a:
array([[1, 12, 3, 0],
[5, 3, 4, 1],
[9, 11, 1, 8],
[7, 10, 0, 1]], dtype=uint8)
a sorted along vertical axis:
array([[1, 3, 0, 0],
[5, 10, 1, 1],
[7, 11, 3, 1],
[9, 12, 4, 8]], dtype=uint8)
a sorted along horizontal axis:
array([[0, 1, 3, 12],
[1, 3, 4, 5],
[1, 8, 9, 11],
[0, 1, 7, 10]], dtype=uint8)
flattened a sorted:
array([0, 0, 1, ..., 10, 11, 12], dtype=uint8)
Unary operators¶
With the exception of len
, which returns a single number, all unary
operators manipulate the underlying data element-wise.
len¶
This operator takes a single argument, and returns either the length (for row vectors), or the number of rows (for matrices) of its argument.
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3, 4, 5], dtype=np.uint8)
b = np.array([range(5), range(5), range(5), range(5)], dtype=np.uint8)
print("a:\t", a)
print("length of a: ", len(a))
print("shape of a: ", a.shape())
print("\nb:\t", b)
print("length of b: ", len(b))
print("shape of b: ", b.shape())
a: array([1, 2, 3, 4, 5], dtype=uint8)
length of a: 5
shape of a: (1, 5)
b: array([[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4]], dtype=uint8)
length of b: 4
shape of b: (4, 5)
The number returned by len
is also the length of the iterations,
when the array supplies the elements for an iteration (see later).
invert¶
The function function is defined for integer data types (uint8
,
int8
, uint16
, and int16
) only, takes a single argument, and
returns the element-by-element, bit-wise inverse of the array. If a
float
is supplied, the function raises a ValueError
exception.
With signed integers (int8
, and int16
), the results might be
unexpected, as in the example below:
# code to be run in micropython
import ulab as np
a = np.array([0, -1, -100], dtype=np.int8)
print("a:\t\t", a)
print("inverse of a:\t", ~a)
a = np.array([0, 1, 254, 255], dtype=np.uint8)
print("\na:\t\t", a)
print("inverse of a:\t", ~a)
a: array([0, -1, -100], dtype=int8)
inverse of a: array([-1, 0, 99], dtype=int8)
a: array([0, 1, 254, 255], dtype=uint8)
inverse of a: array([255, 254, 1, 0], dtype=uint8)
abs¶
This function takes a single argument, and returns the
element-by-element absolute value of the array. When the data type is
unsigned (uint8
, or uint16
), a copy of the array will be
returned immediately, and no calculation takes place.
# code to be run in micropython
import ulab as np
a = np.array([0, -1, -100], dtype=np.int8)
print("a:\t\t\t ", a)
print("absolute value of a:\t ", abs(a))
a: array([0, -1, -100], dtype=int8)
absolute value of a: array([0, 1, 100], dtype=int8)
neg¶
This operator takes a single argument, and changes the sign of each element in the array. Unsigned values are wrapped.
# code to be run in micropython
import ulab as np
a = np.array([10, -1, 1], dtype=np.int8)
print("a:\t\t", a)
print("negative of a:\t", -a)
b = np.array([0, 100, 200], dtype=np.uint8)
print("\nb:\t\t", b)
print("negative of b:\t", -b)
a: array([10, -1, 1], dtype=int8)
negative of a: array([-10, 1, -1], dtype=int8)
b: array([0, 100, 200], dtype=uint8)
negative of b: array([0, 156, 56], dtype=uint8)
pos¶
This function takes a single argument, and simply returns a copy of the array.
# code to be run in micropython
import ulab as np
a = np.array([10, -1, 1], dtype=np.int8)
print("a:\t\t", a)
print("positive of a:\t", +a)
a: array([10, -1, 1], dtype=int8)
positive of a: array([10, -1, 1], dtype=int8)
Binary operators¶
All binary operators work element-wise. This also means that the operands either must have the same shape, or one of them must be a scalar.
WARNING: numpy
also allows operations between a matrix, and a
row vector, if the row vector has exactly as many elements, as many
columns the matrix has. This feature will be added in future versions of
ulab
.
# code to be run in CPython
a = array([[1, 2, 3], [4, 5, 6], [7, 8, 6]])
b = array([10, 20, 30])
a+b
array([[11, 22, 33],
[14, 25, 36],
[17, 28, 36]])
Upcasting¶
Binary operations require special attention, because two arrays with
different typecodes can be the operands of an operation, in which case
it is not trivial, what the typecode of the result is. This decision on
the result’s typecode is called upcasting. Since the number of typecodes
in ulab
is significantly smaller than in numpy
, we have to
define new upcasting rules. Where possible, I followed numpy
’s
conventions.
ulab
observes the following upcasting rules:
- Operations with two
ndarray
s of the samedtype
preserve theirdtype
, even when the results overflow. - if either of the operands is a float, the result is automatically a float
- When the right hand side of a binary operator is a micropython
variable,
mp_obj_int
, ormp_obj_float
, then the result will be promoted todtype
float
. This is necessary, because a micropython integer can be 31 bites wide. Other micropython types (e.g., lists, tuples, etc.) raise aTypeError
exception.
left hand side | right hand side | ulab result | numpy result |
---|---|---|---|
uint8 |
int8 |
int16 |
int16 |
uint8 |
int16 |
int16 |
int16 |
uint8 |
uint16 |
uint16 |
uint16 |
int8 |
int16 |
int16 |
int16 |
int8 |
uint16 |
uint16 |
int32 |
uint16 |
int16 |
float |
int32 |
Note that the last two operations are promoted to int32
in
numpy
.
WARNING: Due to the lower number of available data types, the
upcasting rules of ulab
are slightly different to those of
numpy
. Watch out for this, when porting code!
Upcasting can be seen in action in the following snippet:
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3, 4], dtype=np.uint8)
b = np.array([1, 2, 3, 4], dtype=np.int8)
print("a:\t", a)
print("b:\t", b)
print("a+b:\t", a+b)
c = np.array([1, 2, 3, 4], dtype=np.float)
print("\na:\t", a)
print("c:\t", c)
print("a*c:\t", a*c)
a: array([1, 2, 3, 4], dtype=uint8)
b: array([1, 2, 3, 4], dtype=int8)
a+b: array([2, 4, 6, 8], dtype=int16)
a: array([1, 2, 3, 4], dtype=uint8)
c: array([1.0, 2.0, 3.0, 4.0], dtype=float)
a*c: array([1.0, 4.0, 9.0, 16.0], dtype=float)
WARNING: If a binary operation involves an ndarray
and a
micropython type (integer, or float), then the array must be on the left
hand side.
# code to be run in micropython
import ulab as np
# this is going to work
a = np.array([1, 2, 3, 4], dtype=np.uint8)
b = 12
print("a:\t", a)
print("b:\t", b)
print("a+b:\t", a+b)
# but this will spectacularly fail
print("b+a:\t", b+a)
a: array([1, 2, 3, 4], dtype=uint8)
b: 12
a+b: array([13, 14, 15, 16], dtype=uint8)
Traceback (most recent call last):
File "/dev/shm/micropython.py", line 12, in <module>
TypeError: unsupported types for __add__: 'int', 'ndarray'
The reason for this lies in how micropython resolves binary operators,
and this means that a fix can only be implemented, if micropython itself
changes the corresponding function(s). Till then, keep ndarray
s on
the left hand side.
Benchmarks¶
The following snippet compares the performance of binary operations to a possible implementation in python. For the time measurement, we will take the following snippet from the micropython manual:
# code to be run in micropython
def timeit(f, *args, **kwargs):
func_name = str(f).split(' ')[1]
def new_func(*args, **kwargs):
t = utime.ticks_us()
result = f(*args, **kwargs)
print('execution time: ', utime.ticks_diff(utime.ticks_us(), t), ' us')
return result
return new_func
# code to be run in micropython
import ulab as np
@timeit
def py_add(a, b):
return [a[i]+b[i] for i in range(1000)]
@timeit
def py_multiply(a, b):
return [a[i]*b[i] for i in range(1000)]
@timeit
def ulab_add(a, b):
return a + b
@timeit
def ulab_multiply(a, b):
return a * b
a = [0.0]*1000
b = range(1000)
print('python add:')
py_add(a, b)
print('\npython multiply:')
py_multiply(a, b)
a = np.linspace(0, 10, num=1000)
b = np.ones(1000)
print('\nulab add:')
ulab_add(a, b)
print('\nulab multiply:')
ulab_multiply(a, b)
python add:
execution time: 10051 us
python multiply:
execution time: 14175 us
ulab add:
execution time: 222 us
ulab multiply:
execution time: 213 us
I do not claim that the python implementation above is perfect, and
certainly, there is much room for improvement. However, the factor of 50
difference in execution time is very spectacular. This is nothing but a
consequence of the fact that the ulab
functions run C
code, with
very little python overhead. The factor of 50 appears to be quite
universal: the FFT routine obeys similar scaling (see Speed of
FFTs), and this number came up with font rendering,
too: fast font rendering on graphical
displays.
Comparison operators¶
The smaller than, greater than, smaller or equal, and greater or equal
operators return a vector of Booleans indicating the positions
(True
), where the condition is satisfied.
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3, 4, 5, 6, 7, 8], dtype=np.uint8)
print(a < 5)
[True, True, True, True, False, False, False, False]
WARNING: Note that numpy
returns an array of Booleans. For most
use cases this fact should not make a difference.
# code to be run in CPython
a = array([1, 2, 3, 4, 5, 6, 7, 8])
a < 5
array([ True, True, True, True, False, False, False, False])
These operators work with matrices, too, in which case a list of lists of Booleans will be returned:
# code to be run in micropython
import ulab as np
a = np.array([range(0, 5, 1), range(1, 6, 1), range(2, 7, 1)], dtype=np.uint8)
print(a)
print(a < 5)
array([[0, 1, 2, 3, 4],
[1, 2, 3, 4, 5],
[2, 3, 4, 5, 6]], dtype=uint8)
[[True, True, True, True, True], [True, True, True, True, False], [True, True, True, False, False]]
Iterating over arrays¶
ndarray
s are iterable, which means that their elements can also be
accessed as can the elements of a list, tuple, etc. If the array is
one-dimensional, the iterator returns scalars, otherwise a new
one-dimensional ndarray
, which is simply a copy of the corresponding
row of the matrix, i.e, its data type will be inherited.
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3, 4, 5], dtype=np.uint8)
b = np.array([range(5), range(10, 15, 1), range(20, 25, 1), range(30, 35, 1)], dtype=np.uint8)
print("a:\t", a)
for i, _a in enumerate(a):
print("element %d in a:"%i, _a)
print("\nb:\t", b)
for i, _b in enumerate(b):
print("element %d in b:"%i, _b)
a: array([1, 2, 3, 4, 5], dtype=uint8)
element 0 in a: 1
element 1 in a: 2
element 2 in a: 3
element 3 in a: 4
element 4 in a: 5
b: array([[0, 1, 2, 3, 4],
[10, 11, 12, 13, 14],
[20, 21, 22, 23, 24],
[30, 31, 32, 33, 34]], dtype=uint8)
element 0 in b: array([0, 1, 2, 3, 4], dtype=uint8)
element 1 in b: array([10, 11, 12, 13, 14], dtype=uint8)
element 2 in b: array([20, 21, 22, 23, 24], dtype=uint8)
element 3 in b: array([30, 31, 32, 33, 34], dtype=uint8)
Slicing and indexing¶
Copies of sub-arrays can be created by indexing, and slicing.
Indexing¶
The simplest form of indexing is specifying a single integer between the square brackets as in
# code to be run in micropython
import ulab as np
a = np.array(range(10), dtype=np.uint8)
print("a:\t\t\t\t\t\t", a)
print("the first, and first from right element of a:\t", a[0], a[-1])
print("the second, and second from right element of a:\t", a[1], a[-2])
a: array([0, 1, 2, ..., 7, 8, 9], dtype=uint8)
the first, and first from right element of a: 0 9
the second, and second from right element of a: 1 8
Indices are (not necessarily non-negative) integers, or a list of
Booleans. By using a Boolean list, we can select those elements of an
array that satisfy a specific condition. At the moment, such indexing is
defined for row vectors only, for matrices the function raises a
ValueError
exception, though this will be rectified in a future
version of ulab
.
# code to be run in micropython
import ulab as np
a = np.array(range(9), dtype=np.float)
print("a:\t", a)
print("a < 5:\t", a[a < 5])
a: array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
a < 5: array([0.0, 1.0, 2.0, 3.0, 4.0], dtype=float)
Indexing with Boolean arrays can take more complicated expressions. This is a very concise way of comparing two vectors, e.g.:
# code to be run in micropython
import ulab as np
a = np.array(range(9), dtype=np.uint8)
b = np.array([4, 4, 4, 3, 3, 3, 13, 13, 13], dtype=np.uint8)
print("a:\t", a)
print("\na**2:\t", a*a)
print("\nb:\t", b)
print("\n100*sin(b):\t", np.sin(b)*100.0)
print("\na[a*a > np.sin(b)*100.0]:\t", a[a*a > np.sin(b)*100.0])
a: array([0, 1, 2, 3, 4, 5, 6, 7, 8], dtype=uint8)
a**2: array([0, 1, 4, 9, 16, 25, 36, 49, 64], dtype=uint8)
b: array([4, 4, 4, 3, 3, 3, 13, 13, 13], dtype=uint8)
100*sin(b): array([-75.68025, -75.68025, -75.68025, 14.112, 14.112, 14.112, 42.01671, 42.01671, 42.01671], dtype=float)
a[a*a > np.sin(b)*100.0]: array([0, 1, 2, 4, 5, 7, 8], dtype=uint8)
Slicing and assigning to slices¶
You can also generate sub-arrays by specifying slices as the index of an array. Slices are special python objects of the form
slice = start:end:stop
where start
, end
, and stop
are (not necessarily
non-negative) integers. Not all of these three numbers must be specified
in an index, in fact, all three of them can be missing. The interpreter
takes care of filling in the missing values. (Note that slices cannot be
defined in this way, only there, where an index is expected.) For a good
explanation on how slices work in python, you can read the stackoverflow
question
https://stackoverflow.com/questions/509211/understanding-slice-notation.
Slices work on both axes:
# code to be run in micropython
import ulab as np
a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=np.uint8)
print('a:\n', a)
# the first row
print('\na[0]:\n', a[0])
# the first two elements of the first row
print('\na[0,:2]:\n', a[0,:2])
# the zeroth element in each row (also known as the zeroth column)
print('\na[:,0]:\n', a[:,0])
# the last but one row
print('\na[-1]:\n', a[-1])
# the last two rows backwards
print('\na[::1]:\n', a[::-1])
a:
array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]], dtype=uint8)
a[0]:
array([1, 2, 3], dtype=uint8)
a[0,:2]:
array([1, 2], dtype=uint8)
a[:,0]:
array([1, 4, 7], dtype=uint8)
a[-1]:
array([7, 8, 9], dtype=uint8)
a[::1]:
array([[7, 8, 9],
[4, 5, 6]], dtype=uint8)
Assignment to slices can be done for the whole slice, per row, and per column. A couple of examples should make these statements clearer:
# code to be run in micropython
import ulab as np
zero_list = [0, 0, 0]
a = np.array([zero_list, zero_list, zero_list], dtype=np.uint8)
print('a:\n', a)
# assigning to the whole row
a[0] = 1
print('\na[0] = 1\n', a)
# assigning to the whole row
a[0] = np.array([1, 2, -333], dtype=np.float)
print('\na[0] = np.array([1, 2, 3])\n', a)
# assigning to a column
a[:,2] = 3.0
print('\na[:,0]:\n', a)
a:
array([[0, 0, 0],
[0, 0, 0],
[0, 0, 0]], dtype=uint8)
a[0] = 1
array([[1, 1, 1],
[0, 0, 0],
[0, 0, 0]], dtype=uint8)
a[0] = np.array([1, 2, 3])
array([[1, 2, 179],
[0, 0, 0],
[0, 0, 0]], dtype=uint8)
a[:,0]:
array([[1, 2, 3],
[0, 0, 3],
[0, 0, 3]], dtype=uint8)
Universal functions¶
Standard mathematical functions can be calculated on any scalar-valued
iterable (ranges, lists, tuples containing numbers), and on
ndarray
s without having to change the call signature. In all cases
the functions return a new ndarray
of typecode float
(since
these functions usually generate float values, anyway). The functions
execute faster with ndarray
arguments than with iterables, because
the values of the input vector can be extracted faster.
At present, the following functions are supported:
acos
, acosh
, asin
, asinh
, atan
, atanh
, ceil
,
cos
, erf
, erfc
, exp
, expm1
, floor
, tgamma
,
lgamma
, log
, log10
, log2
, sin
, sinh
, sqrt
,
tan
, tanh
.
These functions are applied element-wise to the arguments, thus, e.g., the exponential of a matrix cannot be calculated in this way.
# code to be run in micropython
import ulab as np
a = range(9)
b = np.array(a)
# works with ranges, lists, tuples etc.
print('a:\t', a)
print('exp(a):\t', np.exp(a))
# with 1D arrays
print('\nb:\t', b)
print('exp(b):\t', np.exp(b))
# as well as with matrices
c = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print('\nc:\t', c)
print('exp(c):\t', np.exp(c))
a: range(0, 9)
exp(a): array([1.0, 2.718282, 7.389056, 20.08554, 54.59816, 148.4132, 403.4288, 1096.633, 2980.958], dtype=float)
b: array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
exp(b): array([1.0, 2.718282, 7.389056, 20.08554, 54.59816, 148.4132, 403.4288, 1096.633, 2980.958], dtype=float)
c: array([[1.0, 2.0, 3.0],
[4.0, 5.0, 6.0],
[7.0, 8.0, 9.0]], dtype=float)
exp(c): array([[2.718282, 7.389056, 20.08554],
[54.59816, 148.4132, 403.4288],
[1096.633, 2980.958, 8103.084]], dtype=float)
Computation expenses¶
The overhead for calculating with micropython iterables is quite
significant: for the 1000 samples below, the difference is more than 800
microseconds, because internally the function has to create the
ndarray
for the output, has to fetch the iterable’s items of unknown
type, and then convert them to floats. All these steps are skipped for
ndarray
s, because these pieces of information are already known.
# code to be run in micropython
import ulab as np
a = [0]*1000
b = np.array(a)
@timeit
def measure_run_time(x):
return np.exp(x)
measure_run_time(a)
measure_run_time(b)
execution time: 1259 us
execution time: 408 us
Of course, such a time saving is reasonable only, if the data are
already available as an ndarray
. If one has to initialise the
ndarray
from the list, then there is no gain, because the iterator
was simply pushed into the initialisation function.
Numerical¶
linspace¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.linspace.html
This function returns an array, whose elements are uniformly spaced
between the start
, and stop
points. The number of intervals is
determined by the num
keyword argument, whose default value is 50.
With the endpoint
keyword argument (defaults to True
) one can
include stop
in the sequence. In addition, the dtype
keyword can
be supplied to force type conversion of the output. The default is
float
. Note that, when dtype
is of integer type, the sequence is
not necessarily evenly spaced. This is not an error, rather a
consequence of rounding. (This is also the numpy
behaviour.)
# code to be run in micropython
import ulab as np
# generate a sequence with defaults
print('default sequence:\t', np.linspace(0, 10))
# num=5
print('num=5:\t\t\t', np.linspace(0, 10, num=5))
# num=5, endpoint=False
print('num=5:\t\t\t', np.linspace(0, 10, num=5, endpoint=False))
# num=5, endpoint=False, dtype=uint8
print('num=5:\t\t\t', np.linspace(0, 5, num=7, endpoint=False, dtype=np.uint8))
default sequence: array([0.0, 0.2040816396474838, 0.4081632792949677, ..., 9.591833114624023, 9.795914649963379, 9.999996185302734], dtype=float)
num=5: array([0.0, 2.5, 5.0, 7.5, 10.0], dtype=float)
num=5: array([0.0, 2.0, 4.0, 6.0, 8.0], dtype=float)
num=5: array([0, 0, 1, 2, 2, 3, 4], dtype=uint8)
min, argmin, max, argmax¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.min.html
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmax.html
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.max.html
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmax.html
WARNING: Difference to numpy
: the out
keyword argument is
not implemented.
These functions follow the same pattern, and work with generic
iterables, and ndarray
s. min
, and max
return the minimum
or maximum of a sequence. If the input array is two-dimensional, the
axis
keyword argument can be supplied, in which case the
minimum/maximum along the given axis will be returned. If axis=None
(this is also the default value), the minimum/maximum of the flattened
array will be determined.
argmin/argmax
return the position (index) of the minimum/maximum in
the sequence.
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 0, 1, 10])
print('a:', a)
print('min of a:', np.min(a))
print('argmin of a:', np.argmin(a))
b = np.array([[1, 2, 0], [1, 10, -1]])
print('\nb:\n', b)
print('min of b (flattened):', np.min(b))
print('min of b (axis=0):', np.min(b, axis=0))
print('min of b (axis=1):', np.min(b, axis=1))
a: array([1.0, 2.0, 0.0, 1.0, 10.0], dtype=float)
min of a: 0.0
argmin of a: 2
b:
array([[1.0, 2.0, 0.0],
[1.0, 10.0, -1.0]], dtype=float)
min of b (flattened): -1.0
min of b (axis=0): array([1.0, 2.0, -1.0], dtype=float)
min of b (axis=1): array([0.0, -1.0], dtype=float)
sum, std, mean¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.sum.html
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.std.html
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.mean.html
These three functions follow the same pattern: if the axis keyword is
not specified, it assumes the default value of None
, and returns the
result of the computation for the flattened array. Otherwise, the
calculation is along the given axis.
# code to be run in micropython
import ulab as np
a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print('a: \n', a)
print('sum, flat array: ', np.sum(a))
print('mean, horizontal: ', np.mean(a, axis=1))
print('std, vertical: ', np.std(a, axis=0))
a:
array([[1.0, 2.0, 3.0],
[4.0, 5.0, 6.0],
[7.0, 8.0, 9.0]], dtype=float)
sum, flat array: 45.0
mean, horizontal: array([2.0, 5.0, 8.0], dtype=float)
std, vertical: array([2.44949, 2.44949, 2.44949], dtype=float)
roll¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.roll.html
The roll function shifts the content of a vector by the positions given
as the second argument. If the axis
keyword is supplied, the shift
is applied to the given axis.
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3, 4, 5, 6, 7, 8])
print("a:\t\t\t", a)
np.roll(a, 2)
print("a rolled to the left:\t", a)
# this should be the original vector
np.roll(a, -2)
print("a rolled to the right:\t", a)
a: array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
a rolled to the left: array([3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 1.0, 2.0], dtype=float)
a rolled to the right: array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
Rolling works with matrices, too. If the axis
keyword is 0, the
matrix is rolled along its vertical axis, otherwise, horizontally.
Horizontal rolls are faster, because they require fewer steps, and
larger memory chunks are copied, however, they also require more RAM:
basically the whole row must be stored internally. Most expensive are
the None
keyword values, because with axis = None
, the array is
flattened first, hence the row’s length is the size of the whole matrix.
Vertical rolls require two internal copies of single columns.
# code to be run in micropython
import ulab as np
a = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])
print("a:\n", a)
np.roll(a, 2)
print("\na rolled to the left:\n", a)
np.roll(a, -1, axis=1)
print("\na rolled up:\n", a)
np.roll(a, 1, axis=None)
print("\na rolled with None:\n", a)
a:
array([[1.0, 2.0, 3.0, 4.0],
[5.0, 6.0, 7.0, 8.0]], dtype=float)
a rolled to the left:
array([[3.0, 4.0, 5.0, 6.0],
[7.0, 8.0, 1.0, 2.0]], dtype=float)
a rolled up:
array([[6.0, 3.0, 4.0, 5.0],
[2.0, 7.0, 8.0, 1.0]], dtype=float)
a rolled with None:
array([[3.0, 4.0, 5.0, 2.0],
[7.0, 8.0, 1.0, 6.0]], dtype=float)
Simple running weighted average¶
As a demonstration of the conciseness of ulab/numpy
operations, we
will calculate an exponentially weighted running average of a
measurement vector in just a couple of lines. I chose this particular
example, because I think that this can indeed be used in real-life
applications.
# code to be run in micropython
import ulab as np
def dummy_adc():
# dummy adc function, so that the results are reproducible
return 2
n = 10
# These are the normalised weights; the last entry is the most dominant
weight = np.exp([1, 2, 3, 4, 5])
weight = weight/np.sum(weight)
print(weight)
# initial array of samples
samples = np.array([0]*n)
for i in range(n):
# a new datum is inserted on the right hand side. This simply overwrites whatever was in the last slot
samples[-1] = dummy_adc()
print(np.mean(samples[-5:]*weight))
print(samples[-5:])
# the data are shifted by one position to the left
np.roll(samples, 1)
array([0.01165623031556606, 0.03168492019176483, 0.08612854033708572, 0.234121635556221, 0.6364086270332336], dtype=float)
0.2545634508132935
array([0.0, 0.0, 0.0, 0.0, 2.0], dtype=float)
0.3482121050357819
array([0.0, 0.0, 0.0, 2.0, 2.0], dtype=float)
0.3826635211706161
array([0.0, 0.0, 2.0, 2.0, 2.0], dtype=float)
0.3953374892473221
array([0.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
flip¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.flip.html
The flip
function takes one positional, an ndarray
, and one
keyword argument, axis = None
, and reverses the order of elements
along the given axis. If the keyword argument is None
, the matrix’
entries are flipped along all axes. flip
returns a new copy of the
array.
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3, 4, 5])
print("a: \t", a)
print("a flipped:\t", np.flip(a))
a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=np.uint8)
print("\na flipped horizontally\n", np.flip(a, axis=1))
print("\na flipped vertically\n", np.flip(a, axis=0))
print("\na flipped horizontally+vertically\n", np.flip(a))
a: array([1.0, 2.0, 3.0, 4.0, 5.0], dtype=float)
a flipped: array([5.0, 4.0, 3.0, 2.0, 1.0], dtype=float)
a flipped horizontally
array([[3, 2, 1],
[6, 5, 4],
[9, 8, 7]], dtype=uint8)
a flipped vertically
array([[7, 8, 9],
[4, 5, 6],
[1, 2, 3]], dtype=uint8)
a flipped horizontally+vertically
array([[9, 8, 7],
[6, 5, 4],
[3, 2, 1]], dtype=uint8)
diff¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.diff.html
The diff
function returns the numerical derivative of the forward
scheme, or more accurately, the differences of an ndarray
along a
given axis. The order of derivative can be stipulated with the n
keyword argument, which should be between 0, and 9. Default is 1. If
higher order derivatives are required, they can be gotten by repeated
calls to the function. The axis
keyword argument should be -1 (last
axis, in ulab
equivalent to the second axis, and this also happens
to be the default value), 0, or 1.
Beyond the output array, the function requires only a couple of bytes of
extra RAM for the differentiation stencil. (The stencil is an int8
array, one byte longer than n
. This also explains, why the highest
order is 9: the coefficients of a ninth-order stencil all fit in signed
bytes, while 10 would require int16
.) Note that as usual in
numerical differentiation (and also in numpy
), the length of the
respective axis will be reduced by n
after the operation. If n
is larger than, or equal to the length of the axis, an empty array will
be returned.
WARNING: the diff
function does not implement the prepend
and append
keywords that can be found in numpy
.
# code to be run in micropython
import ulab as np
a = np.array(range(9), dtype=np.uint8)
print('a:\n', a)
print('\nfirst derivative:\n', np.diff(a, n=1))
print('\nsecond derivative:\n', np.diff(a, n=2))
c = np.array([[1, 2, 3, 4], [4, 3, 2, 1], [1, 4, 9, 16], [0, 0, 0, 0]])
print('\nc:\n', c)
print('\nfirst derivative, first axis:\n', np.diff(c, axis=0))
print('\nfirst derivative, second axis:\n', np.diff(c, axis=1))
a:
array([0, 1, 2, 3, 4, 5, 6, 7, 8], dtype=uint8)
first derivative:
array([1, 1, 1, 1, 1, 1, 1, 1], dtype=uint8)
second derivative:
array([0, 0, 0, 0, 0, 0, 0], dtype=uint8)
c:
array([[1.0, 2.0, 3.0, 4.0],
[4.0, 3.0, 2.0, 1.0],
[1.0, 4.0, 9.0, 16.0],
[0.0, 0.0, 0.0, 0.0]], dtype=float)
first derivative, first axis:
array([[3.0, 1.0, -1.0, -3.0],
[-3.0, 1.0, 7.0, 15.0],
[-1.0, -4.0, -9.0, -16.0]], dtype=float)
first derivative, second axis:
array([[1.0, 1.0, 1.0],
[-1.0, -1.0, -1.0],
[3.0, 5.0, 7.0],
[0.0, 0.0, 0.0]], dtype=float)
sort¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.sort.html
The sort function takes an ndarray, and sorts its elements in ascending
order along the specified axis using a heap sort algorithm. As opposed
to the .sort()
method discussed earlier, this function creates a
copy of its input before sorting, and at the end, returns this copy.
Sorting takes place in place, without auxiliary storage. The axis
keyword argument takes on the possible values of -1 (the last axis, in
ulab
equivalent to the second axis, and this also happens to be the
default value), 0, 1, or None
. The first three cases are identical
to those in diff, while the last one flattens the array
before sorting.
If descending order is required, the result can simply be flip
ped,
see flip.
WARNING: numpy
defines the kind
, and order
keyword
arguments that are not implemented here. The function in ulab
always
uses heap sort, and since ulab
does not have the concept of data
fields, the order
keyword argument would have no meaning.
# code to be run in micropython
import ulab as np
a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.float)
print('\na:\n', a)
b = np.sort(a, axis=0)
print('\na sorted along vertical axis:\n', b)
c = np.sort(a, axis=1)
print('\na sorted along horizontal axis:\n', c)
c = np.sort(a, axis=None)
print('\nflattened a sorted:\n', c)
a:
array([[1.0, 12.0, 3.0, 0.0],
[5.0, 3.0, 4.0, 1.0],
[9.0, 11.0, 1.0, 8.0],
[7.0, 10.0, 0.0, 1.0]], dtype=float)
a sorted along vertical axis:
array([[1.0, 3.0, 0.0, 0.0],
[5.0, 10.0, 1.0, 1.0],
[7.0, 11.0, 3.0, 1.0],
[9.0, 12.0, 4.0, 8.0]], dtype=float)
a sorted along horizontal axis:
array([[0.0, 1.0, 3.0, 12.0],
[1.0, 3.0, 4.0, 5.0],
[1.0, 8.0, 9.0, 11.0],
[0.0, 1.0, 7.0, 10.0]], dtype=float)
flattened a sorted:
array([0.0, 0.0, 1.0, ..., 10.0, 11.0, 12.0], dtype=float)
Heap sort requires \(\sim N\log N\) operations, and notably, the worst case costs only 20% more time than the average. In order to get an order-of-magnitude estimate, we will take the sine of 1000 uniformly spaced numbers between 0, and two pi, and sort them:
# code to be run in micropython
import ulab as np
@timeit
def sort_time(array):
return np.sort(array)
b = np.sin(np.linspace(0, 6.28, num=1000))
print('b: ', b)
sort_time(b)
print('\nb sorted:\n', b)
argsort¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argsort.html
Similarly to sort, argsort
takes a positional, and a
keyword argument, and returns an unsigned short index array of type
ndarray
with the same dimensions as the input, or, if axis=None
,
as a row vector with length equal to the number of elements in the input
(i.e., the flattened array). The indices in the output sort the input in
ascending order. The routine in argsort
is the same as in sort
,
therefore, the comments on computational expenses (time and RAM) also
apply. In particular, since no copy of the original data is required,
virtually no RAM beyond the output array is used.
Since the underlying container of the output array is of type
uint16_t
, neither of the output dimensions should be larger than
65535.
# code to be run in micropython
import ulab as np
a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.float)
print('\na:\n', a)
b = np.argsort(a, axis=0)
print('\na sorted along vertical axis:\n', b)
c = np.argsort(a, axis=1)
print('\na sorted along horizontal axis:\n', c)
c = np.argsort(a, axis=None)
print('\nflattened a sorted:\n', c)
a:
array([[1.0, 12.0, 3.0, 0.0],
[5.0, 3.0, 4.0, 1.0],
[9.0, 11.0, 1.0, 8.0],
[7.0, 10.0, 0.0, 1.0]], dtype=float)
a sorted along vertical axis:
array([[0, 1, 3, 0],
[1, 3, 2, 1],
[3, 2, 0, 3],
[2, 0, 1, 2]], dtype=uint16)
a sorted along horizontal axis:
array([[3, 0, 2, 1],
[3, 1, 2, 0],
[2, 3, 0, 1],
[2, 3, 0, 1]], dtype=uint16)
flattened a sorted:
array([3, 14, 0, ..., 13, 9, 1], dtype=uint16)
Since during the sorting, only the indices are shuffled, argsort
does not modify the input array, as one can verify this by the following
example:
# code to be run in micropython
import ulab as np
a = np.array([0, 5, 1, 3, 2, 4], dtype=np.uint8)
print('\na:\n', a)
b = np.argsort(a, axis=1)
print('\nsorting indices:\n', b)
print('\nthe original array:\n', a)
a:
array([0, 5, 1, 3, 2, 4], dtype=uint8)
sorting indices:
array([0, 2, 4, 3, 5, 1], dtype=uint16)
the original array:
array([0, 5, 1, 3, 2, 4], dtype=uint8)
Linalg¶
size¶
size
takes a single argument, the axis, whose size is to be
returned. Depending on the value of the argument, the following
information will be returned:
- argument is 0: the number of elements of the array
- argument is 1: the number of rows
- argument is 2: the number of columns
# code to be run in micropython
import ulab as np
a = np.array([1, 2, 3, 4], dtype=np.int8)
print("a:\n", a)
print("size of a:", np.size(a, axis=None), ",", np.size(a, axis=0))
b= np.array([[1, 2], [3, 4]], dtype=np.int8)
print("\nb:\n", b)
print("size of b:", np.size(b, axis=None), ",", np.size(b, axis=0), ",", np.size(b, axis=1))
a:
array([1, 2, 3, 4], dtype=int8)
size of a: 4 , 4
b:
array([[1, 2],
[3, 4]], dtype=int8)
size of b: 4 , 2 , 2
ones, zeros¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.zeros.html
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.ones.html
A couple of special arrays and matrices can easily be initialised by
calling one of the ones
, or zeros
functions. ones
and
zeros
follow the same pattern, and have the call signature
ones(shape, dtype=float)
zeros(shape, dtype=float)
where shape is either an integer, or a 2-tuple.
# code to be run in micropython
import ulab as np
print(np.ones(6, dtype=np.uint8))
print(np.zeros((6, 4)))
array([1, 1, 1, 1, 1, 1], dtype=uint8)
array([[0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0]], dtype=float)
eye¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.eye.html
Another special array method is the eye
function, whose call
signature is
eye(N, M, k=0, dtype=float)
where N
(M
) specify the dimensions of the matrix (if only N
is supplied, then we get a square matrix, otherwise one with M
rows,
and N
columns), and k
is the shift of the ones (the main
diagonal corresponds to k=0
). Here are a couple of examples.
With a single argument¶
# code to be run in micropython
import ulab as np
print(np.eye(5))
array([[1.0, 0.0, 0.0, 0.0, 0.0],
[0.0, 1.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.0, 1.0]], dtype=float)
Specifying the dimensions of the matrix¶
# code to be run in micropython
import ulab as np
print(np.eye(4, M=6, dtype=np.int8))
array([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1],
[0, 0, 0, 0],
[0, 0, 0, 0]], dtype=int8)
Shifting the diagonal¶
# code to be run in micropython
import ulab as np
print(np.eye(4, M=6, k=-1, dtype=np.int16))
array([[0, 0, 0, 0],
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1],
[0, 0, 0, 0]], dtype=int16)
inv¶
A square matrix, provided that it is not singular, can be inverted by
calling the inv
function that takes a single argument. The inversion
is based on successive elimination of elements in the lower left
triangle, and raises a ValueError
exception, if the matrix turns out
to be singular (i.e., one of the diagonal entries is zero).
# code to be run in micropython
import ulab as np
m = np.array([[1, 2, 3, 4], [4, 5, 6, 4], [7, 8.6, 9, 4], [3, 4, 5, 6]])
print(np.inv(m))
array([[-2.166666, 1.499999, -0.8333326, 1.0],
[1.666666, -3.333331, 1.666666, -4.768516e-08],
[0.1666672, 2.166666, -0.8333327, -1.0],
[-0.1666666, -0.3333334, 4.96705e-08, 0.5]], dtype=float)
Computation expenses¶
Note that the cost of inverting a matrix is approximately twice as many floats (RAM), as the number of entries in the original matrix, and approximately as many operations, as the number of entries. Here are a couple of numbers:
# code to be run in micropython
import ulab as np
@timeit
def invert_matrix(m):
return np.inv(m)
m = np.array([[1, 2,], [4, 5]])
print('2 by 2 matrix:')
invert_matrix(m)
m = np.array([[1, 2, 3, 4], [4, 5, 6, 4], [7, 8.6, 9, 4], [3, 4, 5, 6]])
print('\n4 by 4 matrix:')
invert_matrix(m)
m = np.array([[1, 2, 3, 4, 5, 6, 7, 8], [0, 5, 6, 4, 5, 6, 4, 5],
[0, 0, 9, 7, 8, 9, 7, 8], [0, 0, 0, 10, 11, 12, 11, 12],
[0, 0, 0, 0, 4, 6, 7, 8], [0, 0, 0, 0, 0, 5, 6, 7],
[0, 0, 0, 0, 0, 0, 7, 6], [0, 0, 0, 0, 0, 0, 0, 2]])
print('\n8 by 8 matrix:')
invert_matrix(m)
2 by 2 matrix:
execution time: 65 us
4 by 4 matrix:
execution time: 105 us
8 by 8 matrix:
execution time: 299 us
The above-mentioned scaling is not obeyed strictly. The reason for the discrepancy is that the function call is still the same for all three cases: the input must be inspected, the output array must be created, and so on.
dot¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.dot.html
WARNING: numpy applies upcasting rules for the multiplication of
matrices, while ulab
simply returns a float matrix.
Once you can invert a matrix, you might want to know, whether the
inversion is correct. You can simply take the original matrix and its
inverse, and multiply them by calling the dot
function, which takes
the two matrices as its arguments. If the matrix dimensions do not
match, the function raises a ValueError
. The result of the
multiplication is expected to be the unit matrix, which is demonstrated
below.
# code to be run in micropython
import ulab as np
m = np.array([[1, 2, 3], [4, 5, 6], [7, 10, 9]], dtype=np.uint8)
n = np.inv(m)
print("m:\n", m)
print("\nm^-1:\n", n)
# this should be the unit matrix
print("\nm*m^-1:\n", np.dot(m, n))
m:
array([[1, 2, 3],
[4, 5, 6],
[7, 10, 9]], dtype=uint8)
m^-1:
array([[-1.25, 1.0, -0.25],
[0.5, -1.0, 0.5],
[0.4166667, 0.3333334, -0.25]], dtype=float)
m*m^-1:
array([[1.0, 2.384186e-07, -1.490116e-07],
[-2.980232e-07, 1.000001, -4.172325e-07],
[-3.278255e-07, 1.311302e-06, 0.9999992]], dtype=float)
Note that for matrix multiplication you don’t necessarily need square matrices, it is enough, if their dimensions are compatible (i.e., the the left-hand-side matrix has as many columns, as does the right-hand-side matrix rows):
# code to be run in micropython
import ulab as np
m = np.array([[1, 2, 3, 4], [5, 6, 7, 8]], dtype=np.uint8)
n = np.array([[1, 2], [3, 4], [5, 6], [7, 8]], dtype=np.uint8)
print(m)
print(n)
print(np.dot(m, n))
array([[1, 2, 3, 4],
[5, 6, 7, 8]], dtype=uint8)
array([[1, 2],
[3, 4],
[5, 6],
[7, 8]], dtype=uint8)
array([[7.0, 10.0],
[23.0, 34.0]], dtype=float)
det¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.det.html
The det
function takes a square matrix as its single argument, and
calculates the determinant. The calculation is based on successive
elimination of the matrix elements, and the return value is a float,
even if the input array was of integer type.
# code to be run in micropython
import ulab as np
a = np.array([[1, 2], [3, 4]], dtype=np.uint8)
print(np.det(a))
-2.0
Benchmark¶
Since the routine for calculating the determinant is pretty much the same as for finding the inverse of a matrix, the execution times are similar:
# code to be run in micropython
@timeit
def matrix_det(m):
return np.inv(m)
m = np.array([[1, 2, 3, 4, 5, 6, 7, 8], [0, 5, 6, 4, 5, 6, 4, 5],
[0, 0, 9, 7, 8, 9, 7, 8], [0, 0, 0, 10, 11, 12, 11, 12],
[0, 0, 0, 0, 4, 6, 7, 8], [0, 0, 0, 0, 0, 5, 6, 7],
[0, 0, 0, 0, 0, 0, 7, 6], [0, 0, 0, 0, 0, 0, 0, 2]])
matrix_det(m)
execution time: 294 us
eig¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.eig.html
The eig
function calculates the eigenvalues and the eigenvectors of
a real, symmetric square matrix. If the matrix is not symmetric, a
ValueError
will be raised. The function takes a single argument, and
returns a tuple with the eigenvalues, and eigenvectors. With the help of
the eigenvectors, amongst other things, you can implement sophisticated
stabilisation routines for robots.
# code to be run in micropython
import ulab as np
a = np.array([[1, 2, 1, 4], [2, 5, 3, 5], [1, 3, 6, 1], [4, 5, 1, 7]], dtype=np.uint8)
x, y = np.eig(a)
print('eigenvectors of a:\n', x)
print('\neigenvalues of a:\n', y)
eigenvectors of a:
array([-1.165288, 0.8029362, 5.585626, 13.77673], dtype=float)
eigenvalues of a:
array([[0.8151754, -0.4499267, -0.1643907, 0.3256237],
[0.2211193, 0.7847154, 0.08373602, 0.5729892],
[-0.1340859, -0.3100657, 0.8742685, 0.3486182],
[-0.5182822, -0.2926556, -0.4490192, 0.6664218]], dtype=float)
The same matrix diagonalised with numpy
yields:
# code to be run in CPython
a = array([[1, 2, 1, 4], [2, 5, 3, 5], [1, 3, 6, 1], [4, 5, 1, 7]], dtype=np.uint8)
x, y = eig(a)
print('eigenvectors of a:\n', x)
print('\neigenvalues of a:\n', y)
eigenvectors of a:
[13.77672606 -1.16528837 0.80293655 5.58562576]
eigenvalues of a:
[[ 0.32561419 0.815156 0.44994112 -0.16446602]
[ 0.57300777 0.22113342 -0.78469926 0.08372081]
[ 0.34861093 -0.13401142 0.31007764 0.87427868]
[ 0.66641421 -0.51832581 0.29266348 -0.44897499]]
When comparing results, we should keep two things in mind:
- the eigenvalues and eigenvectors are not necessarily sorted in the same way
- an eigenvector can be multiplied by an arbitrary non-zero scalar, and
it is still an eigenvector with the same eigenvalue. This is why all
signs of the eigenvector belonging to 5.58, and 0.80 are flipped in
ulab
with respect tonumpy
. This difference, however, is of absolutely no consequence.
Computation expenses¶
Since the function is based on Givens rotations and runs till convergence is achieved, or till the maximum number of allowed rotations is exhausted, there is no universal estimate for the time required to find the eigenvalues. However, an order of magnitude can, at least, be guessed based on the measurement below:
# code to be run in micropython
import ulab as np
@timeit
def matrix_eig(a):
return np.eig(a)
a = np.array([[1, 2, 1, 4], [2, 5, 3, 5], [1, 3, 6, 1], [4, 5, 1, 7]], dtype=np.uint8)
matrix_eig(a)
execution time: 111 us
Polynomials¶
polyval¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.polyval.html
polyval takes two arguments, both arrays or other iterables.
# code to be run in micropython
import ulab as np
p = [1, 1, 1, 0]
x = [0, 1, 2, 3, 4]
print('coefficients: ', p)
print('independent values: ', x)
print('\nvalues of p(x): ', np.polyval(p, x))
# the same works with one-dimensional ndarrays
a = np.array(x)
print('\nndarray (a): ', a)
print('value of p(a): ', np.polyval(p, a))
coefficients: [1, 1, 1, 0]
independent values: [0, 1, 2, 3, 4]
values of p(x): array([0.0, 3.0, 14.0, 39.0, 84.0], dtype=float)
ndarray (a): array([0.0, 1.0, 2.0, 3.0, 4.0], dtype=float)
value of p(a): array([0.0, 3.0, 14.0, 39.0, 84.0], dtype=float)
polyfit¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.polyfit.html
polyfit takes two, or three arguments. The last one is the degree of the
polynomial that will be fitted, the last but one is an array or iterable
with the y
(dependent) values, and the first one, an array or
iterable with the x
(independent) values, can be dropped. If that is
the case, x
will be generated in the function, assuming uniform
sampling.
If the length of x
, and y
are not the same, the function raises
a ValueError
.
# code to be run in micropython
import ulab as np
x = np.array([0, 1, 2, 3, 4, 5, 6])
y = np.array([9, 4, 1, 0, 1, 4, 9])
print('independent values:\t', x)
print('dependent values:\t', y)
print('fitted values:\t\t', np.polyfit(x, y, 2))
# the same with missing x
print('\ndependent values:\t', y)
print('fitted values:\t\t', np.polyfit(y, 2))
independent values: array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0], dtype=float)
dependent values: array([9.0, 4.0, 1.0, 0.0, 1.0, 4.0, 9.0], dtype=float)
fitted values: array([1.0, -6.0, 9.000000000000004], dtype=float)
dependent values: array([9.0, 4.0, 1.0, 0.0, 1.0, 4.0, 9.0], dtype=float)
fitted values: array([1.0, -6.0, 9.000000000000004], dtype=float)
Execution time¶
polyfit
is based on the inversion of a matrix (there is more on the
background in https://en.wikipedia.org/wiki/Polynomial_regression), and
it requires the intermediate storage of 2*N*(deg+1)
floats, where
N
is the number of entries in the input array, and deg
is the
fit’s degree. The additional computation costs of the matrix inversion
discussed in inv also apply. The example from above needs
around 150 microseconds to return:
# code to be run in micropython
import ulab as np
@timeit
def time_polyfit(x, y, n):
return np.polyfit(x, y, n)
x = np.array([0, 1, 2, 3, 4, 5, 6])
y = np.array([9, 4, 1, 0, 1, 4, 9])
time_polyfit(x, y, 2)
execution time: 153 us
Fourier transforms¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.fft.ifft.html
fft¶
Since ulab
’s ndarray
does not support complex numbers, the
invocation of the Fourier transform differs from that in numpy
. In
numpy
, you can simply pass an array or iterable to the function, and
it will be treated as a complex array:
# code to be run in CPython
fft.fft([1, 2, 3, 4, 1, 2, 3, 4])
array([20.+0.j, 0.+0.j, -4.+4.j, 0.+0.j, -4.+0.j, 0.+0.j, -4.-4.j,
0.+0.j])
WARNING: The array that is returned is also complex, i.e., the real
and imaginary components are cast together. In ulab
, the real and
imaginary parts are treated separately: you have to pass two
ndarray
s 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 ndarray
s, holding the real and imaginary
parts of the transform separately.
# code to be run in micropython
import ulab as np
from ulab import numerical
from ulab import vector
from ulab import fft
from ulab import linalg
x = numerical.linspace(0, 10, num=1024)
y = vector.sin(x)
z = linalg.zeros(len(x))
a, b = fft.fft(x)
print('real part:\t', a)
print('\nimaginary part:\t', b)
c, d = fft.fft(x, z)
print('\nreal part:\t', c)
print('\nimaginary part:\t', d)
real part: array([5119.996, -5.004663, -5.004798, ..., -5.005482, -5.005643, -5.006577], dtype=float)
imaginary part: array([0.0, 1631.333, 815.659, ..., -543.764, -815.6588, -1631.333], dtype=float)
real part: array([5119.996, -5.004663, -5.004798, ..., -5.005482, -5.005643, -5.006577], dtype=float)
imaginary part: array([0.0, 1631.333, 815.659, ..., -543.764, -815.6588, -1631.333], dtype=float)
ifft¶
The above-mentioned rules apply to the inverse Fourier transform. The
inverse is also normalised by N
, the number of elements, as is
customary in numpy
. With the normalisation, we can ascertain that
the inverse of the transform is equal to the original array.
# code to be run in micropython
import ulab as np
x = np.linspace(0, 10, num=1024)
y = np.sin(x)
a, b = np.fft(y)
print('original vector:\t', y)
y, z = np.ifft(a, b)
# the real part should be equal to y
print('\nreal part of inverse:\t', y)
# the imaginary part should be equal to zero
print('\nimaginary part of inverse:\t', z)
original vector: array([0.0, 0.009775016, 0.0195491, ..., -0.5275068, -0.5357859, -0.5440139], dtype=float)
real part of inverse: array([-2.980232e-08, 0.0097754, 0.0195494, ..., -0.5275064, -0.5357857, -0.5440133], dtype=float)
imaginary part of inverse: array([-2.980232e-08, -1.451171e-07, 3.693752e-08, ..., 6.44871e-08, 9.34986e-08, 2.18336e-07], dtype=float)
Note that unlike in numpy
, the length of the array on which the
Fourier transform is carried out must be a power of 2. If this is not
the case, the function raises a ValueError
exception.
spectrum¶
In addition to the Fourier transform and its inverse, ulab
also
sports a function called spectrum
, which returns the absolute value
of the Fourier transform. This could be used to find the dominant
spectral component in a time series. The arguments are treated in the
same way as in fft
, and ifft
.
# code to be run in micropython
import ulab as np
x = np.linspace(0, 10, num=1024)
y = np.sin(x)
a = np.spectrum(y)
print('original vector:\t', y)
print('\nspectrum:\t', a)
original vector: array([0.0, 0.009775016, 0.0195491, ..., -0.5275068, -0.5357859, -0.5440139], dtype=float)
spectrum: array([187.8641, 315.3125, 347.8804, ..., 84.4587, 347.8803, 315.3124], dtype=float)
As such, spectrum
is really just a shorthand for
np.sqrt(a*a + b*b)
:
# code to be run in micropython
import ulab as np
x = np.linspace(0, 10, num=1024)
y = np.sin(x)
a, b = np.fft(y)
print('\nspectrum calculated the hard way:\t', np.sqrt(a*a + b*b))
a = np.spectrum(y)
print('\nspectrum calculated the lazy way:\t', a)
spectrum calculated the hard way: array([187.8641, 315.3125, 347.8804, ..., 84.4587, 347.8803, 315.3124], dtype=float)
spectrum calculated the lazy way: array([187.8641, 315.3125, 347.8804, ..., 84.4587, 347.8803, 315.3124], dtype=float)
Computation and storage costs¶
RAM¶
The FFT routine of ulab
calculates the transform in place. This
means that beyond reserving space for the two ndarray
s that will
be returned (the computation uses these two as intermediate storage
space), only a handful of temporary variables, all floats or 32-bit
integers, are required.
Speed of FFTs¶
A comment on the speed: a 1024-point transform implemented in python would cost around 90 ms, and 13 ms in assembly, if the code runs on the pyboard, v.1.1. You can gain a factor of four by moving to the D series https://github.com/peterhinch/micropython-fourier/blob/master/README.md#8-performance.
# code to be run in micropython
import ulab as np
x = np.linspace(0, 10, num=1024)
y = np.sin(x)
np.fft(y)
@timeit
def np_fft(y):
return np.fft(y)
a, b = np_fft(y)
execution time: 1985 us
The C implementation runs in less than 2 ms on the pyboard (we have just measured that), and has been reported to run in under 0.8 ms on the D series board. That is an improvement of at least a factor of four.
Calculating FFTs of real signals¶
Now, if you have real signals, and you are really pressed for time, you can still gain a bit on speed without sacrificing anything at all.
If you take the FFT of a real-valued signal, the real part of the transform will be symmetric, while the imaginary part will be anti-symmetric in frequency.
If, on the other hand, the signal is imaginary-valued, then the real part of the transform will be anti-symmetric, and the imaginary part will be symmetric in frequency. These two statements follow from the definition of the Fourier transform.
By combining the two observations above, if you place the first signal, \(y_1(t)\), into the real part, and the second signal, \(y_2(t)\), into the imaginary part of your input vector, i.e., \(y(t) = y_1(t) + iy_2(t)\), and take the Fourier transform of the combined signal, then the Fourier transforms of the two components can be recovered as
:raw-latex:`\begin{eqnarray} Y_1(k) &=& \frac{1}{2}\left(Y(k) + Y^*(N-k)\right) \\ Y_2(k) &=& -\frac{i}{2}\left(Y(k) - Y^*(N-k)\right) \end{eqnarray}` where \(N\) is the length of \(y_1\), and \(Y_1, Y_2\), and \(Y\), respectively, are the Fourier transforms of \(y_1, y_2\), and \(y = y_1 + iy_2\).
Filter routines¶
numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.convolve.html
convolve¶
Returns the discrete, linear convolution of two one-dimensional sequences.
Only the full
mode is supported, and the mode
named parameter is
not accepted. Note that all other modes can be had by slicing a full
result.
# code to be run in micropython
import ulab as np
x = np.array((1,2,3))
y = np.array((1,10,100,1000))
print(np.convolve(x, y))
array([1.0, 12.0, 123.0, 1230.0, 2300.0, 3000.0], dtype=float)
Extending ulab¶
As mentioned at the beginning, ulab
is a set of sub-modules, out of
which one, extra
is explicitly reserved for user code. You should
implement your functions in this sub-module, or sub-modules thereof.
The new functions can easily be added to extra
in a couple of simple
steps. At the C level, the type definition of an ndarray
is as
follows:
typedef struct _ndarray_obj_t {
mp_obj_base_t base;
size_t m, n;
mp_obj_array_t *array;
size_t bytes;
} ndarray_obj_t;
Creating a new ndarray¶
A new ndarray
can be created by calling
ndarray_obj_t *new_ndarray = create_new_ndarray(m, n, typecode);
where m
, and n
are the number of rows and columns, respectively,
and typecode
is one of the values from
enum NDARRAY_TYPE {
NDARRAY_UINT8 = 'B',
NDARRAY_INT8 = 'b',
NDARRAY_UINT16 = 'H',
NDARRAY_INT16 = 'h',
NDARRAY_FLOAT = 'f',
};
or
enum NDARRAY_TYPE {
NDARRAY_UINT8 = 'B',
NDARRAY_INT8 = 'b',
NDARRAY_UINT16 = 'H',
NDARRAY_INT16 = 'h',
NDARRAY_FLOAT = 'd',
};
The ambiguity is caused by the fact that not all platforms implement
double
, and there one has to take float
s. But you haven’t
actually got to be concerned by this, because at the very beginning of
ndarray.h
, this is already taken care of: the pre-processor figures
out, what the float
implementation of the hardware platform is, and
defines the NDARRAY_FLOAT
typecode accordingly. All you have to keep
in mind is that wherever you would use float
or double
, you have
to use mp_float_t
. That type is defined in py/mpconfig.h
of the
micropython code base.
Therefore, a 4-by-5 matrix of type float can be created as
ndarray_obj_t *new_ndarray = create_new_ndarray(4, 5, NDARRAY_FLOAT);
This function also fills in the ndarray
structure’s m
, n
,
and bytes
members, as well initialises the array
member with
zeros.
Alternatively, a one-to-one copy of an ndarray
can be gotten by
calling
mp_obj_t copy_of_input_object = ndarray_copy(object_in);
Note, however, that this function takes an input object of type
mp_obj_t
, and returns a copy of type mp_obj_t
, i.e., something
that can be taken from, and can immediately be returned to the
interpreter. If you want to work on the data in the copy, you still have
to create a pointer to it
ndarray_obj_t *ndarray = MP_OBJ_TO_PTR(copy_of_input_object);
The values stored in array
can be modified or retrieved by accessing
array->items
. Note that array->items
is a void pointer,
therefore, it must be cast before trying to access the elements.
array
has at least two useful members. One of them is len
, which
is the number of elements that the array holds, while the second one is
the typecode
that we passed to create_new_ndarray
earlier.
Thus, staying with our example of a 4-by-5 float matrix, we can loop through all entries as
mp_float_t *items = (mp_float_t *)new_ndarray->array->items;
mp_float_t item;
for(size_t i=0; i < new_ndarray->array->len; i++) {
item = items[i];
// do something with item...
}
or, since the data are stored in the pointer in a C-like fashion, as
mp_float_t *items = (mp_float_t *)new_ndarray->array->items;
mp_float_t item;
for(size_t m=0; m < new_ndarray->m; m++) { // loop through the rows
for(size_t n=0; n < new_ndarray->n; n++) { // loop through the columns
item = items[m*new_ndarray->n+n]; // get the (m,n) entry
// do something with item...
}
}
Accessing data in the ndarray¶
We have already seen, how the entries of an array can be accessed. If the object in question comes from the user (i.e., via the micropython interface), we can get a pointer to it by calling
ndarray_obj_t *ndarray = MP_OBJ_TO_PTR(object_in);
Once the pointer is at our disposal, we can get a pointer to the underlying numerical array as discussed earlier.
If you need to find out the typecode of the array, you can get it by
accessing the typecode
member of array
, i.e.,
ndarray->array->typecode
should be equal to B
, b
, H
, h
, or f
. The size of a
single item is returned by the function
mp_binary_get_size('@', ndarray->array->typecode, NULL)
. This number
is equal to 1, if the typecode is B
, or b
, 2, if the typecode is
H
, or h
, 4, if the typecode is f
, and 8 for d
.
Alternatively, the size can be found out by dividing ndarray->bytes
with the product of m
, and n
, i.e.,
ndarray->bytes/(ndarray->m*ndarray*n)
is equal to mp_binary_get_size('@', ndarray->array->typecode, NULL)
.
Making certain that we have an ndarray¶
A number of operations make sense for ndarray
s only, therefore,
before doing any heavy work on the data, it might be reasonable to
check, whether the input argument is of the proper type. This you do by
evaluating
mp_obj_is_type(object_in, &ulab_ndarray_type)
which should return true
.
Boilerplate of sorts¶
To summarise the contents of the previous three sections, here is a
useless function that prints out the size (m
, and n
) of an
array, creates a copy of the input, and fills up the resulting matrix
with 13.
mp_obj_t useless_function(mp_obj_t object_in) {
if(!mp_obj_is_type(object_in, &ulab_ndarray_type)) {
mp_raise_TypeError("useless_function takes only ndarray arguments");
}
mp_obj_t object_out = ndarray_copy(object_int);
ndarray_obj_t *ndarray = MP_OBJ_TO_PTR(object_out);
printf("\nsize (m): %ld, size (n): %ld\n", ndarray->m, ndarray->n);
printf("\nlength (len): %ld, typecode: %d\n", ndarray->array->len, ndarray->array->typecode);
if(ndarray->array->typecode == NDARRAY_UINT8) {
// cast the pointer to the items
uint8_t *items = (uint8_t *)ndarray->array->items;
// retrieve the length of the array, and loop over the elements
for(size_t i=0; i < ndarray->array->len; i++) items[i] = 13;
} else if(ndarray->array->typecode == NDARRAY_INT8) {
int8_t *items = (int8_t *)ndarray->array->items;
for(size_t i=0; i < ndarray->array->len; i++) items[i] = 13;
} else if(ndarray->array->typecode == NDARRAY_UINT16) {
uint16_t *items = (uint16_t *)ndarray->array->items;
for(size_t i=0; i < ndarray->array->len; i++) items[i] = 13;
} else if(ndarray->array->typecode == NDARRAY_INT16) {
int16_t *items = (int16_t *)ndarray->array->items;
for(size_t i=0; i < ndarray->array->len; i++) items[i] = 13;
} else {
float *items = (mp_float_t *)ndarray->array->items;
for(size_t i=0; i < ndarray->array->len; i++) items[i] = 13;
}
return object_out;
}
If, on the other hand, you want to create an ndarray
from scratch,
and return that, you could work along the following lines:
mp_obj_t useless_function(mp_obj_t object_in) {
uint16_t m = mp_obj_get_int(object_in);
ndarray_obj_t *ndarray = create_new_ndarray(1, m, NDARRAY_UINT8);
uint8_t *items = (uint8_t *)ndarray->array->items;
// do something with the array's entries
// ...
// and at the very end, return an mp_object_t
return MP_PTR_TO_OBJ(ndarray);
}
Once the function implementation is done, you should add the function
object to the globals dictionary of the extra
sub-module as
...
MP_DEFINE_CONST_FUN_OBJ_1(useless_function_obj, userless_function);
...
STATIC const mp_map_elem_t extra_globals_table[] = {
...
{ MP_OBJ_NEW_QSTR(MP_QSTR_useless), (mp_obj_t)&useless_function_obj },
...
};
The exact form of the function object definition
MP_DEFINE_CONST_FUN_OBJ_1(useless_function_obj, userless_function);
depends naturally on what exactly you implemented, i.e., how many arguments the function takes, whether only positional arguments allowed and so on. For a thorough discussion on how function objects have to be defined, see, e.g., the user module programming manual.
And with that, you are done. You can simply call the compiler as
make BOARD=PYBV11 USER_C_MODULES=../../../ulab all
and the rest is taken care of.
In the boilerplate above, we cast the pointer to array->items
to the
required type. There are certain operations, however, when you do not
need the casting. If you do not want to change the array’s values, only
their position within the array, you can get away with copying the
memory content, regardless the type. A good example for such a scenario
is the transpose function in
https://github.com/v923z/micropython-ulab/blob/master/code/linalg.c.