# Lobatto and Chebyshev interpolation



In [1]:

from __future__ import print_function, division
import numpy as np
import matplotlib.pyplot as plt
import sympy as sym
from sympy.core import S, Dummy, pi
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.trigonometric import sin, cos
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.special.gamma_functions import gamma
from sympy.polys.orthopolys import legendre_poly, laguerre_poly, hermite_poly, jacobi_poly
from sympy.polys.rootoftools import RootOf




In [2]:

def gauss_lobatto(n, n_digits):
r"""
Computes the Gauss-Legendre quadrature [1]_ points and weights.

The Gauss-Lobatto quadrature approximates the integral:

.. math::
\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)

The nodes x_i of an order n quadrature rule are the roots of P'_(n-1)
and the weights w_i are given by:

.. math::
w_i = \frac{2}{n(n-1) \left[P_{n-1}(x_i)\right]^2},\quad x\neq\pm 1
w_i = \frac{2}{n(n-1)},\quad x=\pm 1

Parameters
==========

n : the order of quadrature

n_digits : number of significant digits of the points and weights to return

Returns
=======

(x, w) : the x and w are lists of points and weights as Floats.
The points x_i and weights w_i are returned as (x, w)
tuple of lists.

Examples
========

>>> #from sympy.integrals.quadrature import gauss_lobatto
>>> x, w = gauss_lobatto(3, 5)
>>> x
[-1, 0, 1]
>>> w
[0.33333, 1.3333, 0.33333]
>>> x, w = gauss_lobatto(4, 5)
>>> x
[-1, -0.44721, 0.44721, 1]
>>> w
[0.16667, 0.83333, 0.83333, 0.16667]

========

gauss_legendre,gauss_laguerre, gauss_gen_laguerre, gauss_hermite, gauss_chebyshev_t,\
gauss_chebyshev_u, gauss_jacobi

References
==========

.. [2] http://people.math.sfu.ca/~cbm/aands/page_888.htm
"""
x = Dummy("x")
p = legendre_poly(n-1, x, polys=True)
pd = p.diff(x)
xi = []
w  = []
for r in pd.real_roots():
if isinstance(r, RootOf):
r = r.eval_rational(S(1)/10**(n_digits+2))
xi.append(r.n(n_digits))
w.append((2/(n*(n-1) * p.subs(x, r)**2)).n(n_digits))

xi.insert(0, -1)
xi.append(1)
w.insert(0,(S(2)/(n*(n-1))).n(n_digits))
w.append((S(2)/(n*(n-1))).n(n_digits))

return xi, w




In [3]:

%matplotlib notebook
plt.style.use("ggplot")




In [4]:

y = sym.symbols('y')



Gauss-Lobatto quadrature can integrate exactly polynomials up to $2n-3$, where $n$ is the number of points used

## Fourth order polynomial



In [5]:

poly = 4*y**4 + 3*y**3 + y - 1
sym.plot(poly, (y, -1, 1));




var element = $('#888a49d1-9ebf-419b-a1bc-23a6238a2590'); /* Put everything inside the global mpl namespace */ window.mpl = {}; mpl.get_websocket_type = function() { if (typeof(WebSocket) !== 'undefined') { return WebSocket; } else if (typeof(MozWebSocket) !== 'undefined') { return MozWebSocket; } else { alert('Your browser does not have WebSocket support.' + 'Please try Chrome, Safari or Firefox ≥ 6. ' + 'Firefox 4 and 5 are also supported but you ' + 'have to enable WebSockets in about:config.'); }; } mpl.figure = function(figure_id, websocket, ondownload, parent_element) { this.id = figure_id; this.ws = websocket; this.supports_binary = (this.ws.binaryType != undefined); if (!this.supports_binary) { var warnings = document.getElementById("mpl-warnings"); if (warnings) { warnings.style.display = 'block'; warnings.textContent = ( "This browser does not support binary websocket messages. " + "Performance may be slow."); } } this.imageObj = new Image(); this.context = undefined; this.message = undefined; this.canvas = undefined; this.rubberband_canvas = undefined; this.rubberband_context = undefined; this.format_dropdown = undefined; this.image_mode = 'full'; this.root =$('<div/>');
this._root_extra_style(this.root)
this.root.attr('style', 'display: inline-block');

$(parent_element).append(this.root); this._init_header(this); this._init_canvas(this); this._init_toolbar(this); var fig = this; this.waiting = false; this.ws.onopen = function () { fig.send_message("supports_binary", {value: fig.supports_binary}); fig.send_message("send_image_mode", {}); fig.send_message("refresh", {}); } this.imageObj.onload = function() { if (fig.image_mode == 'full') { // Full images could contain transparency (where diff images // almost always do), so we need to clear the canvas so that // there is no ghosting. fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height); } fig.context.drawImage(fig.imageObj, 0, 0); }; this.imageObj.onunload = function() { this.ws.close(); } this.ws.onmessage = this._make_on_message_function(this); this.ondownload = ondownload; } mpl.figure.prototype._init_header = function() { var titlebar =$(
'<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); this.header = titletext[0]; } 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 = $('<canvas/>'); canvas.addClass('mpl-canvas'); canvas.attr('style', "left: 0; top: 0; z-index: 0; outline: 0") this.canvas = canvas[0]; this.context = canvas[0].getContext("2d"); var rubberband =$('<canvas/>');
rubberband.attr('style', "position: absolute; left: 0; top: 0; z-index: 1;")

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fig.request_resize(ui.size.width, ui.size.height);
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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);
canvas.attr('height', height);

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/>');

var tooltip_span = $('<span/>'); tooltip_span.addClass('ui-button-text'); tooltip_span.html(tooltip); button.append(icon_img); button.append(tooltip_span); nav_element.append(button); } var fmt_picker_span =$('<span/>');

var fmt_picker = $('<select/>'); fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content'); fmt_picker_span.append(fmt_picker); nav_element.append(fmt_picker_span); this.format_dropdown = fmt_picker[0]; for (var ind in mpl.extensions) { var fmt = mpl.extensions[ind]; var option =$(
'<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;
}

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'];
var y0 = fig.canvas.height - msg['y0'];
var x1 = msg['x1'];
var y1 = fig.canvas.height - msg['y1'];
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.
}

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;
var y = canvas_pos.y;

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) {
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 overriden (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 = $("#" + id); var ws_proxy = comm_websocket_adapter(comm) function ondownload(figure, format) { window.open(figure.imageObj.src); } var fig = new mpl.figure(id, ws_proxy, ondownload, element.get(0)); // Call onopen now - mpl needs it, as it is assuming we've passed it a real // web socket which is closed, not our websocket->open comm proxy. ws_proxy.onopen(); fig.parent_element = element.get(0); fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>"); if (!fig.cell_info) { console.error("Failed to find cell for figure", id, fig); return; } var output_index = fig.cell_info[2] var cell = fig.cell_info[0]; }; mpl.figure.prototype.handle_close = function(fig, msg) { fig.root.unbind('remove') // Update the output cell to use the data from the current canvas. fig.push_to_output(); var dataURL = fig.canvas.toDataURL(); // Re-enable the keyboard manager in IPython - without this line, in FF, // the notebook keyboard shortcuts fail. IPython.keyboard_manager.enable()$(fig.parent_element).html('<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '">');
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 dataURL = this.canvas.toDataURL();
this.cell_info[1]['text/html'] = '<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '">';
}

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) {
}

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 [6]:

print(sym.integrate(poly,(y,-1,1)))
print(sym.integrate(poly,(y,-1,1)).n(8))




-2/5
-0.40000000




In [7]:

for k in range(2,10):
xi, w = gauss_lobatto(k, 8)
poly_int = sum(w[j]*poly.subs(y, xi[j]) for j in range(k))
print(k, poly_int)




2 6.0000000
3 0.66666667
4 -0.40000000
5 -0.40000000
6 -0.40000000
7 -0.40000000
8 -0.40000000
9 -0.40000000



## Fifth order polynomial



In [8]:

poly = 2*y**5 - 2*y**4 + 3*y**3 - 4*y**2  + 5*y - 1
sym.plot(poly, (y, -1, 1));




var element = $('#8e9a4403-2440-4505-89be-05a956e95789'); /* Put everything inside the global mpl namespace */ window.mpl = {}; mpl.get_websocket_type = function() { if (typeof(WebSocket) !== 'undefined') { return WebSocket; } else if (typeof(MozWebSocket) !== 'undefined') { return MozWebSocket; } else { alert('Your browser does not have WebSocket support.' + 'Please try Chrome, Safari or Firefox ≥ 6. ' + 'Firefox 4 and 5 are also supported but you ' + 'have to enable WebSockets in about:config.'); }; } mpl.figure = function(figure_id, websocket, ondownload, parent_element) { this.id = figure_id; this.ws = websocket; this.supports_binary = (this.ws.binaryType != undefined); if (!this.supports_binary) { var warnings = document.getElementById("mpl-warnings"); if (warnings) { warnings.style.display = 'block'; warnings.textContent = ( "This browser does not support binary websocket messages. " + "Performance may be slow."); } } this.imageObj = new Image(); this.context = undefined; this.message = undefined; this.canvas = undefined; this.rubberband_canvas = undefined; this.rubberband_context = undefined; this.format_dropdown = undefined; this.image_mode = 'full'; this.root =$('<div/>');
this._root_extra_style(this.root)
this.root.attr('style', 'display: inline-block');

$(parent_element).append(this.root); this._init_header(this); this._init_canvas(this); this._init_toolbar(this); var fig = this; this.waiting = false; this.ws.onopen = function () { fig.send_message("supports_binary", {value: fig.supports_binary}); fig.send_message("send_image_mode", {}); fig.send_message("refresh", {}); } this.imageObj.onload = function() { if (fig.image_mode == 'full') { // Full images could contain transparency (where diff images // almost always do), so we need to clear the canvas so that // there is no ghosting. fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height); } fig.context.drawImage(fig.imageObj, 0, 0); }; this.imageObj.onunload = function() { this.ws.close(); } this.ws.onmessage = this._make_on_message_function(this); this.ondownload = ondownload; } mpl.figure.prototype._init_header = function() { var titlebar =$(
'<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); this.header = titletext[0]; } 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 = $('<canvas/>'); canvas.addClass('mpl-canvas'); canvas.attr('style', "left: 0; top: 0; z-index: 0; outline: 0") this.canvas = canvas[0]; this.context = canvas[0].getContext("2d"); var rubberband =$('<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);
canvas.attr('height', height);

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/>');

var tooltip_span = $('<span/>'); tooltip_span.addClass('ui-button-text'); tooltip_span.html(tooltip); button.append(icon_img); button.append(tooltip_span); nav_element.append(button); } var fmt_picker_span =$('<span/>');

var fmt_picker = $('<select/>'); fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content'); fmt_picker_span.append(fmt_picker); nav_element.append(fmt_picker_span); this.format_dropdown = fmt_picker[0]; for (var ind in mpl.extensions) { var fmt = mpl.extensions[ind]; var option =$(
'<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;
}

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'];
var y0 = fig.canvas.height - msg['y0'];
var x1 = msg['x1'];
var y1 = fig.canvas.height - msg['y1'];
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.
}

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;
var y = canvas_pos.y;

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) {
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 overriden (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 = $("#" + id); var ws_proxy = comm_websocket_adapter(comm) function ondownload(figure, format) { window.open(figure.imageObj.src); } var fig = new mpl.figure(id, ws_proxy, ondownload, element.get(0)); // Call onopen now - mpl needs it, as it is assuming we've passed it a real // web socket which is closed, not our websocket->open comm proxy. ws_proxy.onopen(); fig.parent_element = element.get(0); fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>"); if (!fig.cell_info) { console.error("Failed to find cell for figure", id, fig); return; } var output_index = fig.cell_info[2] var cell = fig.cell_info[0]; }; mpl.figure.prototype.handle_close = function(fig, msg) { fig.root.unbind('remove') // Update the output cell to use the data from the current canvas. fig.push_to_output(); var dataURL = fig.canvas.toDataURL(); // Re-enable the keyboard manager in IPython - without this line, in FF, // the notebook keyboard shortcuts fail. IPython.keyboard_manager.enable()$(fig.parent_element).html('<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '">');
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 dataURL = this.canvas.toDataURL();
this.cell_info[1]['text/html'] = '<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '">';
}

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) {
}

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 [9]:

print(sym.integrate(poly,(y,-1,1)))
print(sym.integrate(poly,(y,-1,1)).n(8))




-82/15
-5.4666667




In [10]:

for k in range(2,10):
xi, w = gauss_lobatto(k, 8)
poly_int = sum(w[j]*poly.subs(y, xi[j]) for j in range(k))
print(k, poly_int)




2 -14.000000
3 -6.0000000
4 -5.4666667
5 -5.4666667
6 -5.4666667
7 -5.4666667
8 -5.4666667
9 -5.4666667



## Trascendent function



In [11]:

fun = y*sym.sin(5*y) + y**2
sym.plot(fun, (y, -1, 1));




var element = $('#3c4e6a32-a47c-416a-b21c-560dd32cd2bc'); /* Put everything inside the global mpl namespace */ window.mpl = {}; mpl.get_websocket_type = function() { if (typeof(WebSocket) !== 'undefined') { return WebSocket; } else if (typeof(MozWebSocket) !== 'undefined') { return MozWebSocket; } else { alert('Your browser does not have WebSocket support.' + 'Please try Chrome, Safari or Firefox ≥ 6. ' + 'Firefox 4 and 5 are also supported but you ' + 'have to enable WebSockets in about:config.'); }; } mpl.figure = function(figure_id, websocket, ondownload, parent_element) { this.id = figure_id; this.ws = websocket; this.supports_binary = (this.ws.binaryType != undefined); if (!this.supports_binary) { var warnings = document.getElementById("mpl-warnings"); if (warnings) { warnings.style.display = 'block'; warnings.textContent = ( "This browser does not support binary websocket messages. " + "Performance may be slow."); } } this.imageObj = new Image(); this.context = undefined; this.message = undefined; this.canvas = undefined; this.rubberband_canvas = undefined; this.rubberband_context = undefined; this.format_dropdown = undefined; this.image_mode = 'full'; this.root =$('<div/>');
this._root_extra_style(this.root)
this.root.attr('style', 'display: inline-block');

$(parent_element).append(this.root); this._init_header(this); this._init_canvas(this); this._init_toolbar(this); var fig = this; this.waiting = false; this.ws.onopen = function () { fig.send_message("supports_binary", {value: fig.supports_binary}); fig.send_message("send_image_mode", {}); fig.send_message("refresh", {}); } this.imageObj.onload = function() { if (fig.image_mode == 'full') { // Full images could contain transparency (where diff images // almost always do), so we need to clear the canvas so that // there is no ghosting. fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height); } fig.context.drawImage(fig.imageObj, 0, 0); }; this.imageObj.onunload = function() { this.ws.close(); } this.ws.onmessage = this._make_on_message_function(this); this.ondownload = ondownload; } mpl.figure.prototype._init_header = function() { var titlebar =$(
'<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); this.header = titletext[0]; } 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 = $('<canvas/>'); canvas.addClass('mpl-canvas'); canvas.attr('style', "left: 0; top: 0; z-index: 0; outline: 0") this.canvas = canvas[0]; this.context = canvas[0].getContext("2d"); var rubberband =$('<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);
canvas.attr('height', height);

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/>');

var tooltip_span = $('<span/>'); tooltip_span.addClass('ui-button-text'); tooltip_span.html(tooltip); button.append(icon_img); button.append(tooltip_span); nav_element.append(button); } var fmt_picker_span =$('<span/>');

var fmt_picker = $('<select/>'); fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content'); fmt_picker_span.append(fmt_picker); nav_element.append(fmt_picker_span); this.format_dropdown = fmt_picker[0]; for (var ind in mpl.extensions) { var fmt = mpl.extensions[ind]; var option =$(
'<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;
}

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'];
var y0 = fig.canvas.height - msg['y0'];
var x1 = msg['x1'];
var y1 = fig.canvas.height - msg['y1'];
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.
}

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;
var y = canvas_pos.y;

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) {
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 overriden (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 = $("#" + id); var ws_proxy = comm_websocket_adapter(comm) function ondownload(figure, format) { window.open(figure.imageObj.src); } var fig = new mpl.figure(id, ws_proxy, ondownload, element.get(0)); // Call onopen now - mpl needs it, as it is assuming we've passed it a real // web socket which is closed, not our websocket->open comm proxy. ws_proxy.onopen(); fig.parent_element = element.get(0); fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>"); if (!fig.cell_info) { console.error("Failed to find cell for figure", id, fig); return; } var output_index = fig.cell_info[2] var cell = fig.cell_info[0]; }; mpl.figure.prototype.handle_close = function(fig, msg) { fig.root.unbind('remove') // Update the output cell to use the data from the current canvas. fig.push_to_output(); var dataURL = fig.canvas.toDataURL(); // Re-enable the keyboard manager in IPython - without this line, in FF, // the notebook keyboard shortcuts fail. IPython.keyboard_manager.enable()$(fig.parent_element).html('<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '">');
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 dataURL = this.canvas.toDataURL();
this.cell_info[1]['text/html'] = '<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '">';
}

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) {
}

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 [12]:

fun_int = sym.integrate(fun,(y,-1,1))
print(fun_int)
print(fun_int.n(8))




-2*cos(5)/5 + 2*sin(5)/25 + 2/3
0.47648785




In [13]:

print("{}\t{}\t{}".format('n', 'Integral', 'Relative error'))
print("===========================================")
for k in range(2,20):
xi, w = gauss_lobatto(k, 8)
poly_int = sum(w[j]*fun.subs(y, xi[j]) for j in range(k))
print("{}\t{}\t{}".format(k, float(poly_int), float((fun_int - poly_int)/poly_int)))




n	Integral	Relative error
===========================================
2	0.0821514506737	4.80011488259
3	0.027383816904	16.4003446056
4	0.933433420263	-0.48953204331
5	0.381288445645	0.249678180132
6	0.48623530045	-0.0200467748469
7	0.475875933472	0.00128587588386
8	0.476514071062	-5.50250332342e-05
9	0.476487031109	1.72040244723e-06
10	0.476487871346	-4.29955194288e-08
11	0.47648784975	2.32630894947e-09
12	0.476487849619	2.54300561481e-09
13	0.476487850528	6.94897244055e-10
14	0.47648785095	-2.48946810218e-10
15	0.476487850553	7.00193794494e-10
16	0.476487850467	9.38803083512e-10
17	0.476487850257	1.14711144529e-09
18	0.476487850304	1.22253587094e-09
19	0.47648785107	-5.01521965971e-10



## Comparison of Lobatto and Chebyshev nodes



In [14]:

norder = 10

min_cheb = []
max_cheb = []
min_lob = []
max_lob = []
plt.figure(figsize=(10, 4))
plt.subplot(1, 2, 1)
x_cheb = []
x_lob = []
y = []
for n in range(1, norder + 1):
x_ch = [np.cos((2*i - 1)/(2*n)*np.pi) for i in range(n, 0, -1)]
x_ch = [-1] + x_ch + [1]
x_cheb += x_ch
x_lo, _ = gauss_lobatto(n + 2, 16)
x_lob += x_lo
dx_cheb = [x_ch[k + 1] - x_ch[k] for k in range(0, n + 1)]
dx_lob = [x_lo[k + 1] - x_lo[k] for k in range(0, n + 1)]
min_cheb.append(min(dx_cheb))
max_cheb.append(max(dx_cheb))
min_lob.append(min(dx_lob))
max_lob.append(max(dx_lob))
y += [n for i in range(n + 2)]

plt.plot(x_cheb, y, marker='s', lw=0)
plt.plot(x_lob, y, marker='v', lw=0)

#%%
plt.xlim(-1.2, 1.2)
plt.ylim(0, norder)
plt.subplot(1, 2, 2)
plt.plot(range(2, norder + 2), min_cheb, label="Min distance Chebyshev",
marker="s", lw=0)
plt.plot(range(2, norder + 2), max_cheb, label="Max distance Chebyshev",
marker="o", lw=0)
plt.plot(range(2, norder + 2), min_lob, label="Min distance Lobatto",
marker="v", lw=0)
plt.plot(range(2, norder + 2), max_lob, label="Max distance Lobatto",
marker="^", lw=0)
plt.legend(loc="best")
plt.show()




var element = $('#4eaef0ec-e377-43cd-bc93-e35edb6a2b96'); /* Put everything inside the global mpl namespace */ window.mpl = {}; mpl.get_websocket_type = function() { if (typeof(WebSocket) !== 'undefined') { return WebSocket; } else if (typeof(MozWebSocket) !== 'undefined') { return MozWebSocket; } else { alert('Your browser does not have WebSocket support.' + 'Please try Chrome, Safari or Firefox ≥ 6. ' + 'Firefox 4 and 5 are also supported but you ' + 'have to enable WebSockets in about:config.'); }; } mpl.figure = function(figure_id, websocket, ondownload, parent_element) { this.id = figure_id; this.ws = websocket; this.supports_binary = (this.ws.binaryType != undefined); if (!this.supports_binary) { var warnings = document.getElementById("mpl-warnings"); if (warnings) { warnings.style.display = 'block'; warnings.textContent = ( "This browser does not support binary websocket messages. " + "Performance may be slow."); } } this.imageObj = new Image(); this.context = undefined; this.message = undefined; this.canvas = undefined; this.rubberband_canvas = undefined; this.rubberband_context = undefined; this.format_dropdown = undefined; this.image_mode = 'full'; this.root =$('<div/>');
this._root_extra_style(this.root)
this.root.attr('style', 'display: inline-block');

$(parent_element).append(this.root); this._init_header(this); this._init_canvas(this); this._init_toolbar(this); var fig = this; this.waiting = false; this.ws.onopen = function () { fig.send_message("supports_binary", {value: fig.supports_binary}); fig.send_message("send_image_mode", {}); fig.send_message("refresh", {}); } this.imageObj.onload = function() { if (fig.image_mode == 'full') { // Full images could contain transparency (where diff images // almost always do), so we need to clear the canvas so that // there is no ghosting. fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height); } fig.context.drawImage(fig.imageObj, 0, 0); }; this.imageObj.onunload = function() { this.ws.close(); } this.ws.onmessage = this._make_on_message_function(this); this.ondownload = ondownload; } mpl.figure.prototype._init_header = function() { var titlebar =$(
'<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); this.header = titletext[0]; } 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 = $('<canvas/>'); canvas.addClass('mpl-canvas'); canvas.attr('style', "left: 0; top: 0; z-index: 0; outline: 0") this.canvas = canvas[0]; this.context = canvas[0].getContext("2d"); var rubberband =$('<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);
canvas.attr('height', height);

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/>');

var tooltip_span = $('<span/>'); tooltip_span.addClass('ui-button-text'); tooltip_span.html(tooltip); button.append(icon_img); button.append(tooltip_span); nav_element.append(button); } var fmt_picker_span =$('<span/>');

var fmt_picker = $('<select/>'); fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content'); fmt_picker_span.append(fmt_picker); nav_element.append(fmt_picker_span); this.format_dropdown = fmt_picker[0]; for (var ind in mpl.extensions) { var fmt = mpl.extensions[ind]; var option =$(
'<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;
}

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'];
var y0 = fig.canvas.height - msg['y0'];
var x1 = msg['x1'];
var y1 = fig.canvas.height - msg['y1'];
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.
}

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;
var y = canvas_pos.y;

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) {
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 overriden (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 = $("#" + id); var ws_proxy = comm_websocket_adapter(comm) function ondownload(figure, format) { window.open(figure.imageObj.src); } var fig = new mpl.figure(id, ws_proxy, ondownload, element.get(0)); // Call onopen now - mpl needs it, as it is assuming we've passed it a real // web socket which is closed, not our websocket->open comm proxy. ws_proxy.onopen(); fig.parent_element = element.get(0); fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>"); if (!fig.cell_info) { console.error("Failed to find cell for figure", id, fig); return; } var output_index = fig.cell_info[2] var cell = fig.cell_info[0]; }; mpl.figure.prototype.handle_close = function(fig, msg) { fig.root.unbind('remove') // Update the output cell to use the data from the current canvas. fig.push_to_output(); var dataURL = fig.canvas.toDataURL(); // Re-enable the keyboard manager in IPython - without this line, in FF, // the notebook keyboard shortcuts fail. IPython.keyboard_manager.enable()$(fig.parent_element).html('<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '">');
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 dataURL = this.canvas.toDataURL();
this.cell_info[1]['text/html'] = '<amp-img layout="responsive" width="500" height="300" src="' + dataURL + '">';
}

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) {
}

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 [15]:

from IPython.core.display import HTML
def css_styling():
styles = open('./styles/custom_barba.css', 'r').read()
return HTML(styles)
css_styling()




Out[15]:

/* 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 [ ]: