{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Lecture 27: Power/inverse iteration\n", "\n", "## Rayleigh quotient\n", "\n", "Suppose $A$ is $m\\times m$. We can define a function called the **Rayleigh quotient**,\n", "\n", "$$r(v) = \\frac{v^*Av}{v^*v},$$\n", "\n", "for all $v\\in\\mathbb{C}^m$. As shown in the text, the stationary points of $r$ are the eigenvectors of $A$. More specifically, if $v$ is close to an eigenvector $x$ with eigenvalue $\\lambda$, then \n", "\n", "$$|r(v)-\\lambda| = O\\bigl( \\|v-x\\|_2 \\bigr)$$\n", "\n", "in general, and \n", "\n", "$$|r(v)-\\lambda| = O\\bigl( \\|v-x\\|_2^2 \\bigr)$$\n", "\n", "if $A$ is hermitian. Thus we can turn an eigenvector estimate into an eigenvalue estimate. \n", "\n", "For example, let" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A =\n", "\n", " 2 1 1\n", " 1 3 1\n", " 1 1 4\n" ] } ], "source": [ "A = [2 1 1; 1 3 1; 1 1 4]\n", "[V,D] = eig(A);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We'll color the unit sphere according to the value of the Rayleigh quotient at each point on it." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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r62Pvznz/3KPmT9nvP/dVAwAUYwglSOOWiVaO4bo3tt+5tW1rpk+jVo1u6dQW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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "[X,Y,Z] = sphere(100);\n", "v = [X(:) Y(:) Z(:)]';\n", "Av = A*v;\n", "Ray = sum( conj(v).*Av, 1 );\n", "surf(X,Y,Z,reshape(Ray,size(X)))\n", "hold on\n", "plot3(V(1,:),V(2,:),V(3,:),'k*')\n", "\n", "title('Rayleigh quotient values on the unit sphere')\n", "shading interp, axis square\n", "view(107,20)\n", "xlabel('x'), ylabel('y'), zlabel('z')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The eigenvectors (their projections onto the sphere) are shown with black markers. The max of $r(v)$ is in the middle of the yellow spot. It's the third eigenvector of $A$ as computed above, and close to $(0.4,0.5,0.75)$. " ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "evec_err =\n", "\n", " 0.0216\n" ] } ], "source": [ "x = V(:,3); \n", "v = [0.4;0.5;0.75];\n", "evec_err = norm(x-v)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "eval_est =\n", "\n", " 5.2134\n", "\n", "\n", "eval_err =\n", "\n", " 9.5213e-04\n" ] } ], "source": [ "eval_est = v'*A*v/(v'*v)\n", "eval_err = norm(eval_est-D(3,3))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Power iteration\n", "\n", "Now that we can turn an eigenvector estimate into an (even better, in the hermitian case) eigenvalue estimate, we turn to finding such eigenvectors. Clearly, if $A=XDX^{-1}$ and $k$ is a positive integer, then $A^k=XD^kX^{-1}$. That is, $A$ and $A^k$ share the same eigenvectors, but the eigenvalues of $A^k$ are raised to the $k$th power. Note that if there is an eigenvalue $\\lambda_1$ such that $|\\lambda_1|>|\\lambda_j|$ for all $j>1$, the ratio $|\\lambda_1/\\lambda_j|^k\\to 0$ as $k\\to\\infty$, and in that case $A^kv$ is dominated by the eigenvector $x_1$ that belongs with $\\lambda_1$, for practically *any* vector $v$. This leads us to the **power iteration**." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "v = rand(3,1);\n", "for k = 1:20\n", " v = v/norm(v);\n", " evec_err(k) = min(norm(v-x),norm(v+x));\n", " v = A*v;\n", "end\n", "semilogy(evec_err) \n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since $A$ is symmetric, we should expect around 12 accurate digits in an eigenvalue estimate derived from a Rayleigh quotient with our final $v$." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "ans =\n", "\n", " 1.8918e-13\n" ] } ], "source": [ "abs( v'*A*v/(v'*v) - D(3,3) )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now how much would you pay? But wait--there's more!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Inverse iteration\n", "\n", "If $\\lambda_J$ is an eigenvalue of $A$, then $(\\lambda_J-\\mu)^{-1}$ is an eigenvalue of $(A-\\mu I)^{-1}$, with the same eigenvector. If $\\mu$ is closer to $\\lambda_J$ than to any other $\\lambda_j$, a power iteration on $(A-\\mu I)^{-1}$ should converge very quickly. This amounts to repeatedly solving a linear system with the matrix $(A-\\mu I)$." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "for mu = [4 4.5 5.1]\n", " [L,U] = lu(A-mu*eye(3));\n", " v = rand(3,1);\n", " for k = 1:20\n", " v = v/norm(v);\n", " evec_err(k) = min(norm(v-x),norm(v+x));\n", " v = U\\(L\\v);\n", " end\n", " semilogy(evec_err)\n", " hold on\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Rayleigh quotient iteration\n", "\n", "The coup de grâce: a better shift means a better eigenvector estimate, which means a better eigenvalue estimate, which is an even better shift, etc. This feedback loop causes *cubic* convergence in the hermitian case, which is rare--and fast! It's so fast that we pretty well need extended precision to see it." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "lambda =\n", " \n", " 5.21431974337753518741549770084858048890791963721949943433138\n", " 2.46081112718911088347412409730147999190011289045787329828077\n", " 1.32486912943335392911037820184993951919196747232262726738785\n" ] } ], "source": [ "digits(60)\n", "A = vpa(A);\n", "lambda = eig(A)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "error in eigenvalue = -0.21431974337753517678706316473835613578557968139648\n", "error in eigenvalue = -0.00120498927917453164254668607213716313708573579788\n", "error in eigenvalue = -0.00000000019350335636704886234483316195391670519466\n", "error in eigenvalue = -0.00000000000000000000000000000086272041087647542775\n", "error in eigenvalue = 0.00000000000000000000000000000000000000000000000000\n", "[\bWarning: The system is inconsistent. Solution does not exist.]\b \n", "[\b> In symengine\n", " In sym/privBinaryOp (line 937)\n", " In \\ (line 335)\n", " In pymat_eval (line 31)\n", " In matlabserver (line 24)]\b\n" ] } ], "source": [ "v = vpa(ones(3,1));\n", "for k = 1:5\n", " v = v/norm(v);\n", " mu = v'*A*v; \n", " fprintf('error in eigenvalue = %.50f\\n',min(mu-lambda))\n", " v = (A-mu*eye(3))\\v; \n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Count 'em: The number of leading zeros *at least triples* with each iteration, and in 5 iterations we get 60 digits of accuracy. 😎" ] } ], "metadata": { "kernelspec": { "display_name": "Matlab", "language": "matlab", "name": "matlab" }, "language_info": { "codemirror_mode": "octave", "file_extension": ".m", "help_links": [ { "text": "MetaKernel Magics", "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" } ], "mimetype": "text/x-octave", "name": "matlab", "version": "0.11.0" } }, "nbformat": 4, "nbformat_minor": 1 }