{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Symbolic solution of ODEs with sympy\n",
"Intro to sympy variables in previous notebook."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"import sympy as sym\n",
"sym.init_printing() # for LaTeX formatted output\n",
"\n",
"import numpy as np\n",
"\n",
"import matplotlib as mpl\n",
"import matplotlib.pyplot as plt"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"# Following is an Ipython magic command that puts figures in the notebook.\n",
"%matplotlib notebook\n",
"\n",
"# M.L. modification of matplotlib defaults\n",
"# Changes can also be put in matplotlibrc file, \n",
"# or effected using mpl.rcParams[]\n",
"mpl.style.use('classic')\n",
"plt.rc('figure', figsize = (6, 4.5)) # Reduces overall size of figures\n",
"plt.rc('axes', labelsize=16, titlesize=14)\n",
"plt.rc('figure', autolayout = True) # Adjusts supblot parameters for new size"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"x = sym.symbols('x') \n",
"f, g = sym.symbols('f g', cls=sym.Function)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle f{\\left(x \\right)}$"
],
"text/plain": [
"f(x)"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f(x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Define the differential equation as a sym.Eq()"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle f{\\left(x \\right)} - 2 \\frac{d}{d x} f{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)} = \\sin{\\left(x \\right)}$"
],
"text/plain": [
" 2 \n",
" d d \n",
"f(x) - 2⋅──(f(x)) + ───(f(x)) = sin(x)\n",
" dx 2 \n",
" dx "
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"diffeq = sym.Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sym.sin(x))\n",
"diffeq"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Solve differential equation"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle f{\\left(x \\right)} = \\left(C_{1} + C_{2} x\\right) e^{x} + \\frac{\\cos{\\left(x \\right)}}{2}$"
],
"text/plain": [
" x cos(x)\n",
"f(x) = (C₁ + C₂⋅x)⋅ℯ + ──────\n",
" 2 "
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"soln = sym.dsolve(diffeq,f(x))\n",
"soln"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Boundary conditions\n",
"This isn't implemented yet in `dsolve` -- it's on the \"to do\" list
\n",
"For now, solve for contants on your own. For example, if\n",
"$$ f(0) = 1\\quad\\mbox{and}\\quad\\left.\\frac{df}{dx}\\right|_0 = 0, $$\n",
"solve the following equations:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left\\{ C_{1} : \\frac{1}{2}, \\ C_{2} : - \\frac{1}{2}\\right\\}$"
],
"text/plain": [
"{C₁: 1/2, C₂: -1/2}"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"constants = sym.solve([soln.rhs.subs(x,0) - 1, soln.rhs.diff(x,1).subs(x,0)- 0])\n",
"constants"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle f{\\left(x \\right)} = \\left(\\frac{1}{2} - \\frac{x}{2}\\right) e^{x} + \\frac{\\cos{\\left(x \\right)}}{2}$"
],
"text/plain": [
" ⎛1 x⎞ x cos(x)\n",
"f(x) = ⎜─ - ─⎟⋅ℯ + ──────\n",
" ⎝2 2⎠ 2 "
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"C1, C2 = sym.symbols('C1,C2')\n",
"soln = soln.subs(constants)\n",
"soln"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Convert soln to python function for numerical evaluation/plotting\n",
"I'm not sure why I had to specify the modulue for conversion of sympy functions.
\n",
"See http://docs.sympy.org/latest/modules/utilities/lambdify.html
\n",
"In previous examples, sympy figured out a good module \"on its own.\" "
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [],
"source": [
"func = sym.lambdify(x,soln.rhs,'numpy')"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
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"/* global mpl */\n",
"window.mpl = {};\n",
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" if (typeof WebSocket !== 'undefined') {\n",
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" } else if (typeof MozWebSocket !== 'undefined') {\n",
" return MozWebSocket;\n",
" } else {\n",
" alert(\n",
" 'Your browser does not have WebSocket support. ' +\n",
" 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
" 'Firefox 4 and 5 are also supported but you ' +\n",
" 'have to enable WebSockets in about:config.'\n",
" );\n",
" }\n",
"};\n",
"\n",
"mpl.figure = function (figure_id, websocket, ondownload, parent_element) {\n",
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" if (warnings) {\n",
" warnings.style.display = 'block';\n",
" warnings.textContent =\n",
" 'This browser does not support binary websocket messages. ' +\n",
" 'Performance may be slow.';\n",
" }\n",
" }\n",
"\n",
" this.imageObj = new Image();\n",
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"};\n",
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"mpl.figure.prototype._init_header = function () {\n",
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" titlebar.classList =\n",
" 'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';\n",
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"};\n",
"\n",
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"\n",
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"\n",
"mpl.figure.prototype._init_canvas = function () {\n",
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"\n",
" var canvas_div = (this.canvas_div = document.createElement('div'));\n",
" canvas_div.setAttribute(\n",
" 'style',\n",
" 'border: 1px solid #ddd;' +\n",
" 'box-sizing: content-box;' +\n",
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" 'outline: 0;' +\n",
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" 'resize: both;'\n",
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"\n",
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" return function (event) {\n",
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" }\n",
"\n",
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" 'keydown',\n",
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" var canvas = (this.canvas = document.createElement('canvas'));\n",
" canvas.classList.add('mpl-canvas');\n",
" canvas.setAttribute('style', 'box-sizing: content-box;');\n",
"\n",
" this.context = canvas.getContext('2d');\n",
"\n",
" var backingStore =\n",
" this.context.backingStorePixelRatio ||\n",
" this.context.webkitBackingStorePixelRatio ||\n",
" this.context.mozBackingStorePixelRatio ||\n",
" this.context.msBackingStorePixelRatio ||\n",
" this.context.oBackingStorePixelRatio ||\n",
" this.context.backingStorePixelRatio ||\n",
" 1;\n",
"\n",
" mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
"\n",
" var rubberband_canvas = (this.rubberband_canvas = document.createElement(\n",
" 'canvas'\n",
" ));\n",
" rubberband_canvas.setAttribute(\n",
" 'style',\n",
" 'box-sizing: content-box; position: absolute; left: 0; top: 0; z-index: 1;'\n",
" );\n",
"\n",
" var resizeObserver = new ResizeObserver(function (entries) {\n",
" var nentries = entries.length;\n",
" for (var i = 0; i < nentries; i++) {\n",
" var entry = entries[i];\n",
" var width, height;\n",
" if (entry.contentBoxSize) {\n",
" width = entry.contentBoxSize.inlineSize;\n",
" height = entry.contentBoxSize.blockSize;\n",
" } else {\n",
" width = entry.contentRect.width;\n",
" height = entry.contentRect.height;\n",
" }\n",
"\n",
" // Keep the size of the canvas and rubber band canvas in sync with\n",
" // the canvas container.\n",
" canvas.setAttribute('width', width * mpl.ratio);\n",
" canvas.setAttribute('height', height * mpl.ratio);\n",
" canvas.setAttribute(\n",
" 'style',\n",
" 'width: ' + width + 'px; height: ' + height + 'px;'\n",
" );\n",
"\n",
" rubberband_canvas.setAttribute('width', width);\n",
" rubberband_canvas.setAttribute('height', height);\n",
"\n",
" // And update the size in Python. We ignore the initial 0/0 size\n",
" // that occurs as the element is placed into the DOM, which should\n",
" // otherwise not happen due to the minimum size styling.\n",
" if (width != 0 && height != 0) {\n",
" fig.request_resize(width, height);\n",
" }\n",
" }\n",
" });\n",
" resizeObserver.observe(canvas_div);\n",
"\n",
" function on_mouse_event_closure(name) {\n",
" return function (event) {\n",
" return fig.mouse_event(event, name);\n",
" };\n",
" }\n",
"\n",
" rubberband_canvas.addEventListener(\n",
" 'mousedown',\n",
" on_mouse_event_closure('button_press')\n",
" );\n",
" rubberband_canvas.addEventListener(\n",
" 'mouseup',\n",
" on_mouse_event_closure('button_release')\n",
" );\n",
" // Throttle sequential mouse events to 1 every 20ms.\n",
" rubberband_canvas.addEventListener(\n",
" 'mousemove',\n",
" on_mouse_event_closure('motion_notify')\n",
" );\n",
"\n",
" rubberband_canvas.addEventListener(\n",
" 'mouseenter',\n",
" on_mouse_event_closure('figure_enter')\n",
" );\n",
" rubberband_canvas.addEventListener(\n",
" 'mouseleave',\n",
" on_mouse_event_closure('figure_leave')\n",
" );\n",
"\n",
" canvas_div.addEventListener('wheel', function (event) {\n",
" if (event.deltaY < 0) {\n",
" event.step = 1;\n",
" } else {\n",
" event.step = -1;\n",
" }\n",
" on_mouse_event_closure('scroll')(event);\n",
" });\n",
"\n",
" canvas_div.appendChild(canvas);\n",
" canvas_div.appendChild(rubberband_canvas);\n",
"\n",
" this.rubberband_context = rubberband_canvas.getContext('2d');\n",
" this.rubberband_context.strokeStyle = '#000000';\n",
"\n",
" this._resize_canvas = function (width, height, forward) {\n",
" if (forward) {\n",
" canvas_div.style.width = width + 'px';\n",
" canvas_div.style.height = height + 'px';\n",
" }\n",
" };\n",
"\n",
" // Disable right mouse context menu.\n",
" this.rubberband_canvas.addEventListener('contextmenu', function (_e) {\n",
" return false;\n",
" });\n",
"\n",
" function set_focus() {\n",
" canvas.focus();\n",
" canvas_div.focus();\n",
" }\n",
"\n",
" window.setTimeout(set_focus, 100);\n",
"};\n",
"\n",
"mpl.figure.prototype._init_toolbar = function () {\n",
" var fig = this;\n",
"\n",
" var toolbar = document.createElement('div');\n",
" toolbar.classList = 'mpl-toolbar';\n",
" this.root.appendChild(toolbar);\n",
"\n",
" function on_click_closure(name) {\n",
" return function (_event) {\n",
" return fig.toolbar_button_onclick(name);\n",
" };\n",
" }\n",
"\n",
" function on_mouseover_closure(tooltip) {\n",
" return function (event) {\n",
" if (!event.currentTarget.disabled) {\n",
" return fig.toolbar_button_onmouseover(tooltip);\n",
" }\n",
" };\n",
" }\n",
"\n",
" fig.buttons = {};\n",
" var buttonGroup = document.createElement('div');\n",
" buttonGroup.classList = 'mpl-button-group';\n",
" for (var toolbar_ind in mpl.toolbar_items) {\n",
" var name = mpl.toolbar_items[toolbar_ind][0];\n",
" var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
" var image = mpl.toolbar_items[toolbar_ind][2];\n",
" var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
"\n",
" if (!name) {\n",
" /* Instead of a spacer, we start a new button group. */\n",
" if (buttonGroup.hasChildNodes()) {\n",
" toolbar.appendChild(buttonGroup);\n",
" }\n",
" buttonGroup = document.createElement('div');\n",
" buttonGroup.classList = 'mpl-button-group';\n",
" continue;\n",
" }\n",
"\n",
" var button = (fig.buttons[name] = document.createElement('button'));\n",
" button.classList = 'mpl-widget';\n",
" button.setAttribute('role', 'button');\n",
" button.setAttribute('aria-disabled', 'false');\n",
" button.addEventListener('click', on_click_closure(method_name));\n",
" button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n",
"\n",
" var icon_img = document.createElement('img');\n",
" icon_img.src = '_images/' + image + '.png';\n",
" icon_img.srcset = '_images/' + image + '_large.png 2x';\n",
" icon_img.alt = tooltip;\n",
" button.appendChild(icon_img);\n",
"\n",
" buttonGroup.appendChild(button);\n",
" }\n",
"\n",
" if (buttonGroup.hasChildNodes()) {\n",
" toolbar.appendChild(buttonGroup);\n",
" }\n",
"\n",
" var fmt_picker = document.createElement('select');\n",
" fmt_picker.classList = 'mpl-widget';\n",
" toolbar.appendChild(fmt_picker);\n",
" this.format_dropdown = fmt_picker;\n",
"\n",
" for (var ind in mpl.extensions) {\n",
" var fmt = mpl.extensions[ind];\n",
" var option = document.createElement('option');\n",
" option.selected = fmt === mpl.default_extension;\n",
" option.innerHTML = fmt;\n",
" fmt_picker.appendChild(option);\n",
" }\n",
"\n",
" var status_bar = document.createElement('span');\n",
" status_bar.classList = 'mpl-message';\n",
" toolbar.appendChild(status_bar);\n",
" this.message = status_bar;\n",
"};\n",
"\n",
"mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {\n",
" // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
" // which will in turn request a refresh of the image.\n",
" this.send_message('resize', { width: x_pixels, height: y_pixels });\n",
"};\n",
"\n",
"mpl.figure.prototype.send_message = function (type, properties) {\n",
" properties['type'] = type;\n",
" properties['figure_id'] = this.id;\n",
" this.ws.send(JSON.stringify(properties));\n",
"};\n",
"\n",
"mpl.figure.prototype.send_draw_message = function () {\n",
" if (!this.waiting) {\n",
" this.waiting = true;\n",
" this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));\n",
" }\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_save = function (fig, _msg) {\n",
" var format_dropdown = fig.format_dropdown;\n",
" var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
" fig.ondownload(fig, format);\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_resize = function (fig, msg) {\n",
" var size = msg['size'];\n",
" if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {\n",
" fig._resize_canvas(size[0], size[1], msg['forward']);\n",
" fig.send_message('refresh', {});\n",
" }\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_rubberband = function (fig, msg) {\n",
" var x0 = msg['x0'] / mpl.ratio;\n",
" var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
" var x1 = msg['x1'] / mpl.ratio;\n",
" var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
" x0 = Math.floor(x0) + 0.5;\n",
" y0 = Math.floor(y0) + 0.5;\n",
" x1 = Math.floor(x1) + 0.5;\n",
" y1 = Math.floor(y1) + 0.5;\n",
" var min_x = Math.min(x0, x1);\n",
" var min_y = Math.min(y0, y1);\n",
" var width = Math.abs(x1 - x0);\n",
" var height = Math.abs(y1 - y0);\n",
"\n",
" fig.rubberband_context.clearRect(\n",
" 0,\n",
" 0,\n",
" fig.canvas.width / mpl.ratio,\n",
" fig.canvas.height / mpl.ratio\n",
" );\n",
"\n",
" fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_figure_label = function (fig, msg) {\n",
" // Updates the figure title.\n",
" fig.header.textContent = msg['label'];\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_cursor = function (fig, msg) {\n",
" var cursor = msg['cursor'];\n",
" switch (cursor) {\n",
" case 0:\n",
" cursor = 'pointer';\n",
" break;\n",
" case 1:\n",
" cursor = 'default';\n",
" break;\n",
" case 2:\n",
" cursor = 'crosshair';\n",
" break;\n",
" case 3:\n",
" cursor = 'move';\n",
" break;\n",
" }\n",
" fig.rubberband_canvas.style.cursor = cursor;\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_message = function (fig, msg) {\n",
" fig.message.textContent = msg['message'];\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_draw = function (fig, _msg) {\n",
" // Request the server to send over a new figure.\n",
" fig.send_draw_message();\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_image_mode = function (fig, msg) {\n",
" fig.image_mode = msg['mode'];\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_history_buttons = function (fig, msg) {\n",
" for (var key in msg) {\n",
" if (!(key in fig.buttons)) {\n",
" continue;\n",
" }\n",
" fig.buttons[key].disabled = !msg[key];\n",
" fig.buttons[key].setAttribute('aria-disabled', !msg[key]);\n",
" }\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {\n",
" if (msg['mode'] === 'PAN') {\n",
" fig.buttons['Pan'].classList.add('active');\n",
" fig.buttons['Zoom'].classList.remove('active');\n",
" } else if (msg['mode'] === 'ZOOM') {\n",
" fig.buttons['Pan'].classList.remove('active');\n",
" fig.buttons['Zoom'].classList.add('active');\n",
" } else {\n",
" fig.buttons['Pan'].classList.remove('active');\n",
" fig.buttons['Zoom'].classList.remove('active');\n",
" }\n",
"};\n",
"\n",
"mpl.figure.prototype.updated_canvas_event = function () {\n",
" // Called whenever the canvas gets updated.\n",
" this.send_message('ack', {});\n",
"};\n",
"\n",
"// A function to construct a web socket function for onmessage handling.\n",
"// Called in the figure constructor.\n",
"mpl.figure.prototype._make_on_message_function = function (fig) {\n",
" return function socket_on_message(evt) {\n",
" if (evt.data instanceof Blob) {\n",
" /* FIXME: We get \"Resource interpreted as Image but\n",
" * transferred with MIME type text/plain:\" errors on\n",
" * Chrome. But how to set the MIME type? It doesn't seem\n",
" * to be part of the websocket stream */\n",
" evt.data.type = 'image/png';\n",
"\n",
" /* Free the memory for the previous frames */\n",
" if (fig.imageObj.src) {\n",
" (window.URL || window.webkitURL).revokeObjectURL(\n",
" fig.imageObj.src\n",
" );\n",
" }\n",
"\n",
" fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
" evt.data\n",
" );\n",
" fig.updated_canvas_event();\n",
" fig.waiting = false;\n",
" return;\n",
" } else if (\n",
" typeof evt.data === 'string' &&\n",
" evt.data.slice(0, 21) === 'data:image/png;base64'\n",
" ) {\n",
" fig.imageObj.src = evt.data;\n",
" fig.updated_canvas_event();\n",
" fig.waiting = false;\n",
" return;\n",
" }\n",
"\n",
" var msg = JSON.parse(evt.data);\n",
" var msg_type = msg['type'];\n",
"\n",
" // Call the \"handle_{type}\" callback, which takes\n",
" // the figure and JSON message as its only arguments.\n",
" try {\n",
" var callback = fig['handle_' + msg_type];\n",
" } catch (e) {\n",
" console.log(\n",
" \"No handler for the '\" + msg_type + \"' message type: \",\n",
" msg\n",
" );\n",
" return;\n",
" }\n",
"\n",
" if (callback) {\n",
" try {\n",
" // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
" callback(fig, msg);\n",
" } catch (e) {\n",
" console.log(\n",
" \"Exception inside the 'handler_\" + msg_type + \"' callback:\",\n",
" e,\n",
" e.stack,\n",
" msg\n",
" );\n",
" }\n",
" }\n",
" };\n",
"};\n",
"\n",
"// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
"mpl.findpos = function (e) {\n",
" //this section is from http://www.quirksmode.org/js/events_properties.html\n",
" var targ;\n",
" if (!e) {\n",
" e = window.event;\n",
" }\n",
" if (e.target) {\n",
" targ = e.target;\n",
" } else if (e.srcElement) {\n",
" targ = e.srcElement;\n",
" }\n",
" if (targ.nodeType === 3) {\n",
" // defeat Safari bug\n",
" targ = targ.parentNode;\n",
" }\n",
"\n",
" // pageX,Y are the mouse positions relative to the document\n",
" var boundingRect = targ.getBoundingClientRect();\n",
" var x = e.pageX - (boundingRect.left + document.body.scrollLeft);\n",
" var y = e.pageY - (boundingRect.top + document.body.scrollTop);\n",
"\n",
" return { x: x, y: y };\n",
"};\n",
"\n",
"/*\n",
" * return a copy of an object with only non-object keys\n",
" * we need this to avoid circular references\n",
" * http://stackoverflow.com/a/24161582/3208463\n",
" */\n",
"function simpleKeys(original) {\n",
" return Object.keys(original).reduce(function (obj, key) {\n",
" if (typeof original[key] !== 'object') {\n",
" obj[key] = original[key];\n",
" }\n",
" return obj;\n",
" }, {});\n",
"}\n",
"\n",
"mpl.figure.prototype.mouse_event = function (event, name) {\n",
" var canvas_pos = mpl.findpos(event);\n",
"\n",
" if (name === 'button_press') {\n",
" this.canvas.focus();\n",
" this.canvas_div.focus();\n",
" }\n",
"\n",
" var x = canvas_pos.x * mpl.ratio;\n",
" var y = canvas_pos.y * mpl.ratio;\n",
"\n",
" this.send_message(name, {\n",
" x: x,\n",
" y: y,\n",
" button: event.button,\n",
" step: event.step,\n",
" guiEvent: simpleKeys(event),\n",
" });\n",
"\n",
" /* This prevents the web browser from automatically changing to\n",
" * the text insertion cursor when the button is pressed. We want\n",
" * to control all of the cursor setting manually through the\n",
" * 'cursor' event from matplotlib */\n",
" event.preventDefault();\n",
" return false;\n",
"};\n",
"\n",
"mpl.figure.prototype._key_event_extra = function (_event, _name) {\n",
" // Handle any extra behaviour associated with a key event\n",
"};\n",
"\n",
"mpl.figure.prototype.key_event = function (event, name) {\n",
" // Prevent repeat events\n",
" if (name === 'key_press') {\n",
" if (event.which === this._key) {\n",
" return;\n",
" } else {\n",
" this._key = event.which;\n",
" }\n",
" }\n",
" if (name === 'key_release') {\n",
" this._key = null;\n",
" }\n",
"\n",
" var value = '';\n",
" if (event.ctrlKey && event.which !== 17) {\n",
" value += 'ctrl+';\n",
" }\n",
" if (event.altKey && event.which !== 18) {\n",
" value += 'alt+';\n",
" }\n",
" if (event.shiftKey && event.which !== 16) {\n",
" value += 'shift+';\n",
" }\n",
"\n",
" value += 'k';\n",
" value += event.which.toString();\n",
"\n",
" this._key_event_extra(event, name);\n",
"\n",
" this.send_message(name, { key: value, guiEvent: simpleKeys(event) });\n",
" return false;\n",
"};\n",
"\n",
"mpl.figure.prototype.toolbar_button_onclick = function (name) {\n",
" if (name === 'download') {\n",
" this.handle_save(this, null);\n",
" } else {\n",
" this.send_message('toolbar_button', { name: name });\n",
" }\n",
"};\n",
"\n",
"mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {\n",
" this.message.textContent = tooltip;\n",
"};\n",
"mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Left button pans, Right button zooms\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
"\n",
"mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
"\n",
"mpl.default_extension = \"png\";/* global mpl */\n",
"\n",
"var comm_websocket_adapter = function (comm) {\n",
" // Create a \"websocket\"-like object which calls the given IPython comm\n",
" // object with the appropriate methods. Currently this is a non binary\n",
" // socket, so there is still some room for performance tuning.\n",
" var ws = {};\n",
"\n",
" ws.close = function () {\n",
" comm.close();\n",
" };\n",
" ws.send = function (m) {\n",
" //console.log('sending', m);\n",
" comm.send(m);\n",
" };\n",
" // Register the callback with on_msg.\n",
" comm.on_msg(function (msg) {\n",
" //console.log('receiving', msg['content']['data'], msg);\n",
" // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
" ws.onmessage(msg['content']['data']);\n",
" });\n",
" return ws;\n",
"};\n",
"\n",
"mpl.mpl_figure_comm = function (comm, msg) {\n",
" // This is the function which gets called when the mpl process\n",
" // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
"\n",
" var id = msg.content.data.id;\n",
" // Get hold of the div created by the display call when the Comm\n",
" // socket was opened in Python.\n",
" var element = document.getElementById(id);\n",
" var ws_proxy = comm_websocket_adapter(comm);\n",
"\n",
" function ondownload(figure, _format) {\n",
" window.open(figure.canvas.toDataURL());\n",
" }\n",
"\n",
" var fig = new mpl.figure(id, ws_proxy, ondownload, element);\n",
"\n",
" // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
" // web socket which is closed, not our websocket->open comm proxy.\n",
" ws_proxy.onopen();\n",
"\n",
" fig.parent_element = element;\n",
" fig.cell_info = mpl.find_output_cell(\"
Software | Version |
---|---|
Python | 3.7.7 64bit [GCC 7.3.0] |
IPython | 7.16.1 |
OS | Linux 3.10.0 1062.9.1.el7.x86_64 x86_64 with centos 7.7.1908 Core |
sympy | 1.6.1 |
numpy | 1.18.5 |
matplotlib | 3.3.0 |
Fri Aug 07 15:53:41 2020 EDT |