Merge pull request #380 from v923z/solve-docs
add docs for solve_triangular
This commit is contained in:
Zoltán Vörös
GitHub
commit
3c39995349
5 changed files with 573 additions and 14 deletions
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@ -27,7 +27,7 @@ copyright = '2019-2021, Zoltán Vörös and contributors'
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author = 'Zoltán Vörös'
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# The full version, including alpha/beta/rc tags
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release = '2.6.0'
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release = '2.7.0'
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# -- General configuration ---------------------------------------------------
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@ -22,6 +22,7 @@ Welcome to the ulab book!
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numpy-universal
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numpy-fft
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numpy-linalg
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scipy-linalg
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scipy-optimize
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scipy-signal
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scipy-special
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120
docs/manual/source/scipy-linalg.rst
Normal file
120
docs/manual/source/scipy-linalg.rst
Normal file
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@ -0,0 +1,120 @@
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linalg
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======
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``scipy``\ ’s ``linalg`` module contains a single function,
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``solve_triangular``, which can be called by prepending it by
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``scipy.linalg.``.
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1. `scipy.linalg.solve_triangular <#solve_triangular>`__
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solve_triangular
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----------------
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``scipy``:
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https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve_triangular.html
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Solve the linear equation
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:raw-latex:`\begin{equation}
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\mathbf{a}\cdot\mathbf{x} = \mathbf{b}
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\end{equation}`
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with the assumption that :math:`\mathbf{a}` is a triangular matrix. The
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two position arguments are :math:`\mathbf{a}`, and :math:`\mathbf{b}`,
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and the optional keyword argument is ``lower`` with a default value of
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``False``. ``lower`` determines, whether data are taken from the lower,
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or upper triangle of :math:`\mathbf{a}`.
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Note that :math:`\mathbf{a}` itself does not have to be a triangular
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matrix: if it is not, then the values are simply taken to be 0 in the
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upper or lower triangle, as dictated by ``lower``. However,
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:math:`\mathbf{a}\cdot\mathbf{x}` will yield :math:`\mathbf{b}` only,
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when :math:`\mathbf{a}` is triangular. You should keep this in mind,
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when trying to establish the validity of the solution by back
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substitution.
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.. code::
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# code to be run in micropython
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from ulab import numpy as np
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from ulab import scipy as spy
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a = np.array([[3, 0, 0, 0], [2, 1, 0, 0], [1, 0, 1, 0], [1, 2, 1, 8]])
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b = np.array([4, 2, 4, 2])
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print('a:\n')
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print(a)
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print('\nb: ', b)
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x = spy.linalg.solve_triangular(a, b, lower=True)
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print('='*20)
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print('x: ', x)
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print('\ndot(a, x): ', np.dot(a, x))
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.. parsed-literal::
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a:
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array([[3.0, 0.0, 0.0, 0.0],
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[2.0, 1.0, 0.0, 0.0],
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[1.0, 0.0, 1.0, 0.0],
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[1.0, 2.0, 1.0, 8.0]], dtype=float64)
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b: array([4.0, 2.0, 4.0, 2.0], dtype=float64)
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====================
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x: array([1.333333333333333, -0.6666666666666665, 2.666666666666667, -0.08333333333333337], dtype=float64)
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dot(a, x): array([4.0, 2.0, 4.0, 2.0], dtype=float64)
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With get the same solution, :math:`\mathbf{x}`, with the following
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matrix, but the dot product of :math:`\mathbf{a}`, and
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:math:`\mathbf{x}` is no longer :math:`\mathbf{b}`:
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.. code::
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# code to be run in micropython
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from ulab import numpy as np
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from ulab import scipy as spy
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a = np.array([[3, 2, 1, 0], [2, 1, 0, 1], [1, 0, 1, 4], [1, 2, 1, 8]])
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b = np.array([4, 2, 4, 2])
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print('a:\n')
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print(a)
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print('\nb: ', b)
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x = spy.linalg.solve_triangular(a, b, lower=True)
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print('='*20)
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print('x: ', x)
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print('\ndot(a, x): ', np.dot(a, x))
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.. parsed-literal::
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a:
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array([[3.0, 2.0, 1.0, 0.0],
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[2.0, 1.0, 0.0, 1.0],
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[1.0, 0.0, 1.0, 4.0],
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[1.0, 2.0, 1.0, 8.0]], dtype=float64)
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b: array([4.0, 2.0, 4.0, 2.0], dtype=float64)
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====================
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x: array([1.333333333333333, -0.6666666666666665, 2.666666666666667, -0.08333333333333337], dtype=float64)
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dot(a, x): array([5.333333333333334, 1.916666666666666, 3.666666666666667, 2.0], dtype=float64)
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.. code::
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# code to be run in CPython
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437
docs/scipy-linalg.ipynb
Normal file
437
docs/scipy-linalg.ipynb
Normal file
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@ -0,0 +1,437 @@
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {
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"ExecuteTime": {
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"end_time": "2021-01-13T18:54:58.722373Z",
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"start_time": "2021-01-13T18:54:57.178438Z"
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}
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Populating the interactive namespace from numpy and matplotlib\n"
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]
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}
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],
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"source": [
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"%pylab inline"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Notebook magic"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {
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"ExecuteTime": {
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"end_time": "2021-05-09T05:37:22.600510Z",
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"start_time": "2021-05-09T05:37:22.595924Z"
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}
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},
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"outputs": [],
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"source": [
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"from IPython.core.magic import Magics, magics_class, line_cell_magic\n",
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"from IPython.core.magic import cell_magic, register_cell_magic, register_line_magic\n",
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"from IPython.core.magic_arguments import argument, magic_arguments, parse_argstring\n",
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"import subprocess\n",
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"import os"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"ExecuteTime": {
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||||
"end_time": "2021-05-09T05:37:26.429136Z",
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"start_time": "2021-05-09T05:37:26.403191Z"
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}
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},
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"outputs": [],
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"source": [
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"@magics_class\n",
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"class PyboardMagic(Magics):\n",
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" @cell_magic\n",
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" @magic_arguments()\n",
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" @argument('-skip')\n",
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" @argument('-unix')\n",
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" @argument('-pyboard')\n",
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" @argument('-file')\n",
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" @argument('-data')\n",
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" @argument('-time')\n",
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" @argument('-memory')\n",
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" def micropython(self, line='', cell=None):\n",
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" args = parse_argstring(self.micropython, line)\n",
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" if args.skip: # doesn't care about the cell's content\n",
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" print('skipped execution')\n",
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" return None # do not parse the rest\n",
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" if args.unix: # tests the code on the unix port. Note that this works on unix only\n",
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" with open('/dev/shm/micropython.py', 'w') as fout:\n",
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" fout.write(cell)\n",
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" proc = subprocess.Popen([\"../../micropython/ports/unix/micropython\", \"/dev/shm/micropython.py\"], \n",
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" stdout=subprocess.PIPE, stderr=subprocess.PIPE)\n",
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" print(proc.stdout.read().decode(\"utf-8\"))\n",
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" print(proc.stderr.read().decode(\"utf-8\"))\n",
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" return None\n",
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" if args.file: # can be used to copy the cell content onto the pyboard's flash\n",
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" spaces = \" \"\n",
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" try:\n",
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" with open(args.file, 'w') as fout:\n",
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" fout.write(cell.replace('\\t', spaces))\n",
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" printf('written cell to {}'.format(args.file))\n",
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" except:\n",
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" print('Failed to write to disc!')\n",
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" return None # do not parse the rest\n",
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" if args.data: # can be used to load data from the pyboard directly into kernel space\n",
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" message = pyb.exec(cell)\n",
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" if len(message) == 0:\n",
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" print('pyboard >>>')\n",
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" else:\n",
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" print(message.decode('utf-8'))\n",
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" # register new variable in user namespace\n",
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" self.shell.user_ns[args.data] = string_to_matrix(message.decode(\"utf-8\"))\n",
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" \n",
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" if args.time: # measures the time of executions\n",
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" pyb.exec('import utime')\n",
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" message = pyb.exec('t = utime.ticks_us()\\n' + cell + '\\ndelta = utime.ticks_diff(utime.ticks_us(), t)' + \n",
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" \"\\nprint('execution time: {:d} us'.format(delta))\")\n",
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" print(message.decode('utf-8'))\n",
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" \n",
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" if args.memory: # prints out memory information \n",
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" message = pyb.exec('from micropython import mem_info\\nprint(mem_info())\\n')\n",
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" print(\"memory before execution:\\n========================\\n\", message.decode('utf-8'))\n",
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" message = pyb.exec(cell)\n",
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" print(\">>> \", message.decode('utf-8'))\n",
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" message = pyb.exec('print(mem_info())')\n",
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" print(\"memory after execution:\\n========================\\n\", message.decode('utf-8'))\n",
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"\n",
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" if args.pyboard:\n",
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" message = pyb.exec(cell)\n",
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" print(message.decode('utf-8'))\n",
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"\n",
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"ip = get_ipython()\n",
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"ip.register_magics(PyboardMagic)"
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]
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||||
},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## pyboard"
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]
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||||
},
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{
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"cell_type": "code",
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"execution_count": 57,
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||||
"metadata": {
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||||
"ExecuteTime": {
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||||
"end_time": "2020-05-07T07:35:35.126401Z",
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||||
"start_time": "2020-05-07T07:35:35.105824Z"
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||||
}
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},
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"outputs": [],
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"source": [
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"import pyboard\n",
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"pyb = pyboard.Pyboard('/dev/ttyACM0')\n",
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"pyb.enter_raw_repl()"
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]
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||||
},
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||||
{
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||||
"cell_type": "code",
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||||
"execution_count": 9,
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||||
"metadata": {
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||||
"ExecuteTime": {
|
||||
"end_time": "2020-05-19T19:11:18.145548Z",
|
||||
"start_time": "2020-05-19T19:11:18.137468Z"
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||||
}
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||||
},
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||||
"outputs": [],
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"source": [
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"pyb.exit_raw_repl()\n",
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"pyb.close()"
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||||
]
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||||
},
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||||
{
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||||
"cell_type": "code",
|
||||
"execution_count": 58,
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||||
"metadata": {
|
||||
"ExecuteTime": {
|
||||
"end_time": "2020-05-07T07:35:38.725924Z",
|
||||
"start_time": "2020-05-07T07:35:38.645488Z"
|
||||
}
|
||||
},
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||||
"outputs": [
|
||||
{
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||||
"name": "stdout",
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||||
"output_type": "stream",
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||||
"text": [
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"\n"
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||||
]
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||||
}
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||||
],
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"source": [
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"%%micropython -pyboard 1\n",
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"\n",
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"import utime\n",
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"import ulab as np\n",
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"\n",
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"def timeit(n=1000):\n",
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||||
" def wrapper(f, *args, **kwargs):\n",
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" func_name = str(f).split(' ')[1]\n",
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||||
" def new_func(*args, **kwargs):\n",
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||||
" run_times = np.zeros(n, dtype=np.uint16)\n",
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||||
" for i in range(n):\n",
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||||
" t = utime.ticks_us()\n",
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||||
" result = f(*args, **kwargs)\n",
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||||
" run_times[i] = utime.ticks_diff(utime.ticks_us(), t)\n",
|
||||
" print('{}() execution times based on {} cycles'.format(func_name, n, (delta2-delta1)/n))\n",
|
||||
" print('\\tbest: %d us'%np.min(run_times))\n",
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||||
" print('\\tworst: %d us'%np.max(run_times))\n",
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||||
" print('\\taverage: %d us'%np.mean(run_times))\n",
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||||
" print('\\tdeviation: +/-%.3f us'%np.std(run_times)) \n",
|
||||
" return result\n",
|
||||
" return new_func\n",
|
||||
" return wrapper\n",
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"\n",
|
||||
"def timeit(f, *args, **kwargs):\n",
|
||||
" func_name = str(f).split(' ')[1]\n",
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||||
" def new_func(*args, **kwargs):\n",
|
||||
" t = utime.ticks_us()\n",
|
||||
" result = f(*args, **kwargs)\n",
|
||||
" print('execution time: ', utime.ticks_diff(utime.ticks_us(), t), ' us')\n",
|
||||
" return result\n",
|
||||
" return new_func"
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||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"__END_OF_DEFS__"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# linalg\n",
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||||
"\n",
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"`scipy`'s `linalg` module contains a single function, `solve_triangular`, which can be called by prepending it by `scipy.linalg.`.\n",
|
||||
"\n",
|
||||
"1. [scipy.linalg.solve_triangular](#solve_triangular)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## solve_triangular\n",
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||||
"\n",
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||||
"`scipy`: https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve_triangular.html \n",
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||||
"\n",
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||||
"Solve the linear equation \n",
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"\n",
|
||||
"\\begin{equation}\n",
|
||||
"\\mathbf{a}\\cdot\\mathbf{x} = \\mathbf{b}\n",
|
||||
"\\end{equation}\n",
|
||||
"\n",
|
||||
"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 $\\mathbf{a}$. \n",
|
||||
"\n",
|
||||
"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."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 14,
|
||||
"metadata": {
|
||||
"ExecuteTime": {
|
||||
"end_time": "2021-05-09T05:56:57.449996Z",
|
||||
"start_time": "2021-05-09T05:56:57.422515Z"
|
||||
}
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"a:\n",
|
||||
"\n",
|
||||
"array([[3.0, 0.0, 0.0, 0.0],\n",
|
||||
" [2.0, 1.0, 0.0, 0.0],\n",
|
||||
" [1.0, 0.0, 1.0, 0.0],\n",
|
||||
" [1.0, 2.0, 1.0, 8.0]], dtype=float64)\n",
|
||||
"\n",
|
||||
"b: array([4.0, 2.0, 4.0, 2.0], dtype=float64)\n",
|
||||
"====================\n",
|
||||
"x: array([1.333333333333333, -0.6666666666666665, 2.666666666666667, -0.08333333333333337], dtype=float64)\n",
|
||||
"\n",
|
||||
"dot(a, x): array([4.0, 2.0, 4.0, 2.0], dtype=float64)\n",
|
||||
"\n",
|
||||
"\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"%%micropython -unix 1\n",
|
||||
"\n",
|
||||
"from ulab import numpy as np\n",
|
||||
"from ulab import scipy as spy\n",
|
||||
"\n",
|
||||
"a = np.array([[3, 0, 0, 0], [2, 1, 0, 0], [1, 0, 1, 0], [1, 2, 1, 8]])\n",
|
||||
"b = np.array([4, 2, 4, 2])\n",
|
||||
"\n",
|
||||
"print('a:\\n')\n",
|
||||
"print(a)\n",
|
||||
"print('\\nb: ', b)\n",
|
||||
"\n",
|
||||
"x = spy.linalg.solve_triangular(a, b, lower=True)\n",
|
||||
"\n",
|
||||
"print('='*20)\n",
|
||||
"print('x: ', x)\n",
|
||||
"print('\\ndot(a, x): ', np.dot(a, x))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"With get the same solution, $\\mathbf{x}$, with the following matrix, but the dot product of $\\mathbf{a}$, and $\\mathbf{x}$ is no longer $\\mathbf{b}$:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 17,
|
||||
"metadata": {
|
||||
"ExecuteTime": {
|
||||
"end_time": "2021-05-09T06:03:30.853054Z",
|
||||
"start_time": "2021-05-09T06:03:30.841500Z"
|
||||
}
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"a:\n",
|
||||
"\n",
|
||||
"array([[3.0, 2.0, 1.0, 0.0],\n",
|
||||
" [2.0, 1.0, 0.0, 1.0],\n",
|
||||
" [1.0, 0.0, 1.0, 4.0],\n",
|
||||
" [1.0, 2.0, 1.0, 8.0]], dtype=float64)\n",
|
||||
"\n",
|
||||
"b: array([4.0, 2.0, 4.0, 2.0], dtype=float64)\n",
|
||||
"====================\n",
|
||||
"x: array([1.333333333333333, -0.6666666666666665, 2.666666666666667, -0.08333333333333337], dtype=float64)\n",
|
||||
"\n",
|
||||
"dot(a, x): array([5.333333333333334, 1.916666666666666, 3.666666666666667, 2.0], dtype=float64)\n",
|
||||
"\n",
|
||||
"\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"%%micropython -unix 1\n",
|
||||
"\n",
|
||||
"from ulab import numpy as np\n",
|
||||
"from ulab import scipy as spy\n",
|
||||
"\n",
|
||||
"a = np.array([[3, 2, 1, 0], [2, 1, 0, 1], [1, 0, 1, 4], [1, 2, 1, 8]])\n",
|
||||
"b = np.array([4, 2, 4, 2])\n",
|
||||
"\n",
|
||||
"print('a:\\n')\n",
|
||||
"print(a)\n",
|
||||
"print('\\nb: ', b)\n",
|
||||
"\n",
|
||||
"x = spy.linalg.solve_triangular(a, b, lower=True)\n",
|
||||
"\n",
|
||||
"print('='*20)\n",
|
||||
"print('x: ', x)\n",
|
||||
"print('\\ndot(a, x): ', np.dot(a, x))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.8.5"
|
||||
},
|
||||
"toc": {
|
||||
"base_numbering": 1,
|
||||
"nav_menu": {},
|
||||
"number_sections": true,
|
||||
"sideBar": true,
|
||||
"skip_h1_title": false,
|
||||
"title_cell": "Table of Contents",
|
||||
"title_sidebar": "Contents",
|
||||
"toc_cell": false,
|
||||
"toc_position": {
|
||||
"height": "calc(100% - 180px)",
|
||||
"left": "10px",
|
||||
"top": "150px",
|
||||
"width": "382.797px"
|
||||
},
|
||||
"toc_section_display": true,
|
||||
"toc_window_display": true
|
||||
},
|
||||
"varInspector": {
|
||||
"cols": {
|
||||
"lenName": 16,
|
||||
"lenType": 16,
|
||||
"lenVar": 40
|
||||
},
|
||||
"kernels_config": {
|
||||
"python": {
|
||||
"delete_cmd_postfix": "",
|
||||
"delete_cmd_prefix": "del ",
|
||||
"library": "var_list.py",
|
||||
"varRefreshCmd": "print(var_dic_list())"
|
||||
},
|
||||
"r": {
|
||||
"delete_cmd_postfix": ") ",
|
||||
"delete_cmd_prefix": "rm(",
|
||||
"library": "var_list.r",
|
||||
"varRefreshCmd": "cat(var_dic_list()) "
|
||||
}
|
||||
},
|
||||
"types_to_exclude": [
|
||||
"module",
|
||||
"function",
|
||||
"builtin_function_or_method",
|
||||
"instance",
|
||||
"_Feature"
|
||||
],
|
||||
"window_display": false
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 4
|
||||
}
|
||||
|
|
@ -14,11 +14,11 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"execution_count": 1,
|
||||
"metadata": {
|
||||
"ExecuteTime": {
|
||||
"end_time": "2021-03-23T16:27:42.273017Z",
|
||||
"start_time": "2021-03-23T16:27:42.261057Z"
|
||||
"end_time": "2021-05-09T06:05:50.855683Z",
|
||||
"start_time": "2021-05-09T06:05:50.838482Z"
|
||||
}
|
||||
},
|
||||
"outputs": [
|
||||
|
|
@ -61,7 +61,7 @@
|
|||
"author = 'Zoltán Vörös'\n",
|
||||
"\n",
|
||||
"# The full version, including alpha/beta/rc tags\n",
|
||||
"release = '2.6.0'\n",
|
||||
"release = '2.7.0'\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"# -- General configuration ---------------------------------------------------\n",
|
||||
|
|
@ -151,8 +151,8 @@
|
|||
"execution_count": 2,
|
||||
"metadata": {
|
||||
"ExecuteTime": {
|
||||
"end_time": "2021-03-04T18:23:33.312441Z",
|
||||
"start_time": "2021-03-04T18:23:33.298784Z"
|
||||
"end_time": "2021-05-09T06:06:28.491158Z",
|
||||
"start_time": "2021-05-09T06:06:28.477127Z"
|
||||
}
|
||||
},
|
||||
"outputs": [
|
||||
|
|
@ -190,10 +190,11 @@
|
|||
" numpy-universal\n",
|
||||
" numpy-fft\n",
|
||||
" numpy-linalg\n",
|
||||
" scipy-linalg\n",
|
||||
" scipy-optimize\n",
|
||||
" scipy-signal\n",
|
||||
" scipy-special\n",
|
||||
" ulab-utils\n",
|
||||
" ulab-utils\n",
|
||||
" ulab-programming\n",
|
||||
"\n",
|
||||
"Indices and tables\n",
|
||||
|
|
@ -213,11 +214,11 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"execution_count": 3,
|
||||
"metadata": {
|
||||
"ExecuteTime": {
|
||||
"end_time": "2021-03-23T16:27:30.647984Z",
|
||||
"start_time": "2021-03-23T16:27:28.144633Z"
|
||||
"end_time": "2021-05-09T06:06:35.839960Z",
|
||||
"start_time": "2021-05-09T06:06:33.112686Z"
|
||||
}
|
||||
},
|
||||
"outputs": [],
|
||||
|
|
@ -254,11 +255,11 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"execution_count": 4,
|
||||
"metadata": {
|
||||
"ExecuteTime": {
|
||||
"end_time": "2021-03-23T16:27:35.399254Z",
|
||||
"start_time": "2021-03-23T16:27:30.656938Z"
|
||||
"end_time": "2021-05-09T06:07:00.143083Z",
|
||||
"start_time": "2021-05-09T06:06:56.719659Z"
|
||||
}
|
||||
},
|
||||
"outputs": [],
|
||||
|
|
|
|||
Loading…
Reference in a new issue