# Pseudospectra of matrices

The idea is based on [1], the pseudospectra around the complex number $z$ is found as

$$\sigma_\min(A - zI)$$

the minimum singular value of $(A - zI)$.

## References

• [1]: L.N. Trefethen. Pseudospectra of matrices.


In [2]:

from __future__ import division
import numpy as np
from scipy.sparse import diags
from scipy.sparse.linalg import svds, eigs
import matplotlib.pyplot as plt




In [3]:

%matplotlib notebook




In [4]:

def pseudo_spec(x, y, mat_A):
"""Compute the pseudospectra of mat_A around the point $x + iy""" ps_spec = np.zeros_like(x) for row in range(npts): for col in range(npts): z_num = x[row, col] + 1j*y[row, col] sv_min = np.min(svds((mat_A - z_num*np.eye(N)), k=10, which='SM', return_singular_vectors=False)) ps_spec[row, col] = sv_min return ps_spec   In [5]: def plot_pseudo_spec(x, y, mat_A, ps_spec, levels=8): """Plot the pseudospectra and eigenvalues for matrix mat_A""" n_eigs = np.shape(mat_A)[0] - 2 vals = eigs(mat_A, k=n_eigs, which='SM', return_eigenvectors=False) ps_spec[np.isnan(ps_spec)] = 0 plt.plot(np.real(vals), np.imag(vals), 'k.') plt.contourf(x, y, ps_spec, levels, cmap="YlGnBu_r") C = plt.contour(x, y, ps_spec, levels, colors='k') plt.axis('image') return C   In [6]: npts = 51 N = 200  ## 1. Jordan block $$A_1 = \begin{pmatrix} 0 & 1 & & & & \\ & 0 & 1 & & & \\ & & 0 & 1 & & \\ & & & 0 & 1 & \\ & & & & 0 & 1\\ & & & & & 0 \end{pmatrix}$$  In [7]: x1, y1 = np.mgrid[-1:1:npts*1j, -1:1:npts*1j] A1 = diags([1.], [1], shape=(N, N))   In [8]: ps_spec1 = pseudo_spec(x1, y1, A1)   /home/nguarinz/anaconda3/lib/python3.6/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.py:1849: RuntimeWarning: invalid value encountered in sqrt s = np.sqrt(eigvals) /home/nguarinz/anaconda3/lib/python3.6/site-packages/numpy/core/fromnumeric.py:83: RuntimeWarning: invalid value encountered in reduce return ufunc.reduce(obj, axis, dtype, out, **passkwargs)   In [9]: plt.figure(figsize=(8,8)) plot_pseudo_spec(x1, y1, A1, ps_spec1) plt.show()   var element =$('#6dec037b-7cac-4c22-9936-ee03fd27fa90');
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/* Put everything inside the global mpl namespace */
window.mpl = {};

mpl.get_websocket_type = function() {
if (typeof(WebSocket) !== 'undefined') {
return WebSocket;
} else if (typeof(MozWebSocket) !== 'undefined') {
return MozWebSocket;
} else {
alert('Your browser does not have WebSocket support.' +
'Please try Chrome, Safari or Firefox ≥ 6. ' +
'Firefox 4 and 5 are also supported but you ' +
'have to enable WebSockets in about:config.');
};
}

mpl.figure = function(figure_id, websocket, ondownload, parent_element) {
this.id = figure_id;

this.ws = websocket;

this.supports_binary = (this.ws.binaryType != undefined);

if (!this.supports_binary) {
var warnings = document.getElementById("mpl-warnings");
if (warnings) {
warnings.style.display = 'block';
warnings.textContent = (
"This browser does not support binary websocket messages. " +
"Performance may be slow.");
}
}

this.imageObj = new Image();

this.context = undefined;
this.message = undefined;
this.canvas = undefined;
this.rubberband_canvas = undefined;
this.rubberband_context = undefined;
this.format_dropdown = undefined;

this.image_mode = 'full';

this.root = $('<div/>'); this._root_extra_style(this.root) this.root.attr('style', 'display: inline-block');$(parent_element).append(this.root);

this._init_canvas(this);
this._init_toolbar(this);

var fig = this;

this.waiting = false;

this.ws.onopen =  function () {
fig.send_message("supports_binary", {value: fig.supports_binary});
fig.send_message("send_image_mode", {});
if (mpl.ratio != 1) {
fig.send_message("set_dpi_ratio", {'dpi_ratio': mpl.ratio});
}
fig.send_message("refresh", {});
}

this.imageObj.onload = function() {
if (fig.image_mode == 'full') {
// Full images could contain transparency (where diff images
// almost always do), so we need to clear the canvas so that
// there is no ghosting.
fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);
}
fig.context.drawImage(fig.imageObj, 0, 0);
};

this.imageObj.onunload = function() {
fig.ws.close();
}

this.ws.onmessage = this._make_on_message_function(this);

}

mpl.figure.prototype._init_header = function() {
var titlebar = $( '<div class="ui-dialog-titlebar ui-widget-header ui-corner-all ' + 'ui-helper-clearfix"/>'); var titletext =$(
'<div class="ui-dialog-title" style="width: 100%; ' +
'text-align: center; padding: 3px;"/>');
titlebar.append(titletext)
this.root.append(titlebar);
}

mpl.figure.prototype._canvas_extra_style = function(canvas_div) {

}

mpl.figure.prototype._root_extra_style = function(canvas_div) {

}

mpl.figure.prototype._init_canvas = function() {
var fig = this;

var canvas_div = $('<div/>'); canvas_div.attr('style', 'position: relative; clear: both; outline: 0'); function canvas_keyboard_event(event) { return fig.key_event(event, event['data']); } canvas_div.keydown('key_press', canvas_keyboard_event); canvas_div.keyup('key_release', canvas_keyboard_event); this.canvas_div = canvas_div this._canvas_extra_style(canvas_div) this.root.append(canvas_div); var canvas =$('<canvas/>');
canvas.attr('style', "left: 0; top: 0; z-index: 0; outline: 0")

this.canvas = canvas[0];
this.context = canvas[0].getContext("2d");

var backingStore = this.context.backingStorePixelRatio ||
this.context.webkitBackingStorePixelRatio ||
this.context.mozBackingStorePixelRatio ||
this.context.msBackingStorePixelRatio ||
this.context.oBackingStorePixelRatio ||
this.context.backingStorePixelRatio || 1;

mpl.ratio = (window.devicePixelRatio || 1) / backingStore;

var rubberband = $('<canvas/>'); rubberband.attr('style', "position: absolute; left: 0; top: 0; z-index: 1;") var pass_mouse_events = true; canvas_div.resizable({ start: function(event, ui) { pass_mouse_events = false; }, resize: function(event, ui) { fig.request_resize(ui.size.width, ui.size.height); }, stop: function(event, ui) { pass_mouse_events = true; fig.request_resize(ui.size.width, ui.size.height); }, }); function mouse_event_fn(event) { if (pass_mouse_events) return fig.mouse_event(event, event['data']); } rubberband.mousedown('button_press', mouse_event_fn); rubberband.mouseup('button_release', mouse_event_fn); // Throttle sequential mouse events to 1 every 20ms. rubberband.mousemove('motion_notify', mouse_event_fn); rubberband.mouseenter('figure_enter', mouse_event_fn); rubberband.mouseleave('figure_leave', mouse_event_fn); canvas_div.on("wheel", function (event) { event = event.originalEvent; event['data'] = 'scroll' if (event.deltaY < 0) { event.step = 1; } else { event.step = -1; } mouse_event_fn(event); }); canvas_div.append(canvas); canvas_div.append(rubberband); this.rubberband = rubberband; this.rubberband_canvas = rubberband[0]; this.rubberband_context = rubberband[0].getContext("2d"); this.rubberband_context.strokeStyle = "#000000"; this._resize_canvas = function(width, height) { // Keep the size of the canvas, canvas container, and rubber band // canvas in synch. canvas_div.css('width', width) canvas_div.css('height', height) canvas.attr('width', width * mpl.ratio); canvas.attr('height', height * mpl.ratio); canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;'); rubberband.attr('width', width); rubberband.attr('height', height); } // Set the figure to an initial 600x600px, this will subsequently be updated // upon first draw. this._resize_canvas(600, 600); // Disable right mouse context menu.$(this.rubberband_canvas).bind("contextmenu",function(e){
return false;
});

function set_focus () {
canvas.focus();
canvas_div.focus();
}

window.setTimeout(set_focus, 100);
}

mpl.figure.prototype._init_toolbar = function() {
var fig = this;

var nav_element = $('<div/>') nav_element.attr('style', 'width: 100%'); this.root.append(nav_element); // Define a callback function for later on. function toolbar_event(event) { return fig.toolbar_button_onclick(event['data']); } function toolbar_mouse_event(event) { return fig.toolbar_button_onmouseover(event['data']); } for(var toolbar_ind in mpl.toolbar_items) { var name = mpl.toolbar_items[toolbar_ind][0]; var tooltip = mpl.toolbar_items[toolbar_ind][1]; var image = mpl.toolbar_items[toolbar_ind][2]; var method_name = mpl.toolbar_items[toolbar_ind][3]; if (!name) { // put a spacer in here. continue; } var button =$('<button/>');
button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +
'ui-button-icon-only');
button.attr('role', 'button');
button.attr('aria-disabled', 'false');
button.click(method_name, toolbar_event);
button.mouseover(tooltip, toolbar_mouse_event);

var icon_img = $('<span/>'); icon_img.addClass('ui-button-icon-primary ui-icon'); icon_img.addClass(image); icon_img.addClass('ui-corner-all'); var tooltip_span =$('<span/>');
tooltip_span.html(tooltip);

button.append(icon_img);
button.append(tooltip_span);

nav_element.append(button);
}

var fmt_picker_span = $('<span/>'); var fmt_picker =$('<select/>');
fmt_picker_span.append(fmt_picker);
nav_element.append(fmt_picker_span);
this.format_dropdown = fmt_picker[0];

for (var ind in mpl.extensions) {
var fmt = mpl.extensions[ind];
var option = $( '<option/>', {selected: fmt === mpl.default_extension}).html(fmt); fmt_picker.append(option) } // Add hover states to the ui-buttons$( ".ui-button" ).hover(
function() { $(this).addClass("ui-state-hover");}, function() {$(this).removeClass("ui-state-hover");}
);

var status_bar = $('<span class="mpl-message"/>'); nav_element.append(status_bar); this.message = status_bar[0]; } mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) { // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client, // which will in turn request a refresh of the image. this.send_message('resize', {'width': x_pixels, 'height': y_pixels}); } mpl.figure.prototype.send_message = function(type, properties) { properties['type'] = type; properties['figure_id'] = this.id; this.ws.send(JSON.stringify(properties)); } mpl.figure.prototype.send_draw_message = function() { if (!this.waiting) { this.waiting = true; this.ws.send(JSON.stringify({type: "draw", figure_id: this.id})); } } mpl.figure.prototype.handle_save = function(fig, msg) { var format_dropdown = fig.format_dropdown; var format = format_dropdown.options[format_dropdown.selectedIndex].value; fig.ondownload(fig, format); } mpl.figure.prototype.handle_resize = function(fig, msg) { var size = msg['size']; if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) { fig._resize_canvas(size[0], size[1]); fig.send_message("refresh", {}); }; } mpl.figure.prototype.handle_rubberband = function(fig, msg) { var x0 = msg['x0'] / mpl.ratio; var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio; var x1 = msg['x1'] / mpl.ratio; var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio; x0 = Math.floor(x0) + 0.5; y0 = Math.floor(y0) + 0.5; x1 = Math.floor(x1) + 0.5; y1 = Math.floor(y1) + 0.5; var min_x = Math.min(x0, x1); var min_y = Math.min(y0, y1); var width = Math.abs(x1 - x0); var height = Math.abs(y1 - y0); fig.rubberband_context.clearRect( 0, 0, fig.canvas.width, fig.canvas.height); fig.rubberband_context.strokeRect(min_x, min_y, width, height); } mpl.figure.prototype.handle_figure_label = function(fig, msg) { // Updates the figure title. fig.header.textContent = msg['label']; } mpl.figure.prototype.handle_cursor = function(fig, msg) { var cursor = msg['cursor']; switch(cursor) { case 0: cursor = 'pointer'; break; case 1: cursor = 'default'; break; case 2: cursor = 'crosshair'; break; case 3: cursor = 'move'; break; } fig.rubberband_canvas.style.cursor = cursor; } mpl.figure.prototype.handle_message = function(fig, msg) { fig.message.textContent = msg['message']; } mpl.figure.prototype.handle_draw = function(fig, msg) { // Request the server to send over a new figure. fig.send_draw_message(); } mpl.figure.prototype.handle_image_mode = function(fig, msg) { fig.image_mode = msg['mode']; } mpl.figure.prototype.updated_canvas_event = function() { // Called whenever the canvas gets updated. this.send_message("ack", {}); } // A function to construct a web socket function for onmessage handling. // Called in the figure constructor. mpl.figure.prototype._make_on_message_function = function(fig) { return function socket_on_message(evt) { if (evt.data instanceof Blob) { /* FIXME: We get "Resource interpreted as Image but * transferred with MIME type text/plain:" errors on * Chrome. But how to set the MIME type? It doesn't seem * to be part of the websocket stream */ evt.data.type = "image/png"; /* Free the memory for the previous frames */ if (fig.imageObj.src) { (window.URL || window.webkitURL).revokeObjectURL( fig.imageObj.src); } fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL( evt.data); fig.updated_canvas_event(); fig.waiting = false; return; } else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == "data:image/png;base64") { fig.imageObj.src = evt.data; fig.updated_canvas_event(); fig.waiting = false; return; } var msg = JSON.parse(evt.data); var msg_type = msg['type']; // Call the "handle_{type}" callback, which takes // the figure and JSON message as its only arguments. try { var callback = fig["handle_" + msg_type]; } catch (e) { console.log("No handler for the '" + msg_type + "' message type: ", msg); return; } if (callback) { try { // console.log("Handling '" + msg_type + "' message: ", msg); callback(fig, msg); } catch (e) { console.log("Exception inside the 'handler_" + msg_type + "' callback:", e, e.stack, msg); } } }; } // from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas mpl.findpos = function(e) { //this section is from http://www.quirksmode.org/js/events_properties.html var targ; if (!e) e = window.event; if (e.target) targ = e.target; else if (e.srcElement) targ = e.srcElement; if (targ.nodeType == 3) // defeat Safari bug targ = targ.parentNode; // jQuery normalizes the pageX and pageY // pageX,Y are the mouse positions relative to the document // offset() returns the position of the element relative to the document var x = e.pageX -$(targ).offset().left;
var y = e.pageY - $(targ).offset().top; return {"x": x, "y": y}; }; /* * return a copy of an object with only non-object keys * we need this to avoid circular references * http://stackoverflow.com/a/24161582/3208463 */ function simpleKeys (original) { return Object.keys(original).reduce(function (obj, key) { if (typeof original[key] !== 'object') obj[key] = original[key] return obj; }, {}); } mpl.figure.prototype.mouse_event = function(event, name) { var canvas_pos = mpl.findpos(event) if (name === 'button_press') { this.canvas.focus(); this.canvas_div.focus(); } var x = canvas_pos.x * mpl.ratio; var y = canvas_pos.y * mpl.ratio; this.send_message(name, {x: x, y: y, button: event.button, step: event.step, guiEvent: simpleKeys(event)}); /* This prevents the web browser from automatically changing to * the text insertion cursor when the button is pressed. We want * to control all of the cursor setting manually through the * 'cursor' event from matplotlib */ event.preventDefault(); return false; } mpl.figure.prototype._key_event_extra = function(event, name) { // Handle any extra behaviour associated with a key event } mpl.figure.prototype.key_event = function(event, name) { // Prevent repeat events if (name == 'key_press') { if (event.which === this._key) return; else this._key = event.which; } if (name == 'key_release') this._key = null; var value = ''; if (event.ctrlKey && event.which != 17) value += "ctrl+"; if (event.altKey && event.which != 18) value += "alt+"; if (event.shiftKey && event.which != 16) value += "shift+"; value += 'k'; value += event.which.toString(); this._key_event_extra(event, name); this.send_message(name, {key: value, guiEvent: simpleKeys(event)}); return false; } mpl.figure.prototype.toolbar_button_onclick = function(name) { if (name == 'download') { this.handle_save(this, null); } else { this.send_message("toolbar_button", {name: name}); } }; mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) { this.message.textContent = tooltip; }; 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", "Pan axes with left mouse, zoom with right", "fa fa-arrows icon-move", "pan"], ["Zoom", "Zoom to rectangle", "fa fa-square-o icon-check-empty", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o icon-save", "download"]]; mpl.extensions = ["eps", "jpeg", "pdf", "png", "ps", "raw", "svg", "tif"]; mpl.default_extension = "png";var comm_websocket_adapter = function(comm) { // Create a "websocket"-like object which calls the given IPython comm // object with the appropriate methods. Currently this is a non binary // socket, so there is still some room for performance tuning. var ws = {}; ws.close = function() { comm.close() }; ws.send = function(m) { //console.log('sending', m); comm.send(m); }; // Register the callback with on_msg. comm.on_msg(function(msg) { //console.log('receiving', msg['content']['data'], msg); // Pass the mpl event to the overridden (by mpl) onmessage function. ws.onmessage(msg['content']['data']) }); return ws; } mpl.mpl_figure_comm = function(comm, msg) { // This is the function which gets called when the mpl process // starts-up an IPython Comm through the "matplotlib" channel. var id = msg.content.data.id; // Get hold of the div created by the display call when the Comm // socket was opened in Python. var element =$("#" + id);
var ws_proxy = comm_websocket_adapter(comm)

window.open(figure.imageObj.src);
}

var fig = new mpl.figure(id, ws_proxy,
element.get(0));

// Call onopen now - mpl needs it, as it is assuming we've passed it a real
// web socket which is closed, not our websocket->open comm proxy.
ws_proxy.onopen();

fig.parent_element = element.get(0);
fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
if (!fig.cell_info) {
console.error("Failed to find cell for figure", id, fig);
return;
}

var output_index = fig.cell_info[2]
var cell = fig.cell_info[0];

};

mpl.figure.prototype.handle_close = function(fig, msg) {
var width = fig.canvas.width/mpl.ratio
fig.root.unbind('remove')

// Update the output cell to use the data from the current canvas.
fig.push_to_output();
var dataURL = fig.canvas.toDataURL();
// Re-enable the keyboard manager in IPython - without this line, in FF,
// the notebook keyboard shortcuts fail.
IPython.keyboard_manager.enable()
$(fig.parent_element).html('<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '" width="' + width + '">'); fig.close_ws(fig, msg); } mpl.figure.prototype.close_ws = function(fig, msg){ fig.send_message('closing', msg); // fig.ws.close() } mpl.figure.prototype.push_to_output = function(remove_interactive) { // Turn the data on the canvas into data in the output cell. var width = this.canvas.width/mpl.ratio var dataURL = this.canvas.toDataURL(); this.cell_info[1]['text/html'] = '<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '" width="' + width + '">'; } mpl.figure.prototype.updated_canvas_event = function() { // Tell IPython that the notebook contents must change. IPython.notebook.set_dirty(true); this.send_message("ack", {}); var fig = this; // Wait a second, then push the new image to the DOM so // that it is saved nicely (might be nice to debounce this). setTimeout(function () { fig.push_to_output() }, 1000); } mpl.figure.prototype._init_toolbar = function() { var fig = this; var nav_element =$('<div/>')
nav_element.attr('style', 'width: 100%');
this.root.append(nav_element);

// Define a callback function for later on.
function toolbar_event(event) {
return fig.toolbar_button_onclick(event['data']);
}
function toolbar_mouse_event(event) {
return fig.toolbar_button_onmouseover(event['data']);
}

for(var toolbar_ind in mpl.toolbar_items){
var name = mpl.toolbar_items[toolbar_ind][0];
var tooltip = mpl.toolbar_items[toolbar_ind][1];
var image = mpl.toolbar_items[toolbar_ind][2];
var method_name = mpl.toolbar_items[toolbar_ind][3];

if (!name) { continue; };

var button = $('<button class="btn btn-default" href="#" title="' + name + '"><i class="fa ' + image + ' fa-lg"></i></button>'); button.click(method_name, toolbar_event); button.mouseover(tooltip, toolbar_mouse_event); nav_element.append(button); } // Add the status bar. var status_bar =$('<span class="mpl-message" style="text-align:right; float: right;"/>');
nav_element.append(status_bar);
this.message = status_bar[0];

// Add the close button to the window.
var buttongrp = $('<div class="btn-group inline pull-right"></div>'); var button =$('<button class="btn btn-mini btn-primary" href="#" title="Stop Interaction"><i class="fa fa-power-off icon-remove icon-large"></i></button>');
button.click(function (evt) { fig.handle_close(fig, {}); } );
button.mouseover('Stop Interaction', toolbar_mouse_event);
buttongrp.append(button);
var titlebar = this.root.find($('.ui-dialog-titlebar')); titlebar.prepend(buttongrp); } mpl.figure.prototype._root_extra_style = function(el){ var fig = this el.on("remove", function(){ fig.close_ws(fig, {}); }); } mpl.figure.prototype._canvas_extra_style = function(el){ // this is important to make the div 'focusable el.attr('tabindex', 0) // reach out to IPython and tell the keyboard manager to turn it's self // off when our div gets focus // location in version 3 if (IPython.notebook.keyboard_manager) { IPython.notebook.keyboard_manager.register_events(el); } else { // location in version 2 IPython.keyboard_manager.register_events(el); } } mpl.figure.prototype._key_event_extra = function(event, name) { var manager = IPython.notebook.keyboard_manager; if (!manager) manager = IPython.keyboard_manager; // Check for shift+enter if (event.shiftKey && event.which == 13) { this.canvas_div.blur(); event.shiftKey = false; // Send a "J" for go to next cell event.which = 74; event.keyCode = 74; manager.command_mode(); manager.handle_keydown(event); } } mpl.figure.prototype.handle_save = function(fig, msg) { fig.ondownload(fig, null); } mpl.find_output_cell = function(html_output) { // Return the cell and output element which can be found *uniquely* in the notebook. // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook" // IPython event is triggered only after the cells have been serialised, which for // our purposes (turning an active figure into a static one), is too late. var cells = IPython.notebook.get_cells(); var ncells = cells.length; for (var i=0; i<ncells; i++) { var cell = cells[i]; if (cell.cell_type === 'code'){ for (var j=0; j<cell.output_area.outputs.length; j++) { var data = cell.output_area.outputs[j]; if (data.data) { // IPython >= 3 moved mimebundle to data attribute of output data = data.data; } if (data['text/html'] == html_output) { return [cell, data, j]; } } } } } // Register the function which deals with the matplotlib target/channel. // The kernel may be null if the page has been refreshed. if (IPython.notebook.kernel != null) { IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm); }  ## 3. Grcar matrix The matrix looks like $$A_3 = \begin{pmatrix} 1 & 1 & 1 & 1 & & \\ -1 & 1 & 1 & 1 & 1 & \\ & -1 & 1 & 1 & 1 & 1\\ & & -1 & 1 & 1 & 1\\ & & & -1 & 1 & 1\\ & & & & 1 & 1 \end{pmatrix}$$  In [13]: x3, y3 = np.mgrid[-4:4:npts*1j, -4:4:npts*1j] A3 = diags([-1., 1., 1., 1, 1], [-1, 0, 1, 2, 3], shape=(N, N))   In [14]: ps_spec3 = pseudo_spec(x3, y3, A3)   /home/nguarinz/anaconda3/lib/python3.6/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.py:1849: RuntimeWarning: invalid value encountered in sqrt s = np.sqrt(eigvals) /home/nguarinz/anaconda3/lib/python3.6/site-packages/numpy/core/fromnumeric.py:83: RuntimeWarning: invalid value encountered in reduce return ufunc.reduce(obj, axis, dtype, out, **passkwargs)   In [15]: plt.figure(figsize=(8,8)) plot_pseudo_spec(x3, y3, A3, ps_spec3) plt.show()   var element =$('#35a52d99-5d2a-41f3-9d91-bf8ae60b6ca3');
/* Put everything inside the global mpl namespace */
window.mpl = {};

mpl.get_websocket_type = function() {
if (typeof(WebSocket) !== 'undefined') {
return WebSocket;
} else if (typeof(MozWebSocket) !== 'undefined') {
return MozWebSocket;
} else {
alert('Your browser does not have WebSocket support.' +
'Please try Chrome, Safari or Firefox ≥ 6. ' +
'Firefox 4 and 5 are also supported but you ' +
'have to enable WebSockets in about:config.');
};
}

mpl.figure = function(figure_id, websocket, ondownload, parent_element) {
this.id = figure_id;

this.ws = websocket;

this.supports_binary = (this.ws.binaryType != undefined);

if (!this.supports_binary) {
var warnings = document.getElementById("mpl-warnings");
if (warnings) {
warnings.style.display = 'block';
warnings.textContent = (
"This browser does not support binary websocket messages. " +
"Performance may be slow.");
}
}

this.imageObj = new Image();

this.context = undefined;
this.message = undefined;
this.canvas = undefined;
this.rubberband_canvas = undefined;
this.rubberband_context = undefined;
this.format_dropdown = undefined;

this.image_mode = 'full';

this.root = $('<div/>'); this._root_extra_style(this.root) this.root.attr('style', 'display: inline-block');$(parent_element).append(this.root);

this._init_canvas(this);
this._init_toolbar(this);

var fig = this;

this.waiting = false;

this.ws.onopen =  function () {
fig.send_message("supports_binary", {value: fig.supports_binary});
fig.send_message("send_image_mode", {});
if (mpl.ratio != 1) {
fig.send_message("set_dpi_ratio", {'dpi_ratio': mpl.ratio});
}
fig.send_message("refresh", {});
}

this.imageObj.onload = function() {
if (fig.image_mode == 'full') {
// Full images could contain transparency (where diff images
// almost always do), so we need to clear the canvas so that
// there is no ghosting.
fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);
}
fig.context.drawImage(fig.imageObj, 0, 0);
};

this.imageObj.onunload = function() {
fig.ws.close();
}

this.ws.onmessage = this._make_on_message_function(this);

}

mpl.figure.prototype._init_header = function() {
var titlebar = $( '<div class="ui-dialog-titlebar ui-widget-header ui-corner-all ' + 'ui-helper-clearfix"/>'); var titletext =$(
'<div class="ui-dialog-title" style="width: 100%; ' +
'text-align: center; padding: 3px;"/>');
titlebar.append(titletext)
this.root.append(titlebar);
}

mpl.figure.prototype._canvas_extra_style = function(canvas_div) {

}

mpl.figure.prototype._root_extra_style = function(canvas_div) {

}

mpl.figure.prototype._init_canvas = function() {
var fig = this;

var canvas_div = $('<div/>'); canvas_div.attr('style', 'position: relative; clear: both; outline: 0'); function canvas_keyboard_event(event) { return fig.key_event(event, event['data']); } canvas_div.keydown('key_press', canvas_keyboard_event); canvas_div.keyup('key_release', canvas_keyboard_event); this.canvas_div = canvas_div this._canvas_extra_style(canvas_div) this.root.append(canvas_div); var canvas =$('<canvas/>');
canvas.attr('style', "left: 0; top: 0; z-index: 0; outline: 0")

this.canvas = canvas[0];
this.context = canvas[0].getContext("2d");

var backingStore = this.context.backingStorePixelRatio ||
this.context.webkitBackingStorePixelRatio ||
this.context.mozBackingStorePixelRatio ||
this.context.msBackingStorePixelRatio ||
this.context.oBackingStorePixelRatio ||
this.context.backingStorePixelRatio || 1;

mpl.ratio = (window.devicePixelRatio || 1) / backingStore;

var rubberband = $('<canvas/>'); rubberband.attr('style', "position: absolute; left: 0; top: 0; z-index: 1;") var pass_mouse_events = true; canvas_div.resizable({ start: function(event, ui) { pass_mouse_events = false; }, resize: function(event, ui) { fig.request_resize(ui.size.width, ui.size.height); }, stop: function(event, ui) { pass_mouse_events = true; fig.request_resize(ui.size.width, ui.size.height); }, }); function mouse_event_fn(event) { if (pass_mouse_events) return fig.mouse_event(event, event['data']); } rubberband.mousedown('button_press', mouse_event_fn); rubberband.mouseup('button_release', mouse_event_fn); // Throttle sequential mouse events to 1 every 20ms. rubberband.mousemove('motion_notify', mouse_event_fn); rubberband.mouseenter('figure_enter', mouse_event_fn); rubberband.mouseleave('figure_leave', mouse_event_fn); canvas_div.on("wheel", function (event) { event = event.originalEvent; event['data'] = 'scroll' if (event.deltaY < 0) { event.step = 1; } else { event.step = -1; } mouse_event_fn(event); }); canvas_div.append(canvas); canvas_div.append(rubberband); this.rubberband = rubberband; this.rubberband_canvas = rubberband[0]; this.rubberband_context = rubberband[0].getContext("2d"); this.rubberband_context.strokeStyle = "#000000"; this._resize_canvas = function(width, height) { // Keep the size of the canvas, canvas container, and rubber band // canvas in synch. canvas_div.css('width', width) canvas_div.css('height', height) canvas.attr('width', width * mpl.ratio); canvas.attr('height', height * mpl.ratio); canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;'); rubberband.attr('width', width); rubberband.attr('height', height); } // Set the figure to an initial 600x600px, this will subsequently be updated // upon first draw. this._resize_canvas(600, 600); // Disable right mouse context menu.$(this.rubberband_canvas).bind("contextmenu",function(e){
return false;
});

function set_focus () {
canvas.focus();
canvas_div.focus();
}

window.setTimeout(set_focus, 100);
}

mpl.figure.prototype._init_toolbar = function() {
var fig = this;

var nav_element = $('<div/>') nav_element.attr('style', 'width: 100%'); this.root.append(nav_element); // Define a callback function for later on. function toolbar_event(event) { return fig.toolbar_button_onclick(event['data']); } function toolbar_mouse_event(event) { return fig.toolbar_button_onmouseover(event['data']); } for(var toolbar_ind in mpl.toolbar_items) { var name = mpl.toolbar_items[toolbar_ind][0]; var tooltip = mpl.toolbar_items[toolbar_ind][1]; var image = mpl.toolbar_items[toolbar_ind][2]; var method_name = mpl.toolbar_items[toolbar_ind][3]; if (!name) { // put a spacer in here. continue; } var button =$('<button/>');
button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +
'ui-button-icon-only');
button.attr('role', 'button');
button.attr('aria-disabled', 'false');
button.click(method_name, toolbar_event);
button.mouseover(tooltip, toolbar_mouse_event);

var icon_img = $('<span/>'); icon_img.addClass('ui-button-icon-primary ui-icon'); icon_img.addClass(image); icon_img.addClass('ui-corner-all'); var tooltip_span =$('<span/>');
tooltip_span.html(tooltip);

button.append(icon_img);
button.append(tooltip_span);

nav_element.append(button);
}

var fmt_picker_span = $('<span/>'); var fmt_picker =$('<select/>');
fmt_picker_span.append(fmt_picker);
nav_element.append(fmt_picker_span);
this.format_dropdown = fmt_picker[0];

for (var ind in mpl.extensions) {
var fmt = mpl.extensions[ind];
var option = $( '<option/>', {selected: fmt === mpl.default_extension}).html(fmt); fmt_picker.append(option) } // Add hover states to the ui-buttons$( ".ui-button" ).hover(
function() { $(this).addClass("ui-state-hover");}, function() {$(this).removeClass("ui-state-hover");}
);

var status_bar = $('<span class="mpl-message"/>'); nav_element.append(status_bar); this.message = status_bar[0]; } mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) { // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client, // which will in turn request a refresh of the image. this.send_message('resize', {'width': x_pixels, 'height': y_pixels}); } mpl.figure.prototype.send_message = function(type, properties) { properties['type'] = type; properties['figure_id'] = this.id; this.ws.send(JSON.stringify(properties)); } mpl.figure.prototype.send_draw_message = function() { if (!this.waiting) { this.waiting = true; this.ws.send(JSON.stringify({type: "draw", figure_id: this.id})); } } mpl.figure.prototype.handle_save = function(fig, msg) { var format_dropdown = fig.format_dropdown; var format = format_dropdown.options[format_dropdown.selectedIndex].value; fig.ondownload(fig, format); } mpl.figure.prototype.handle_resize = function(fig, msg) { var size = msg['size']; if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) { fig._resize_canvas(size[0], size[1]); fig.send_message("refresh", {}); }; } mpl.figure.prototype.handle_rubberband = function(fig, msg) { var x0 = msg['x0'] / mpl.ratio; var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio; var x1 = msg['x1'] / mpl.ratio; var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio; x0 = Math.floor(x0) + 0.5; y0 = Math.floor(y0) + 0.5; x1 = Math.floor(x1) + 0.5; y1 = Math.floor(y1) + 0.5; var min_x = Math.min(x0, x1); var min_y = Math.min(y0, y1); var width = Math.abs(x1 - x0); var height = Math.abs(y1 - y0); fig.rubberband_context.clearRect( 0, 0, fig.canvas.width, fig.canvas.height); fig.rubberband_context.strokeRect(min_x, min_y, width, height); } mpl.figure.prototype.handle_figure_label = function(fig, msg) { // Updates the figure title. fig.header.textContent = msg['label']; } mpl.figure.prototype.handle_cursor = function(fig, msg) { var cursor = msg['cursor']; switch(cursor) { case 0: cursor = 'pointer'; break; case 1: cursor = 'default'; break; case 2: cursor = 'crosshair'; break; case 3: cursor = 'move'; break; } fig.rubberband_canvas.style.cursor = cursor; } mpl.figure.prototype.handle_message = function(fig, msg) { fig.message.textContent = msg['message']; } mpl.figure.prototype.handle_draw = function(fig, msg) { // Request the server to send over a new figure. fig.send_draw_message(); } mpl.figure.prototype.handle_image_mode = function(fig, msg) { fig.image_mode = msg['mode']; } mpl.figure.prototype.updated_canvas_event = function() { // Called whenever the canvas gets updated. this.send_message("ack", {}); } // A function to construct a web socket function for onmessage handling. // Called in the figure constructor. mpl.figure.prototype._make_on_message_function = function(fig) { return function socket_on_message(evt) { if (evt.data instanceof Blob) { /* FIXME: We get "Resource interpreted as Image but * transferred with MIME type text/plain:" errors on * Chrome. But how to set the MIME type? It doesn't seem * to be part of the websocket stream */ evt.data.type = "image/png"; /* Free the memory for the previous frames */ if (fig.imageObj.src) { (window.URL || window.webkitURL).revokeObjectURL( fig.imageObj.src); } fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL( evt.data); fig.updated_canvas_event(); fig.waiting = false; return; } else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == "data:image/png;base64") { fig.imageObj.src = evt.data; fig.updated_canvas_event(); fig.waiting = false; return; } var msg = JSON.parse(evt.data); var msg_type = msg['type']; // Call the "handle_{type}" callback, which takes // the figure and JSON message as its only arguments. try { var callback = fig["handle_" + msg_type]; } catch (e) { console.log("No handler for the '" + msg_type + "' message type: ", msg); return; } if (callback) { try { // console.log("Handling '" + msg_type + "' message: ", msg); callback(fig, msg); } catch (e) { console.log("Exception inside the 'handler_" + msg_type + "' callback:", e, e.stack, msg); } } }; } // from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas mpl.findpos = function(e) { //this section is from http://www.quirksmode.org/js/events_properties.html var targ; if (!e) e = window.event; if (e.target) targ = e.target; else if (e.srcElement) targ = e.srcElement; if (targ.nodeType == 3) // defeat Safari bug targ = targ.parentNode; // jQuery normalizes the pageX and pageY // pageX,Y are the mouse positions relative to the document // offset() returns the position of the element relative to the document var x = e.pageX -$(targ).offset().left;
var y = e.pageY - $(targ).offset().top; return {"x": x, "y": y}; }; /* * return a copy of an object with only non-object keys * we need this to avoid circular references * http://stackoverflow.com/a/24161582/3208463 */ function simpleKeys (original) { return Object.keys(original).reduce(function (obj, key) { if (typeof original[key] !== 'object') obj[key] = original[key] return obj; }, {}); } mpl.figure.prototype.mouse_event = function(event, name) { var canvas_pos = mpl.findpos(event) if (name === 'button_press') { this.canvas.focus(); this.canvas_div.focus(); } var x = canvas_pos.x * mpl.ratio; var y = canvas_pos.y * mpl.ratio; this.send_message(name, {x: x, y: y, button: event.button, step: event.step, guiEvent: simpleKeys(event)}); /* This prevents the web browser from automatically changing to * the text insertion cursor when the button is pressed. We want * to control all of the cursor setting manually through the * 'cursor' event from matplotlib */ event.preventDefault(); return false; } mpl.figure.prototype._key_event_extra = function(event, name) { // Handle any extra behaviour associated with a key event } mpl.figure.prototype.key_event = function(event, name) { // Prevent repeat events if (name == 'key_press') { if (event.which === this._key) return; else this._key = event.which; } if (name == 'key_release') this._key = null; var value = ''; if (event.ctrlKey && event.which != 17) value += "ctrl+"; if (event.altKey && event.which != 18) value += "alt+"; if (event.shiftKey && event.which != 16) value += "shift+"; value += 'k'; value += event.which.toString(); this._key_event_extra(event, name); this.send_message(name, {key: value, guiEvent: simpleKeys(event)}); return false; } mpl.figure.prototype.toolbar_button_onclick = function(name) { if (name == 'download') { this.handle_save(this, null); } else { this.send_message("toolbar_button", {name: name}); } }; mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) { this.message.textContent = tooltip; }; 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", "Pan axes with left mouse, zoom with right", "fa fa-arrows icon-move", "pan"], ["Zoom", "Zoom to rectangle", "fa fa-square-o icon-check-empty", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o icon-save", "download"]]; mpl.extensions = ["eps", "jpeg", "pdf", "png", "ps", "raw", "svg", "tif"]; mpl.default_extension = "png";var comm_websocket_adapter = function(comm) { // Create a "websocket"-like object which calls the given IPython comm // object with the appropriate methods. Currently this is a non binary // socket, so there is still some room for performance tuning. var ws = {}; ws.close = function() { comm.close() }; ws.send = function(m) { //console.log('sending', m); comm.send(m); }; // Register the callback with on_msg. comm.on_msg(function(msg) { //console.log('receiving', msg['content']['data'], msg); // Pass the mpl event to the overridden (by mpl) onmessage function. ws.onmessage(msg['content']['data']) }); return ws; } mpl.mpl_figure_comm = function(comm, msg) { // This is the function which gets called when the mpl process // starts-up an IPython Comm through the "matplotlib" channel. var id = msg.content.data.id; // Get hold of the div created by the display call when the Comm // socket was opened in Python. var element =$("#" + id);
var ws_proxy = comm_websocket_adapter(comm)

window.open(figure.imageObj.src);
}

var fig = new mpl.figure(id, ws_proxy,
element.get(0));

// Call onopen now - mpl needs it, as it is assuming we've passed it a real
// web socket which is closed, not our websocket->open comm proxy.
ws_proxy.onopen();

fig.parent_element = element.get(0);
fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
if (!fig.cell_info) {
console.error("Failed to find cell for figure", id, fig);
return;
}

var output_index = fig.cell_info[2]
var cell = fig.cell_info[0];

};

mpl.figure.prototype.handle_close = function(fig, msg) {
var width = fig.canvas.width/mpl.ratio
fig.root.unbind('remove')

// Update the output cell to use the data from the current canvas.
fig.push_to_output();
var dataURL = fig.canvas.toDataURL();
// Re-enable the keyboard manager in IPython - without this line, in FF,
// the notebook keyboard shortcuts fail.
IPython.keyboard_manager.enable()
$(fig.parent_element).html('<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '" width="' + width + '">'); fig.close_ws(fig, msg); } mpl.figure.prototype.close_ws = function(fig, msg){ fig.send_message('closing', msg); // fig.ws.close() } mpl.figure.prototype.push_to_output = function(remove_interactive) { // Turn the data on the canvas into data in the output cell. var width = this.canvas.width/mpl.ratio var dataURL = this.canvas.toDataURL(); this.cell_info[1]['text/html'] = '<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '" width="' + width + '">'; } mpl.figure.prototype.updated_canvas_event = function() { // Tell IPython that the notebook contents must change. IPython.notebook.set_dirty(true); this.send_message("ack", {}); var fig = this; // Wait a second, then push the new image to the DOM so // that it is saved nicely (might be nice to debounce this). setTimeout(function () { fig.push_to_output() }, 1000); } mpl.figure.prototype._init_toolbar = function() { var fig = this; var nav_element =$('<div/>')
nav_element.attr('style', 'width: 100%');
this.root.append(nav_element);

// Define a callback function for later on.
function toolbar_event(event) {
return fig.toolbar_button_onclick(event['data']);
}
function toolbar_mouse_event(event) {
return fig.toolbar_button_onmouseover(event['data']);
}

for(var toolbar_ind in mpl.toolbar_items){
var name = mpl.toolbar_items[toolbar_ind][0];
var tooltip = mpl.toolbar_items[toolbar_ind][1];
var image = mpl.toolbar_items[toolbar_ind][2];
var method_name = mpl.toolbar_items[toolbar_ind][3];

if (!name) { continue; };

var button = $('<button class="btn btn-default" href="#" title="' + name + '"><i class="fa ' + image + ' fa-lg"></i></button>'); button.click(method_name, toolbar_event); button.mouseover(tooltip, toolbar_mouse_event); nav_element.append(button); } // Add the status bar. var status_bar =$('<span class="mpl-message" style="text-align:right; float: right;"/>');
nav_element.append(status_bar);
this.message = status_bar[0];

// Add the close button to the window.
var buttongrp = $('<div class="btn-group inline pull-right"></div>'); var button =$('<button class="btn btn-mini btn-primary" href="#" title="Stop Interaction"><i class="fa fa-power-off icon-remove icon-large"></i></button>');
button.click(function (evt) { fig.handle_close(fig, {}); } );
button.mouseover('Stop Interaction', toolbar_mouse_event);
buttongrp.append(button);
var titlebar = this.root.find($('.ui-dialog-titlebar')); titlebar.prepend(buttongrp); } mpl.figure.prototype._root_extra_style = function(el){ var fig = this el.on("remove", function(){ fig.close_ws(fig, {}); }); } mpl.figure.prototype._canvas_extra_style = function(el){ // this is important to make the div 'focusable el.attr('tabindex', 0) // reach out to IPython and tell the keyboard manager to turn it's self // off when our div gets focus // location in version 3 if (IPython.notebook.keyboard_manager) { IPython.notebook.keyboard_manager.register_events(el); } else { // location in version 2 IPython.keyboard_manager.register_events(el); } } mpl.figure.prototype._key_event_extra = function(event, name) { var manager = IPython.notebook.keyboard_manager; if (!manager) manager = IPython.keyboard_manager; // Check for shift+enter if (event.shiftKey && event.which == 13) { this.canvas_div.blur(); event.shiftKey = false; // Send a "J" for go to next cell event.which = 74; event.keyCode = 74; manager.command_mode(); manager.handle_keydown(event); } } mpl.figure.prototype.handle_save = function(fig, msg) { fig.ondownload(fig, null); } mpl.find_output_cell = function(html_output) { // Return the cell and output element which can be found *uniquely* in the notebook. // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook" // IPython event is triggered only after the cells have been serialised, which for // our purposes (turning an active figure into a static one), is too late. var cells = IPython.notebook.get_cells(); var ncells = cells.length; for (var i=0; i<ncells; i++) { var cell = cells[i]; if (cell.cell_type === 'code'){ for (var j=0; j<cell.output_area.outputs.length; j++) { var data = cell.output_area.outputs[j]; if (data.data) { // IPython >= 3 moved mimebundle to data attribute of output data = data.data; } if (data['text/html'] == html_output) { return [cell, data, j]; } } } } } // Register the function which deals with the matplotlib target/channel. // The kernel may be null if the page has been refreshed. if (IPython.notebook.kernel != null) { IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm); }  ## 4. Wilkinson matrix $$A_4 = \begin{pmatrix} \frac{1}{N} & 1 & & & & \\ & \frac{2}{N} & 1 & & & \\ & & \frac{3}{N} & 1 & & \\ & & & \frac{4}{N} & 1 & \\ & & & & \ddots & 1\\ & & & & & 1 \end{pmatrix}$$  In [16]: x4, y4 = np.mgrid[-1:2:npts*1j, -1.5:1.5:npts*1j] diag_0 = [k/N for k in range(1,N+1)] diag_1 = [1 for k in range(N-1)] A4 = diags([diag_0, diag_1], [0, 1], shape=(N, N))   In [17]: ps_spec4 = pseudo_spec(x4, y4, A4)   /home/nguarinz/anaconda3/lib/python3.6/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.py:1849: RuntimeWarning: invalid value encountered in sqrt s = np.sqrt(eigvals) /home/nguarinz/anaconda3/lib/python3.6/site-packages/numpy/core/fromnumeric.py:83: RuntimeWarning: invalid value encountered in reduce return ufunc.reduce(obj, axis, dtype, out, **passkwargs)   In [18]: plt.figure(figsize=(8,8)) plot_pseudo_spec(x4, y4, A4, ps_spec4) plt.show()   var element =$('#e0560e7a-640e-4af6-8868-d8c813e17a96');
/* Put everything inside the global mpl namespace */
window.mpl = {};

mpl.get_websocket_type = function() {
if (typeof(WebSocket) !== 'undefined') {
return WebSocket;
} else if (typeof(MozWebSocket) !== 'undefined') {
return MozWebSocket;
} else {
alert('Your browser does not have WebSocket support.' +
'Please try Chrome, Safari or Firefox ≥ 6. ' +
'Firefox 4 and 5 are also supported but you ' +
'have to enable WebSockets in about:config.');
};
}

mpl.figure = function(figure_id, websocket, ondownload, parent_element) {
this.id = figure_id;

this.ws = websocket;

this.supports_binary = (this.ws.binaryType != undefined);

if (!this.supports_binary) {
var warnings = document.getElementById("mpl-warnings");
if (warnings) {
warnings.style.display = 'block';
warnings.textContent = (
"This browser does not support binary websocket messages. " +
"Performance may be slow.");
}
}

this.imageObj = new Image();

this.context = undefined;
this.message = undefined;
this.canvas = undefined;
this.rubberband_canvas = undefined;
this.rubberband_context = undefined;
this.format_dropdown = undefined;

this.image_mode = 'full';

this.root = $('<div/>'); this._root_extra_style(this.root) this.root.attr('style', 'display: inline-block');$(parent_element).append(this.root);

this._init_canvas(this);
this._init_toolbar(this);

var fig = this;

this.waiting = false;

this.ws.onopen =  function () {
fig.send_message("supports_binary", {value: fig.supports_binary});
fig.send_message("send_image_mode", {});
if (mpl.ratio != 1) {
fig.send_message("set_dpi_ratio", {'dpi_ratio': mpl.ratio});
}
fig.send_message("refresh", {});
}

this.imageObj.onload = function() {
if (fig.image_mode == 'full') {
// Full images could contain transparency (where diff images
// almost always do), so we need to clear the canvas so that
// there is no ghosting.
fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);
}
fig.context.drawImage(fig.imageObj, 0, 0);
};

this.imageObj.onunload = function() {
fig.ws.close();
}

this.ws.onmessage = this._make_on_message_function(this);

}

mpl.figure.prototype._init_header = function() {
var titlebar = $( '<div class="ui-dialog-titlebar ui-widget-header ui-corner-all ' + 'ui-helper-clearfix"/>'); var titletext =$(
'<div class="ui-dialog-title" style="width: 100%; ' +
'text-align: center; padding: 3px;"/>');
titlebar.append(titletext)
this.root.append(titlebar);
}

mpl.figure.prototype._canvas_extra_style = function(canvas_div) {

}

mpl.figure.prototype._root_extra_style = function(canvas_div) {

}

mpl.figure.prototype._init_canvas = function() {
var fig = this;

var canvas_div = $('<div/>'); canvas_div.attr('style', 'position: relative; clear: both; outline: 0'); function canvas_keyboard_event(event) { return fig.key_event(event, event['data']); } canvas_div.keydown('key_press', canvas_keyboard_event); canvas_div.keyup('key_release', canvas_keyboard_event); this.canvas_div = canvas_div this._canvas_extra_style(canvas_div) this.root.append(canvas_div); var canvas =$('<canvas/>');
canvas.attr('style', "left: 0; top: 0; z-index: 0; outline: 0")

this.canvas = canvas[0];
this.context = canvas[0].getContext("2d");

var backingStore = this.context.backingStorePixelRatio ||
this.context.webkitBackingStorePixelRatio ||
this.context.mozBackingStorePixelRatio ||
this.context.msBackingStorePixelRatio ||
this.context.oBackingStorePixelRatio ||
this.context.backingStorePixelRatio || 1;

mpl.ratio = (window.devicePixelRatio || 1) / backingStore;

var rubberband = $('<canvas/>'); rubberband.attr('style', "position: absolute; left: 0; top: 0; z-index: 1;") var pass_mouse_events = true; canvas_div.resizable({ start: function(event, ui) { pass_mouse_events = false; }, resize: function(event, ui) { fig.request_resize(ui.size.width, ui.size.height); }, stop: function(event, ui) { pass_mouse_events = true; fig.request_resize(ui.size.width, ui.size.height); }, }); function mouse_event_fn(event) { if (pass_mouse_events) return fig.mouse_event(event, event['data']); } rubberband.mousedown('button_press', mouse_event_fn); rubberband.mouseup('button_release', mouse_event_fn); // Throttle sequential mouse events to 1 every 20ms. rubberband.mousemove('motion_notify', mouse_event_fn); rubberband.mouseenter('figure_enter', mouse_event_fn); rubberband.mouseleave('figure_leave', mouse_event_fn); canvas_div.on("wheel", function (event) { event = event.originalEvent; event['data'] = 'scroll' if (event.deltaY < 0) { event.step = 1; } else { event.step = -1; } mouse_event_fn(event); }); canvas_div.append(canvas); canvas_div.append(rubberband); this.rubberband = rubberband; this.rubberband_canvas = rubberband[0]; this.rubberband_context = rubberband[0].getContext("2d"); this.rubberband_context.strokeStyle = "#000000"; this._resize_canvas = function(width, height) { // Keep the size of the canvas, canvas container, and rubber band // canvas in synch. canvas_div.css('width', width) canvas_div.css('height', height) canvas.attr('width', width * mpl.ratio); canvas.attr('height', height * mpl.ratio); canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;'); rubberband.attr('width', width); rubberband.attr('height', height); } // Set the figure to an initial 600x600px, this will subsequently be updated // upon first draw. this._resize_canvas(600, 600); // Disable right mouse context menu.$(this.rubberband_canvas).bind("contextmenu",function(e){
return false;
});

function set_focus () {
canvas.focus();
canvas_div.focus();
}

window.setTimeout(set_focus, 100);
}

mpl.figure.prototype._init_toolbar = function() {
var fig = this;

var nav_element = $('<div/>') nav_element.attr('style', 'width: 100%'); this.root.append(nav_element); // Define a callback function for later on. function toolbar_event(event) { return fig.toolbar_button_onclick(event['data']); } function toolbar_mouse_event(event) { return fig.toolbar_button_onmouseover(event['data']); } for(var toolbar_ind in mpl.toolbar_items) { var name = mpl.toolbar_items[toolbar_ind][0]; var tooltip = mpl.toolbar_items[toolbar_ind][1]; var image = mpl.toolbar_items[toolbar_ind][2]; var method_name = mpl.toolbar_items[toolbar_ind][3]; if (!name) { // put a spacer in here. continue; } var button =$('<button/>');
button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +
'ui-button-icon-only');
button.attr('role', 'button');
button.attr('aria-disabled', 'false');
button.click(method_name, toolbar_event);
button.mouseover(tooltip, toolbar_mouse_event);

var icon_img = $('<span/>'); icon_img.addClass('ui-button-icon-primary ui-icon'); icon_img.addClass(image); icon_img.addClass('ui-corner-all'); var tooltip_span =$('<span/>');
tooltip_span.html(tooltip);

button.append(icon_img);
button.append(tooltip_span);

nav_element.append(button);
}

var fmt_picker_span = $('<span/>'); var fmt_picker =$('<select/>');
fmt_picker_span.append(fmt_picker);
nav_element.append(fmt_picker_span);
this.format_dropdown = fmt_picker[0];

for (var ind in mpl.extensions) {
var fmt = mpl.extensions[ind];
var option = $( '<option/>', {selected: fmt === mpl.default_extension}).html(fmt); fmt_picker.append(option) } // Add hover states to the ui-buttons$( ".ui-button" ).hover(
function() { $(this).addClass("ui-state-hover");}, function() {$(this).removeClass("ui-state-hover");}
);

var status_bar = $('<span class="mpl-message"/>'); nav_element.append(status_bar); this.message = status_bar[0]; } mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) { // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client, // which will in turn request a refresh of the image. this.send_message('resize', {'width': x_pixels, 'height': y_pixels}); } mpl.figure.prototype.send_message = function(type, properties) { properties['type'] = type; properties['figure_id'] = this.id; this.ws.send(JSON.stringify(properties)); } mpl.figure.prototype.send_draw_message = function() { if (!this.waiting) { this.waiting = true; this.ws.send(JSON.stringify({type: "draw", figure_id: this.id})); } } mpl.figure.prototype.handle_save = function(fig, msg) { var format_dropdown = fig.format_dropdown; var format = format_dropdown.options[format_dropdown.selectedIndex].value; fig.ondownload(fig, format); } mpl.figure.prototype.handle_resize = function(fig, msg) { var size = msg['size']; if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) { fig._resize_canvas(size[0], size[1]); fig.send_message("refresh", {}); }; } mpl.figure.prototype.handle_rubberband = function(fig, msg) { var x0 = msg['x0'] / mpl.ratio; var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio; var x1 = msg['x1'] / mpl.ratio; var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio; x0 = Math.floor(x0) + 0.5; y0 = Math.floor(y0) + 0.5; x1 = Math.floor(x1) + 0.5; y1 = Math.floor(y1) + 0.5; var min_x = Math.min(x0, x1); var min_y = Math.min(y0, y1); var width = Math.abs(x1 - x0); var height = Math.abs(y1 - y0); fig.rubberband_context.clearRect( 0, 0, fig.canvas.width, fig.canvas.height); fig.rubberband_context.strokeRect(min_x, min_y, width, height); } mpl.figure.prototype.handle_figure_label = function(fig, msg) { // Updates the figure title. fig.header.textContent = msg['label']; } mpl.figure.prototype.handle_cursor = function(fig, msg) { var cursor = msg['cursor']; switch(cursor) { case 0: cursor = 'pointer'; break; case 1: cursor = 'default'; break; case 2: cursor = 'crosshair'; break; case 3: cursor = 'move'; break; } fig.rubberband_canvas.style.cursor = cursor; } mpl.figure.prototype.handle_message = function(fig, msg) { fig.message.textContent = msg['message']; } mpl.figure.prototype.handle_draw = function(fig, msg) { // Request the server to send over a new figure. fig.send_draw_message(); } mpl.figure.prototype.handle_image_mode = function(fig, msg) { fig.image_mode = msg['mode']; } mpl.figure.prototype.updated_canvas_event = function() { // Called whenever the canvas gets updated. this.send_message("ack", {}); } // A function to construct a web socket function for onmessage handling. // Called in the figure constructor. mpl.figure.prototype._make_on_message_function = function(fig) { return function socket_on_message(evt) { if (evt.data instanceof Blob) { /* FIXME: We get "Resource interpreted as Image but * transferred with MIME type text/plain:" errors on * Chrome. But how to set the MIME type? It doesn't seem * to be part of the websocket stream */ evt.data.type = "image/png"; /* Free the memory for the previous frames */ if (fig.imageObj.src) { (window.URL || window.webkitURL).revokeObjectURL( fig.imageObj.src); } fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL( evt.data); fig.updated_canvas_event(); fig.waiting = false; return; } else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == "data:image/png;base64") { fig.imageObj.src = evt.data; fig.updated_canvas_event(); fig.waiting = false; return; } var msg = JSON.parse(evt.data); var msg_type = msg['type']; // Call the "handle_{type}" callback, which takes // the figure and JSON message as its only arguments. try { var callback = fig["handle_" + msg_type]; } catch (e) { console.log("No handler for the '" + msg_type + "' message type: ", msg); return; } if (callback) { try { // console.log("Handling '" + msg_type + "' message: ", msg); callback(fig, msg); } catch (e) { console.log("Exception inside the 'handler_" + msg_type + "' callback:", e, e.stack, msg); } } }; } // from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas mpl.findpos = function(e) { //this section is from http://www.quirksmode.org/js/events_properties.html var targ; if (!e) e = window.event; if (e.target) targ = e.target; else if (e.srcElement) targ = e.srcElement; if (targ.nodeType == 3) // defeat Safari bug targ = targ.parentNode; // jQuery normalizes the pageX and pageY // pageX,Y are the mouse positions relative to the document // offset() returns the position of the element relative to the document var x = e.pageX -$(targ).offset().left;
var y = e.pageY - $(targ).offset().top; return {"x": x, "y": y}; }; /* * return a copy of an object with only non-object keys * we need this to avoid circular references * http://stackoverflow.com/a/24161582/3208463 */ function simpleKeys (original) { return Object.keys(original).reduce(function (obj, key) { if (typeof original[key] !== 'object') obj[key] = original[key] return obj; }, {}); } mpl.figure.prototype.mouse_event = function(event, name) { var canvas_pos = mpl.findpos(event) if (name === 'button_press') { this.canvas.focus(); this.canvas_div.focus(); } var x = canvas_pos.x * mpl.ratio; var y = canvas_pos.y * mpl.ratio; this.send_message(name, {x: x, y: y, button: event.button, step: event.step, guiEvent: simpleKeys(event)}); /* This prevents the web browser from automatically changing to * the text insertion cursor when the button is pressed. We want * to control all of the cursor setting manually through the * 'cursor' event from matplotlib */ event.preventDefault(); return false; } mpl.figure.prototype._key_event_extra = function(event, name) { // Handle any extra behaviour associated with a key event } mpl.figure.prototype.key_event = function(event, name) { // Prevent repeat events if (name == 'key_press') { if (event.which === this._key) return; else this._key = event.which; } if (name == 'key_release') this._key = null; var value = ''; if (event.ctrlKey && event.which != 17) value += "ctrl+"; if (event.altKey && event.which != 18) value += "alt+"; if (event.shiftKey && event.which != 16) value += "shift+"; value += 'k'; value += event.which.toString(); this._key_event_extra(event, name); this.send_message(name, {key: value, guiEvent: simpleKeys(event)}); return false; } mpl.figure.prototype.toolbar_button_onclick = function(name) { if (name == 'download') { this.handle_save(this, null); } else { this.send_message("toolbar_button", {name: name}); } }; mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) { this.message.textContent = tooltip; }; 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", "Pan axes with left mouse, zoom with right", "fa fa-arrows icon-move", "pan"], ["Zoom", "Zoom to rectangle", "fa fa-square-o icon-check-empty", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o icon-save", "download"]]; mpl.extensions = ["eps", "jpeg", "pdf", "png", "ps", "raw", "svg", "tif"]; mpl.default_extension = "png";var comm_websocket_adapter = function(comm) { // Create a "websocket"-like object which calls the given IPython comm // object with the appropriate methods. Currently this is a non binary // socket, so there is still some room for performance tuning. var ws = {}; ws.close = function() { comm.close() }; ws.send = function(m) { //console.log('sending', m); comm.send(m); }; // Register the callback with on_msg. comm.on_msg(function(msg) { //console.log('receiving', msg['content']['data'], msg); // Pass the mpl event to the overridden (by mpl) onmessage function. ws.onmessage(msg['content']['data']) }); return ws; } mpl.mpl_figure_comm = function(comm, msg) { // This is the function which gets called when the mpl process // starts-up an IPython Comm through the "matplotlib" channel. var id = msg.content.data.id; // Get hold of the div created by the display call when the Comm // socket was opened in Python. var element =$("#" + id);
var ws_proxy = comm_websocket_adapter(comm)

window.open(figure.imageObj.src);
}

var fig = new mpl.figure(id, ws_proxy,
element.get(0));

// Call onopen now - mpl needs it, as it is assuming we've passed it a real
// web socket which is closed, not our websocket->open comm proxy.
ws_proxy.onopen();

fig.parent_element = element.get(0);
fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
if (!fig.cell_info) {
console.error("Failed to find cell for figure", id, fig);
return;
}

var output_index = fig.cell_info[2]
var cell = fig.cell_info[0];

};

mpl.figure.prototype.handle_close = function(fig, msg) {
var width = fig.canvas.width/mpl.ratio
fig.root.unbind('remove')

// Update the output cell to use the data from the current canvas.
fig.push_to_output();
var dataURL = fig.canvas.toDataURL();
// Re-enable the keyboard manager in IPython - without this line, in FF,
// the notebook keyboard shortcuts fail.
IPython.keyboard_manager.enable()
$(fig.parent_element).html('<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '" width="' + width + '">'); fig.close_ws(fig, msg); } mpl.figure.prototype.close_ws = function(fig, msg){ fig.send_message('closing', msg); // fig.ws.close() } mpl.figure.prototype.push_to_output = function(remove_interactive) { // Turn the data on the canvas into data in the output cell. var width = this.canvas.width/mpl.ratio var dataURL = this.canvas.toDataURL(); this.cell_info[1]['text/html'] = '<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '" width="' + width + '">'; } mpl.figure.prototype.updated_canvas_event = function() { // Tell IPython that the notebook contents must change. IPython.notebook.set_dirty(true); this.send_message("ack", {}); var fig = this; // Wait a second, then push the new image to the DOM so // that it is saved nicely (might be nice to debounce this). setTimeout(function () { fig.push_to_output() }, 1000); } mpl.figure.prototype._init_toolbar = function() { var fig = this; var nav_element =$('<div/>')
nav_element.attr('style', 'width: 100%');
this.root.append(nav_element);

// Define a callback function for later on.
function toolbar_event(event) {
return fig.toolbar_button_onclick(event['data']);
}
function toolbar_mouse_event(event) {
return fig.toolbar_button_onmouseover(event['data']);
}

for(var toolbar_ind in mpl.toolbar_items){
var name = mpl.toolbar_items[toolbar_ind][0];
var tooltip = mpl.toolbar_items[toolbar_ind][1];
var image = mpl.toolbar_items[toolbar_ind][2];
var method_name = mpl.toolbar_items[toolbar_ind][3];

if (!name) { continue; };

var button = $('<button class="btn btn-default" href="#" title="' + name + '"><i class="fa ' + image + ' fa-lg"></i></button>'); button.click(method_name, toolbar_event); button.mouseover(tooltip, toolbar_mouse_event); nav_element.append(button); } // Add the status bar. var status_bar =$('<span class="mpl-message" style="text-align:right; float: right;"/>');
nav_element.append(status_bar);
this.message = status_bar[0];

// Add the close button to the window.
var buttongrp = $('<div class="btn-group inline pull-right"></div>'); var button =$('<button class="btn btn-mini btn-primary" href="#" title="Stop Interaction"><i class="fa fa-power-off icon-remove icon-large"></i></button>');
button.click(function (evt) { fig.handle_close(fig, {}); } );
button.mouseover('Stop Interaction', toolbar_mouse_event);
buttongrp.append(button);
var titlebar = this.root.find($('.ui-dialog-titlebar')); titlebar.prepend(buttongrp); } mpl.figure.prototype._root_extra_style = function(el){ var fig = this el.on("remove", function(){ fig.close_ws(fig, {}); }); } mpl.figure.prototype._canvas_extra_style = function(el){ // this is important to make the div 'focusable el.attr('tabindex', 0) // reach out to IPython and tell the keyboard manager to turn it's self // off when our div gets focus // location in version 3 if (IPython.notebook.keyboard_manager) { IPython.notebook.keyboard_manager.register_events(el); } else { // location in version 2 IPython.keyboard_manager.register_events(el); } } mpl.figure.prototype._key_event_extra = function(event, name) { var manager = IPython.notebook.keyboard_manager; if (!manager) manager = IPython.keyboard_manager; // Check for shift+enter if (event.shiftKey && event.which == 13) { this.canvas_div.blur(); event.shiftKey = false; // Send a "J" for go to next cell event.which = 74; event.keyCode = 74; manager.command_mode(); manager.handle_keydown(event); } } mpl.figure.prototype.handle_save = function(fig, msg) { fig.ondownload(fig, null); } mpl.find_output_cell = function(html_output) { // Return the cell and output element which can be found *uniquely* in the notebook. // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook" // IPython event is triggered only after the cells have been serialised, which for // our purposes (turning an active figure into a static one), is too late. var cells = IPython.notebook.get_cells(); var ncells = cells.length; for (var i=0; i<ncells; i++) { var cell = cells[i]; if (cell.cell_type === 'code'){ for (var j=0; j<cell.output_area.outputs.length; j++) { var data = cell.output_area.outputs[j]; if (data.data) { // IPython >= 3 moved mimebundle to data attribute of output data = data.data; } if (data['text/html'] == html_output) { return [cell, data, j]; } } } } } // Register the function which deals with the matplotlib target/channel. // The kernel may be null if the page has been refreshed. if (IPython.notebook.kernel != null) { IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm); }   In [ ]:   In [19]: from IPython.core.display import HTML def css_styling(): styles = open('./styles/custom_barba.css', 'r').read() return HTML(styles) css_styling()   Out[19]: /* Based on Lorena Barba template available at: https://github.com/barbagroup/AeroPython/blob/master/styles/custom.css*/ @font-face { font-family: "Computer Modern"; src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf'); } div.cell{ width:800px; margin-left:16% !important; margin-right:auto; } h1 { font-family: 'Alegreya Sans', sans-serif; } h2 { font-family: 'Fenix', serif; } h3{ font-family: 'Fenix', serif; margin-top:12px; margin-bottom: 3px; } h4{ font-family: 'Fenix', serif; } h5 { font-family: 'Alegreya Sans', sans-serif; } div.text_cell_render{ font-family: 'Alegreya Sans',Computer Modern, "Helvetica Neue", Arial, Helvetica, Geneva, sans-serif; line-height: 135%; font-size: 120%; width:600px; margin-left:auto; margin-right:auto; } .CodeMirror{ font-family: "Source Code Pro"; font-size: 90%; } /* .prompt{ display: None; }*/ .text_cell_render h1 { font-weight: 200; font-size: 50pt; line-height: 100%; color:#CD2305; margin-bottom: 0.5em; margin-top: 0.5em; display: block; } .text_cell_render h5 { font-weight: 300; font-size: 16pt; color: #CD2305; font-style: italic; margin-bottom: .5em; margin-top: 0.5em; display: block; } .warning{ color: rgb( 240, 20, 20 ) } MathJax.Hub.Config({ TeX: { extensions: ["AMSmath.js"] }, tex2jax: { inlineMath: [ ['$','\$'], ["\$","\$"] ],
displayMath: [ ['$$','$$'], ["\$","\$"] ]
},
displayAlign: 'center', // Change this to 'center' to center equations.
"HTML-CSS": {
styles: {'.MathJax_Display': {"margin": 4}}
}
});




In [ ]: