add lect 12

lecture_13/lecture_13.ipynb → lecture_12/.ipynb_checkpoints/lecture_12-checkpoint.ipynb

@@ -768,7 +768,89 @@
     "### *just checked in to see what condition my condition was in*\n",
     "### Matrix norms\n",
     "\n",
-    "The Euclidean norm of a vector is measure of the magnitude (in 3D this would be: $|x|=\\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$) in general t"
+    "The Euclidean norm of a vector is measure of the magnitude (in 3D this would be: $|x|=\\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$) in general the equation is:\n",
+    "\n",
+    "$||x||_{e}=\\sqrt{\\sum_{i=1}^{n}x_{i}^{2}}$\n",
+    "\n",
+    "For a matrix, A, the same norm is called the Frobenius norm:\n",
+    "\n",
+    "$||A||_{f}=\\sqrt{\\sum_{i=1}^{n}\\sum_{i=1}^{m}A_{i,j}^{2}}$\n",
+    "\n",
+    "In general we can calculate any $p$-norm where\n",
+    "\n",
+    "$||A||_{p}=\\sqrt{\\sum_{i=1}^{n}\\sum_{i=1}^{m}A_{i,j}^{p}}$\n",
+    "\n",
+    "so the p=1, 1-norm is \n",
+    "\n",
+    "$||A||_{1}=\\sqrt{\\sum_{i=1}^{n}\\sum_{i=1}^{m}A_{i,j}^{1}}=\\sum_{i=1}^{n}\\sum_{i=1}^{m}|A_{i,j}|$\n",
+    "\n",
+    "$||A||_{\\infty}=\\sqrt{\\sum_{i=1}^{n}\\sum_{i=1}^{m}A_{i,j}^{\\infty}}=\\max_{1\\le i \\le n}\\sum_{j=1}^{m}|A_{i,j}|$\n",
+    "\n",
+    "### Condition of Matrix\n",
+    "\n",
+    "The matrix condition is the product of \n",
+    "\n",
+    "$Cond(A) = ||A||\\cdot||A^{-1}||$ \n",
+    "\n",
+    "So each norm will have a different condition number, but the limit is $Cond(A)\\ge 1$\n",
+    "\n",
+    "An estimate of the rounding error is based on the condition of A:\n",
+    "\n",
+    "$\\frac{||\\Delta x||}{x} \\le Cond(A) \\frac{||\\Delta A||}{||A||}$\n",
+    "\n",
+    "So if the coefficients of A have accuracy to $10^{-t}\n",
+    "\n",
+    "and the condition of A, $Cond(A)=10^{c}$\n",
+    "\n",
+    "then the solution for x can have rounding errors up to $10^{c-t}$\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   1.00000   0.50000   0.33333\n",
+      "   0.50000   0.33333   0.25000\n",
+      "   0.33333   0.25000   0.20000\n",
+      "\n",
+      "L =\n",
+      "\n",
+      "   1.00000   0.00000   0.00000\n",
+      "   0.50000   1.00000   0.00000\n",
+      "   0.33333   1.00000   1.00000\n",
+      "\n",
+      "U =\n",
+      "\n",
+      "   1.00000   0.50000   0.33333\n",
+      "   0.00000   0.08333   0.08333\n",
+      "   0.00000  -0.00000   0.00556\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=[1,1/2,1/3;1/2,1/3,1/4;1/3,1/4,1/5]\n",
+    "[L,U]=LU_naive(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "invA="
    ]
   },
   {

lecture_12/LU_naive.m

@@ -0,0 +1,27 @@
+function [L, U] = LU_naive(A)
+% GaussNaive: naive Gauss elimination
+%   x = GaussNaive(A,b): Gauss elimination without pivoting.
+% input:
+%   A = coefficient matrix
+%   y = right hand side vector
+% output:
+%   x = solution vector
+[m,n] = size(A);
+if m~=n, error('Matrix A must be square'); end
+nb = n;
+L=diag(ones(n,1));
+U=A;
+% forward elimination
+for k = 1:n-1
+  for i = k+1:n
+    fik = U(i,k)/U(k,k);
+    L(i,k)=fik;
+    U(i,k:nb) = U(i,k:nb)-fik*U(k,k:nb);
+  end
+end
+%% back substitution
+%x = zeros(n,1);
+%x(n) = Aug(n,nb)/Aug(n,n);
+%for i = n-1:-1:1
+%  x(i) = (Aug(i,nb)-Aug(i,i+1:n)*x(i+1:n))/Aug(i,i);
+%end

lecture_12/lecture_12.aux

@@ -0,0 +1,27 @@
+\relax 
+\providecommand\hyper@newdestlabel[2]{}
+\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
+\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
+\global\let\oldcontentsline\contentsline
+\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
+\global\let\oldnewlabel\newlabel
+\gdef\newlabel#1#2{\newlabelxx{#1}#2}
+\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
+\AtEndDocument{\ifx\hyper@anchor\@undefined
+\let\contentsline\oldcontentsline
+\let\newlabel\oldnewlabel
+\fi}
+\fi}
+\global\let\hyper@last\relax 
+\gdef\HyperFirstAtBeginDocument#1{#1}
+\providecommand\HyField@AuxAddToFields[1]{}
+\providecommand\HyField@AuxAddToCoFields[2]{}
+\providecommand \oddpage@label [2]{}
+\@writefile{toc}{\contentsline {subsection}{\numberline {0.1}My question from last class}{1}{subsection.0.1}}
+\newlabel{my-question-from-last-class}{{0.1}{1}{My question from last class}{subsection.0.1}{}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces q1\relax }}{1}{figure.caption.1}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {0.2}Your questions from last class}{1}{subsection.0.2}}
+\newlabel{your-questions-from-last-class}{{0.2}{1}{Your questions from last class}{subsection.0.2}{}}
+\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces q2\relax }}{2}{figure.caption.2}}
+\@writefile{toc}{\contentsline {section}{\numberline {1}Matrix Inverse and Condition}{2}{section.1}}
+\newlabel{matrix-inverse-and-condition}{{1}{2}{Matrix Inverse and Condition}{section.1}{}}