{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Fixed point method and iterative methods for solving linear systems\n", "\n", "
\n", "\n", "This problem illustrates how we can used a \"pre-conditioned fixed-point method\" to solve the problem \n", "\n", "\\begin{equation*}\n", "f(x) = b-ax = 0\n", "\\end{equation*}\n", "\n", "using the equivalent fixed point iteration $x = g(x)$, where\n", "\n", "\\begin{equation*}\n", "g(x) \\equiv m^{-1}(b-ax) + x\n", "\\end{equation*}\n", "\n", "for some value of $m$. \n", "\n", "To see how the choice of $m$ affect the convergence of the fixed point method, we rewrite the fixed point method as \n", "\n", "\$$\n", "x = (1-m^{-1}a)x + m^{-1}b\n", "\$$\n", "\n", "The fixed point iteration is then given by\n", "\n", "\$$\n", "x_{k+1} = (1-m^{-1}a)x_k + m^{-1}b.\n", "\$$\n", "\n", "Defining $e_{k} \\equiv x_k - x_{k-1}$, we get\n", "\n", "\$$\n", "e_{k} = (1-m^{-1}a)e_{k-1} = ... = (1-m^{-1}a)^{k} e_0\n", "\$$\n", "\n", "or\n", "\n", "\$$\n", "\\left|e_k\\right| \\le \\left|1 - m^{-1}a\\right|^k e_0\n", "\$$\n", "\n", "Convergence is then assured if \n", "\n", "\$$\n", "\\left|1 - m^{-1}a\\right| < 1\n", "\$$\n", "\n", "or $0 < m^{-1}a < 2$. This is equivalent to the more general condition $|g'(x)| < 1$.\n", "\n", "What is now obvious is that if we can make $\\left|1 - m^{-1}a\\right| \\ll 1$, we can expect fast convergence.\n", "\n", "We will see that this scalar problem can be viewed as an analogue to the linear system\n", "\n", "\$$\n", "F(\\mathbf x) = \\mathbf b-A\\mathbf x\n", "\$$\n", "\n", "For the system, the iteration takes the form\n", "\n", "\$$\n", "\\mathbf x_{k+1} = \\mathbf x_k + M^{-1}\\left( \\mathbf b - A \\mathbf x_k\\right)\n", "\$$\n", "\n", "which we can write as\n", "\n", "\$$\n", "\\mathbf x_{k+1} = \\left(I - M^{-1}A\\right) \\mathbf x_k + M^{-1}\\mathbf b\n", "\$$\n", "\n", "for some non-singular matrix $M$. A sufficient condition for convergence is $\\rho\\left(I - M^{-1}A\\right) < 1$\n", "\n", "The advantages of this approach is that if we can find an $M$ that is both easy to converge, and satisfies a convergence condition, it might be less work computationally to iterate to a solution (possibly stopping early if we are happy with our solution) than applying a direct method. Furthermore, this method only requires that we be able to multiply $\\mathbf x_k$ by $A$.\n", "\n", "\n", "Use the slider below to interactively vary the value of $m$ and see the convergence rate is affected.\n", "\n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Populating the interactive namespace from numpy and matplotlib\n" ] } ], "source": [ "%pylab inline\n", "%matplotlib notebook\n", "\n", "from ipywidgets import interactive, fixed\n", "import warnings\n", "\n", "from fp_tools import *" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $(' ');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", "$(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " this.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " ' ');\n", " var titletext =$(\n", " '
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