{ "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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"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": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "/* global mpl */\n", "window.mpl = {};\n", "\n", "mpl.get_websocket_type = function () {\n", " if (typeof WebSocket !== 'undefined') {\n", " return WebSocket;\n", " } 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", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = this.ws.binaryType !== undefined;\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById('mpl-warnings');\n", " 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", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = document.createElement('div');\n", " this.root.setAttribute('style', 'display: inline-block');\n", " this._root_extra_style(this.root);\n", "\n", " parent_element.appendChild(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message('supports_binary', { value: fig.supports_binary });\n", " fig.send_message('send_image_mode', {});\n", " if (mpl.ratio !== 1) {\n", " fig.send_message('set_dpi_ratio', { dpi_ratio: mpl.ratio });\n", " }\n", " fig.send_message('refresh', {});\n", " };\n", "\n", " this.imageObj.onload = function () {\n", " if (fig.image_mode === 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function () {\n", " fig.ws.close();\n", " };\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "};\n", "\n", "mpl.figure.prototype._init_header = function () {\n", " var titlebar = document.createElement('div');\n", " titlebar.classList =\n", " 'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';\n", " var titletext = document.createElement('div');\n", " titletext.classList = 'ui-dialog-title';\n", " titletext.setAttribute(\n", " 'style',\n", " 'width: 100%; text-align: center; padding: 3px;'\n", " );\n", " titlebar.appendChild(titletext);\n", " this.root.appendChild(titlebar);\n", " this.header = titletext;\n", "};\n", "\n", "mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};\n", "\n", "mpl.figure.prototype._root_extra_style = function (_canvas_div) {};\n", "\n", "mpl.figure.prototype._init_canvas = function () {\n", " var fig = this;\n", "\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", " 'clear: both;' +\n", " 'min-height: 1px;' +\n", " 'min-width: 1px;' +\n", " 'outline: 0;' +\n", " 'overflow: hidden;' +\n", " 'position: relative;' +\n", " 'resize: both;'\n", " );\n", "\n", " function on_keyboard_event_closure(name) {\n", " return function (event) {\n", " return fig.key_event(event, name);\n", " };\n", " }\n", "\n", " canvas_div.addEventListener(\n", " 'keydown',\n", " on_keyboard_event_closure('key_press')\n", " );\n", " canvas_div.addEventListener(\n", " 'keyup',\n", " on_keyboard_event_closure('key_release')\n", " );\n", "\n", " this._canvas_extra_style(canvas_div);\n", " this.root.appendChild(canvas_div);\n", "\n", " 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(\"
\");\n", " if (!fig.cell_info) {\n", " console.error('Failed to find cell for figure', id, fig);\n", " return;\n", " }\n", "};\n", "\n", "mpl.figure.prototype.handle_close = function (fig, msg) {\n", " var width = fig.canvas.width / mpl.ratio;\n", " fig.root.removeEventListener('remove', this._remove_fig_handler);\n", "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", " // Re-enable the keyboard manager in IPython - without this line, in FF,\n", " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable();\n", " fig.parent_element.innerHTML =\n", " '';\n", " fig.close_ws(fig, msg);\n", "};\n", "\n", "mpl.figure.prototype.close_ws = function (fig, msg) {\n", " fig.send_message('closing', msg);\n", " // fig.ws.close()\n", "};\n", "\n", "mpl.figure.prototype.push_to_output = function (_remove_interactive) {\n", " // Turn the data on the canvas into data in the output cell.\n", " var width = this.canvas.width / mpl.ratio;\n", " var dataURL = this.canvas.toDataURL();\n", " this.cell_info[1]['text/html'] =\n", " '';\n", "};\n", "\n", "mpl.figure.prototype.updated_canvas_event = function () {\n", " // Tell IPython that the notebook contents must change.\n", " IPython.notebook.set_dirty(true);\n", " this.send_message('ack', {});\n", " var fig = this;\n", " // Wait a second, then push the new image to the DOM so\n", " // that it is saved nicely (might be nice to debounce this).\n", " setTimeout(function () {\n", " fig.push_to_output();\n", " }, 1000);\n", "};\n", "\n", "mpl.figure.prototype._init_toolbar = function () {\n", " var fig = this;\n", "\n", " var toolbar = document.createElement('div');\n", " toolbar.classList = 'btn-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 = 'btn-group';\n", " var button;\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 = 'btn-group';\n", " continue;\n", " }\n", "\n", " button = fig.buttons[name] = document.createElement('button');\n", " button.classList = 'btn btn-default';\n", " button.href = '#';\n", " button.title = name;\n", " button.innerHTML = '';\n", " button.addEventListener('click', on_click_closure(method_name));\n", " button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n", " buttonGroup.appendChild(button);\n", " }\n", "\n", " if (buttonGroup.hasChildNodes()) {\n", " toolbar.appendChild(buttonGroup);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = document.createElement('span');\n", " status_bar.classList = 'mpl-message pull-right';\n", " toolbar.appendChild(status_bar);\n", " this.message = status_bar;\n", "\n", " // Add the close button to the window.\n", " var buttongrp = document.createElement('div');\n", " buttongrp.classList = 'btn-group inline pull-right';\n", " button = document.createElement('button');\n", " button.classList = 'btn btn-mini btn-primary';\n", " button.href = '#';\n", " button.title = 'Stop Interaction';\n", " button.innerHTML = '';\n", " button.addEventListener('click', function (_evt) {\n", " fig.handle_close(fig, {});\n", " });\n", " button.addEventListener(\n", " 'mouseover',\n", " on_mouseover_closure('Stop Interaction')\n", " );\n", " buttongrp.appendChild(button);\n", " var titlebar = this.root.querySelector('.ui-dialog-titlebar');\n", " titlebar.insertBefore(buttongrp, titlebar.firstChild);\n", "};\n", "\n", "mpl.figure.prototype._remove_fig_handler = function () {\n", " this.close_ws(this, {});\n", "};\n", "\n", "mpl.figure.prototype._root_extra_style = function (el) {\n", " el.style.boxSizing = 'content-box'; // override notebook setting of border-box.\n", " el.addEventListener('remove', this._remove_fig_handler);\n", "};\n", "\n", "mpl.figure.prototype._canvas_extra_style = function (el) {\n", " // this is important to make the div 'focusable\n", " el.setAttribute('tabindex', 0);\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " } else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "};\n", "\n", "mpl.figure.prototype._key_event_extra = function (event, _name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager) {\n", " manager = IPython.keyboard_manager;\n", " }\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which === 13) {\n", " this.canvas_div.blur();\n", " // select the cell after this one\n", " var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n", " IPython.notebook.select(index + 1);\n", " }\n", "};\n", "\n", "mpl.figure.prototype.handle_save = function (fig, _msg) {\n", " fig.ondownload(fig, null);\n", "};\n", "\n", "mpl.find_output_cell = function (html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i = 0; i < ncells; i++) {\n", " var cell = cells[i];\n", " if (cell.cell_type === 'code') {\n", " for (var j = 0; j < cell.output_area.outputs.length; j++) {\n", " var data = cell.output_area.outputs[j];\n", " if (data.data) {\n", " // IPython >= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] === html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "};\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel !== null) {\n", " IPython.notebook.kernel.comm_manager.register_target(\n", " 'matplotlib',\n", " mpl.mpl_figure_comm\n", " );\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "xx = np.linspace(-1,1,201) # name = xx so it won't collide with symbol x\n", "y = func(xx)\n", "plt.figure()\n", "plt.xlabel('$x$')\n", "plt.ylabel('$f(x)$')\n", "plt.plot(xx,y);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Version Information\n", "`version_information` is from J.R. Johansson (jrjohansson at gmail.com); see Introduction to scientific computing with Python for more information and instructions for package installation.\n", "\n", "`version_information` is installed on the linux network at Bucknell" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "%load_ext version_information" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "application/json": { "Software versions": [ { "module": "Python", "version": "3.7.7 64bit [GCC 7.3.0]" }, { "module": "IPython", "version": "7.16.1" }, { "module": "OS", "version": "Linux 3.10.0 1062.9.1.el7.x86_64 x86_64 with centos 7.7.1908 Core" }, { "module": "sympy", "version": "1.6.1" }, { "module": "numpy", "version": "1.18.5" }, { "module": "matplotlib", "version": "3.3.0" } ] }, "text/html": [ "
SoftwareVersion
Python3.7.7 64bit [GCC 7.3.0]
IPython7.16.1
OSLinux 3.10.0 1062.9.1.el7.x86_64 x86_64 with centos 7.7.1908 Core
sympy1.6.1
numpy1.18.5
matplotlib3.3.0
Fri Aug 07 15:53:41 2020 EDT
" ], "text/latex": [ "\\begin{tabular}{|l|l|}\\hline\n", "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n", "Python & 3.7.7 64bit [GCC 7.3.0] \\\\ \\hline\n", "IPython & 7.16.1 \\\\ \\hline\n", "OS & Linux 3.10.0 1062.9.1.el7.x86\\_64 x86\\_64 with centos 7.7.1908 Core \\\\ \\hline\n", "sympy & 1.6.1 \\\\ \\hline\n", "numpy & 1.18.5 \\\\ \\hline\n", "matplotlib & 3.3.0 \\\\ \\hline\n", "\\hline \\multicolumn{2}{|l|}{Fri Aug 07 15:53:41 2020 EDT} \\\\ \\hline\n", "\\end{tabular}\n" ], "text/plain": [ "Software versions\n", "Python 3.7.7 64bit [GCC 7.3.0]\n", "IPython 7.16.1\n", "OS Linux 3.10.0 1062.9.1.el7.x86_64 x86_64 with centos 7.7.1908 Core\n", "sympy 1.6.1\n", "numpy 1.18.5\n", "matplotlib 3.3.0\n", "Fri Aug 07 15:53:41 2020 EDT" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version_information sympy, numpy, matplotlib" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "anaconda-cloud": {}, "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": "3.7.7" } }, "nbformat": 4, "nbformat_minor": 1 }