From aefcae1d1530e35a22c59c4c6d947027ca12b4a0 Mon Sep 17 00:00:00 2001
From: "Ryan C. Cooper" <ryan.c.cooper@uconn.edu>
Date: Tue, 24 Oct 2017 16:08:44 -0400
Subject: [PATCH] updated eigenvalue lecture

---
 11_LU-decomposition/mass_springs.svg          |   13 +-
 .../12_matrix_condition-checkpoint.ipynb      | 4882 -----------------
 12_matrix_condition/12_matrix_condition.ipynb | 4882 -----------------
 .../13_eigenvalues-checkpoint.ipynb           |  168 +-
 13_eigenvalues/13_eigenvalues.ipynb           |  168 +-
 13_eigenvalues/octave-workspace               |  Bin 153 -> 546 bytes
 6 files changed, 288 insertions(+), 9825 deletions(-)

diff --git a/11_LU-decomposition/mass_springs.svg b/11_LU-decomposition/mass_springs.svg
index c6edff5..abfb4ed 100644
--- a/11_LU-decomposition/mass_springs.svg
+++ b/11_LU-decomposition/mass_springs.svg
@@ -15,7 +15,10 @@
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diff --git a/12_matrix_condition/.ipynb_checkpoints/12_matrix_condition-checkpoint.ipynb b/12_matrix_condition/.ipynb_checkpoints/12_matrix_condition-checkpoint.ipynb
index cc25231..932c3cd 100644
--- a/12_matrix_condition/.ipynb_checkpoints/12_matrix_condition-checkpoint.ipynb
+++ b/12_matrix_condition/.ipynb_checkpoints/12_matrix_condition-checkpoint.ipynb
@@ -353,4888 +353,6 @@
     "\n",
     "no need to calculate the inv(K)"
    ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Iterative Methods\n",
-    "\n",
-    "## Gauss-Seidel method\n",
-    "\n",
-    "If we have an intial guess for each value of a vector $x$ that we are trying to solve, then it is easy enough to solve for one component given the others. \n",
-    "\n",
-    "Take a 3$\\times$3 matrix \n",
-    "\n",
-    "$Ax=b$\n",
-    "\n",
-    "$\\left[ \\begin{array}{ccc}\n",
-    "3 & -0.1 & -0.2  \\\\\n",
-    "0.1 & 7 & -0.3  \\\\\n",
-    "0.3 & -0.2 & 10 \\end{array} \\right]\n",
-    "\\left[ \\begin{array}{c}\n",
-    "x_{1} \\\\\n",
-    "x_{2} \\\\\n",
-    "x_{3} \\end{array} \\right]=\n",
-    "\\left[ \\begin{array}{c}\n",
-    "7.85 \\\\\n",
-    "-19.3 \\\\\n",
-    "71.4\\end{array} \\right]$\n",
-    "\n",
-    "$x_{1}=\\frac{7.85+0.1x_{2}+0.2x_{3}}{3}$\n",
-    "\n",
-    "$x_{2}=\\frac{-19.3-0.1x_{1}+0.3x_{3}}{7}$\n",
-    "\n",
-    "$x_{3}=\\frac{71.4+0.1x_{1}+0.2x_{2}}{10}$"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 12,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "A =\n",
-      "\n",
-      "    3.00000   -0.10000   -0.20000\n",
-      "    0.10000    7.00000   -0.30000\n",
-      "    0.30000   -0.20000   10.00000\n",
-      "\n",
-      "b =\n",
-      "\n",
-      "    7.8500\n",
-      "  -19.3000\n",
-      "   71.4000\n",
-      "\n",
-      "x =\n",
-      "\n",
-      "   3.0000\n",
-      "  -2.5000\n",
-      "   7.0000\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "A=[3 -0.1 -0.2;0.1 7 -0.3;0.3 -0.2 10]\n",
-    "b=[7.85;-19.3;71.4]\n",
-    "\n",
-    "x=A\\b"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "collapsed": true
-   },
-   "source": [
-    "### Gauss-Seidel Iterative approach\n",
-    "\n",
-    "As a first guess, we can use $x_{1}=x_{2}=x_{3}=0$\n",
-    "\n",
-    "$x_{1}=\\frac{7.85+0.1(0)+0.3(0)}{3}=2.6167$\n",
-    "\n",
-    "$x_{2}=\\frac{-19.3-0.1(2.6167)+0.3(0)}{7}=-2.7945$\n",
-    "\n",
-    "$x_{3}=\\frac{71.4+0.1(2.6167)+0.2(-2.7945)}{10}=7.0056$\n",
-    "\n",
-    "Then, we update the guess:\n",
-    "\n",
-    "$x_{1}=\\frac{7.85+0.1(-2.7945)+0.3(7.0056)}{3}=2.9906$\n",
-    "\n",
-    "$x_{2}=\\frac{-19.3-0.1(2.9906)+0.3(7.0056)}{7}=-2.4996$\n",
-    "\n",
-    "$x_{3}=\\frac{71.4+0.1(2.9906)+0.2(-2.4966)}{10}=7.00029$\n",
-    "\n",
-    "The results are conveerging to the solution we found with `\\` of $x_{1}=3,~x_{2}=-2.5,~x_{3}=7$\n",
-    "\n",
-    "We could also use an iterative method that solves for all of the x-components in one step:\n",
-    "\n",
-    "### Jacobi method\n",
-    "\n",
-    "$x_{1}^{i}=\\frac{7.85+0.1x_{2}^{i-1}+0.3x_{3}^{i-1}}{3}$\n",
-    "\n",
-    "$x_{2}^{i}=\\frac{-19.3-0.1x_{1}^{i-1}+0.3x_{3}^{i-1}}{7}$\n",
-    "\n",
-    "$x_{3}^{i}=\\frac{71.4+0.1x_{1}^{i-1}+0.2x_{2}^{i-1}}{10}$\n",
-    "\n",
-    "Here the solution is a matrix multiplication and vector addition\n",
-    "\n",
-    "$\\left[ \\begin{array}{c}\n",
-    "x_{1}^{i} \\\\\n",
-    "x_{2}^{i} \\\\\n",
-    "x_{3}^{i} \\end{array} \\right]=\n",
-    "\\left[ \\begin{array}{c}\n",
-    "7.85/3 \\\\\n",
-    "-19.3/7 \\\\\n",
-    "71.4/10\\end{array} \\right]-\n",
-    "\\left[ \\begin{array}{ccc}\n",
-    "0 & 0.1/3 & 0.2/3  \\\\\n",
-    "0.1/7 & 0 & -0.3/7  \\\\\n",
-    "0.3/10 & -0.2/10 & 0 \\end{array} \\right]\n",
-    "\\left[ \\begin{array}{c}\n",
-    "x_{1}^{i-1} \\\\\n",
-    "x_{2}^{i-1} \\\\\n",
-    "x_{3}^{i-1} \\end{array} \\right]$\n",
-    "\n",
-    "|x_{j}|Jacobi method |vs| Gauss-Seidel |\n",
-    "|--------|------------------------------|---|-------------------------------|\n",
-    "|$x_{1}^{i}=$ | $\\frac{7.85+0.1x_{2}^{i-1}+0.3x_{3}^{i-1}}{3}$ | | $\\frac{7.85+0.1x_{2}^{i-1}+0.3x_{3}^{i-1}}{3}$|\n",
-    "|$x_{2}^{i}=$ | $\\frac{-19.3-0.1x_{1}^{i-1}+0.3x_{3}^{i-1}}{7}$ | | $\\frac{-19.3-0.1x_{1}^{i}+0.3x_{3}^{i-1}}{7}$ |\n",
-    "|$x_{3}^{i}=$ | $\\frac{71.4+0.1x_{1}^{i-1}+0.2x_{2}^{i-1}}{10}$ | | $\\frac{71.4+0.1x_{1}^{i}+0.2x_{2}^{i}}{10}$|"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 14,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "ba =\n",
-      "\n",
-      "   2.6167\n",
-      "  -2.7571\n",
-      "   7.1400\n",
-      "\n",
-      "sA =\n",
-      "\n",
-      "   0.00000  -0.10000  -0.20000\n",
-      "   0.10000   0.00000  -0.30000\n",
-      "   0.30000  -0.20000   0.00000\n",
-      "\n",
-      "sA =\n",
-      "\n",
-      "   0.000000  -0.033333  -0.066667\n",
-      "   0.014286   0.000000  -0.042857\n",
-      "   0.030000  -0.020000   0.000000\n",
-      "\n",
-      "x1 =\n",
-      "\n",
-      "   2.6167\n",
-      "  -2.7571\n",
-      "   7.1400\n",
-      "\n",
-      "x2 =\n",
-      "\n",
-      "   3.0008\n",
-      "  -2.4885\n",
-      "   7.0064\n",
-      "\n",
-      "x3 =\n",
-      "\n",
-      "   3.0008\n",
-      "  -2.4997\n",
-      "   7.0002\n",
-      "\n",
-      "solution is converging to [3,-2.5,7]]\n"
-     ]
-    }
-   ],
-   "source": [
-    "ba=b./diag(A) % or ba=b./[A(1,1);A(2,2);A(3,3)]\n",
-    "sA=A-diag(diag(A)) % A with zeros on diagonal\n",
-    "sA(1,:)=sA(1,:)/A(1,1);\n",
-    "sA(2,:)=sA(2,:)/A(2,2);\n",
-    "sA(3,:)=sA(3,:)/A(3,3)\n",
-    "x0=[0;0;0];\n",
-    "x1=ba-sA*x0\n",
-    "x2=ba-sA*x1\n",
-    "x3=ba-sA*x2\n",
-    "fprintf('solution is converging to [3,-2.5,7]]\\n')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 16,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "ans =\n",
-      "\n",
-      "    3\n",
-      "    7\n",
-      "   10\n",
-      "\n",
-      "ans =\n",
-      "\n",
-      "Diagonal Matrix\n",
-      "\n",
-      "    3    0    0\n",
-      "    0    7    0\n",
-      "    0    0   10\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "diag(A)\n",
-    "diag(diag(A))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "This method works if problem is diagonally dominant, \n",
-    "\n",
-    "$|a_{ii}|>\\sum_{j=1,j\\ne i}^{n}|a_{ij}|$\n",
-    "\n",
-    "If this condition is true, then Jacobi or Gauss-Seidel should converge\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 15,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "A =\n",
-      "\n",
-      "   0.10000   1.00000   3.00000\n",
-      "   1.00000   0.20000   3.00000\n",
-      "   5.00000   2.00000   0.30000\n",
-      "\n",
-      "b =\n",
-      "\n",
-      "   12\n",
-      "    2\n",
-      "    4\n",
-      "\n",
-      "ans =\n",
-      "\n",
-      "  -2.9393\n",
-      "   9.1933\n",
-      "   1.0336\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "A=[0.1,1,3;1,0.2,3;5,2,0.3]\n",
-    "b=[12;2;4]\n",
-    "A\\b"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 16,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "ba =\n",
-      "\n",
-      "   120.000\n",
-      "    10.000\n",
-      "    13.333\n",
-      "\n",
-      "sA =\n",
-      "\n",
-      "   0   1   3\n",
-      "   1   0   3\n",
-      "   5   2   0\n",
-      "\n",
-      "sA =\n",
-      "\n",
-      "    0.00000   10.00000   30.00000\n",
-      "    5.00000    0.00000   15.00000\n",
-      "   16.66667    6.66667    0.00000\n",
-      "\n",
-      "x1 =\n",
-      "\n",
-      "   120.000\n",
-      "    10.000\n",
-      "    13.333\n",
-      "\n",
-      "x2 =\n",
-      "\n",
-      "   -380.00\n",
-      "   -790.00\n",
-      "  -2053.33\n",
-      "\n",
-      "x3 =\n",
-      "\n",
-      "   6.9620e+04\n",
-      "   3.2710e+04\n",
-      "   1.1613e+04\n",
-      "\n",
-      "solution is not converging to [-2.93,9.19,1.03]\n"
-     ]
-    }
-   ],
-   "source": [
-    "ba=b./diag(A) % or ba=b./[A(1,1);A(2,2);A(3,3)]\n",
-    "sA=A-diag(diag(A)) % A with zeros on diagonal\n",
-    "sA(1,:)=sA(1,:)/A(1,1);\n",
-    "sA(2,:)=sA(2,:)/A(2,2);\n",
-    "sA(3,:)=sA(3,:)/A(3,3)\n",
-    "x0=[0;0;0];\n",
-    "x1=ba-sA*x0\n",
-    "x2=ba-sA*x1\n",
-    "x3=ba-sA*x2\n",
-    "fprintf('solution is not converging to [-2.93,9.19,1.03]\\n')"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Gauss-Seidel with Relaxation\n",
-    "\n",
-    "In order to force the solution to converge faster, we can introduce a relaxation term $\\lambda$. \n",
-    "\n",
-    "where the new x values are weighted between the old and new:\n",
-    "\n",
-    "$x^{i}=\\lambda x^{i}+(1-\\lambda)x^{i-1}$\n",
-    "\n",
-    "after solving for x, lambda weights the current approximation with the previous approximation for the updated x\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 17,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "A =\n",
-      "\n",
-      "    3.00000   -0.10000   -0.20000\n",
-      "    0.10000    7.00000   -0.30000\n",
-      "    0.30000   -0.20000   10.00000\n",
-      "\n",
-      "b =\n",
-      "\n",
-      "    7.8500\n",
-      "  -19.3000\n",
-      "   71.4000\n",
-      "\n"
-     ]
-    },
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-    "% rearrange A and b\n",
-    "A=[3 -0.1 -0.2;0.1 7 -0.3;0.3 -0.2 10]\n",
-    "b=[7.85;-19.3;71.4]\n",
-    "\n",
-    "iters=zeros(100,1);\n",
-    "for i=1:100\n",
-    "    lambda=2/100*i;\n",
-    "    [x,ea,iters(i)]=Jacobi_rel(A,b,lambda);\n",
-    "end\n",
-    "plot([1:100]*2/100,iters)    "
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-   "execution_count": 107,
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-     "text": [
-      "l =  0.99158\r\n"
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-   ],
-   "source": [
-    "l=fminbnd(@(l) lambda_fcn(A,b,l),0.5,1.5)"
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-   "execution_count": 108,
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-      "ans =\n",
-      "\n",
-      "   3.0000\n",
-      "  -2.5000\n",
-      "   7.0000\n",
-      "\n"
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-    "A\\b"
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-      "x =\n",
-      "\n",
-      "   3.0000\n",
-      "  -2.5000\n",
-      "   7.0000\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   1.8289e-07\n",
-      "   2.1984e-08\n",
-      "   2.3864e-08\n",
-      "\n",
-      "iter =  8\n",
-      "x =\n",
-      "\n",
-      "   3.0000\n",
-      "  -2.5000\n",
-      "   7.0000\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   1.9130e-08\n",
-      "   7.6449e-08\n",
-      "   3.3378e-08\n",
-      "\n",
-      "iter =  8\n"
-     ]
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-    "[x,ea,iter]=Jacobi_rel(A,b,l,0.000001)\n",
-    "[x,ea,iter]=Jacobi_rel(A,b,1,0.000001)\n"
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-    "## Nonlinear Systems\n",
-    "\n",
-    "Consider two simultaneous nonlinear equations with two unknowns:\n",
-    "\n",
-    "$x_{1}^{2}+x_{1}x_{2}=10$\n",
-    "\n",
-    "$x_{2}+3x_{1}x_{2}^{2}=57$\n",
-    "\n",
-    "Graphically, we are looking for the solution:\n"
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-   "cell_type": "code",
-   "execution_count": 19,
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-    "x11=linspace(0.5,3);\n",
-    "x12=(10-x11.^2)./x11;\n",
-    "\n",
-    "x22=linspace(2,8);\n",
-    "x21=(57-x22).*x22.^-2/3;\n",
-    "\n",
-    "plot(x11,x12,x21,x22)\n",
-    "% Solution at x_1=2, x_2=3\n",
-    "hold on;\n",
-    "plot(2,3,'o')\n",
-    "xlabel('x_1')\n",
-    "ylabel('x_2')"
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-    "## Newton-Raphson part II\n",
-    "\n",
-    "Remember the first order approximation for the next point in a function is:\n",
-    "\n",
-    "$f(x_{i+1})=f(x_{i})+(x_{i+1}-x_{i})f'(x_{i})$\n",
-    "\n",
-    "then, $f(x_{i+1})=0$ so we are left with:\n",
-    "\n",
-    "$x_{i+1}=x_{i}-\\frac{f(x_{i})}{f'(x_{i})}$\n",
-    "\n",
-    "We can use the same formula, but now we have multiple dimensions so we need to determine the Jacobian\n",
-    "\n",
-    "$[J]=\\left[ \\begin{array}{cccc}\n",
-    "\\frac{\\partial f_{1,i}}{\\partial x_{1}} & \\frac{\\partial f_{1,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{1,i}}{\\partial x_{n}}  \\\\\n",
-    "\\frac{\\partial f_{2,i}}{\\partial x_{1}} & \\frac{\\partial f_{2,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{2,i}}{\\partial x_{n}}  \\\\\n",
-    "\\vdots & \\vdots & & \\vdots \\\\\n",
-    "\\frac{\\partial f_{n,i}}{\\partial x_{1}} & \\frac{\\partial f_{n,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{n,i}}{\\partial x_{n}}  \\\\\n",
-    "\\end{array} \\right]$\n",
-    "\n",
-    "$\\left[ \\begin{array}{c}\n",
-    "f_{1,i+1} \\\\\n",
-    "f_{2,i+1} \\\\\n",
-    "\\vdots \\\\\n",
-    "f_{n,i+1}\\end{array} \\right]=\n",
-    "\\left[ \\begin{array}{c}\n",
-    "f_{1,i} \\\\\n",
-    "f_{2,i} \\\\\n",
-    "\\vdots \\\\\n",
-    "f_{n,i}\\end{array} \\right]+\n",
-    "\\left[ \\begin{array}{cccc}\n",
-    "\\frac{\\partial f_{1,i}}{\\partial x_{1}} & \\frac{\\partial f_{1,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{1,i}}{\\partial x_{n}}  \\\\\n",
-    "\\frac{\\partial f_{2,i}}{\\partial x_{1}} & \\frac{\\partial f_{2,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{2,i}}{\\partial x_{n}}  \\\\\n",
-    "\\vdots & \\vdots & & \\vdots \\\\\n",
-    "\\frac{\\partial f_{n,i}}{\\partial x_{1}} & \\frac{\\partial f_{n,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{n,i}}{\\partial x_{n}}  \\\\\n",
-    "\\end{array} \\right]\n",
-    "\\left( \\left[ \\begin{array}{c}\n",
-    "x_{i+1} \\\\\n",
-    "x_{i+1} \\\\\n",
-    "\\vdots \\\\\n",
-    "x_{i+1}\\end{array} \\right]-\n",
-    "\\left[ \\begin{array}{c}\n",
-    "x_{1,i} \\\\\n",
-    "x_{2,i} \\\\\n",
-    "\\vdots \\\\\n",
-    "x_{n,i}\\end{array} \\right]\\right)$\n",
-    "\n",
-    "### Solution is again in the form Ax=b\n",
-    "\n",
-    "$[J]([x_{i+1}]-[x_{i}])=-[f]$\n",
-    "\n",
-    "so\n",
-    "\n",
-    "$[x_{i+1}]= [x_{i}]-[J]^{-1}[f]$\n",
-    "\n",
-    "## Example of Jacobian calculation\n",
-    "\n",
-    "### Nonlinear springs supporting two masses in series\n",
-    "\n",
-    "Two springs are connected to two masses, with $m_1$=1 kg and $m_{2}$=2 kg. The springs are identical, but they have nonlinear spring constants, of $k_1$=100 N/m and $k_2$=-10 N/m\n",
-    "\n",
-    "We want to solve for the final position of the masses ($x_1$ and $x_2$)\n",
-    "\n",
-    "$m_{1}g+k_{1}(x_{2}-x_{1})+k_{2}(x_{2}-x_{1})^{2}+k_{1}x_{1}+k_{2}x_{1}^{2}=0$\n",
-    "\n",
-    "$m_{2}g-k_{1}(x_{2}-x_{1})-k_{2}(x_2-x_1)^{2}=0$\n",
-    "\n",
-    "$J(1,1)=\\frac{\\partial f_{1}}{\\partial x_{1}}=-k_{1}-2k_{2}(x_{2}-x_{1})+k_{1}+2k_{2}x_{1}$\n",
-    "\n",
-    "$J(1,2)=\\frac{\\partial f_1}{\\partial x_{2}}=k_{1}+2k_{2}(x_{2}-x_{1})$\n",
-    "\n",
-    "$J(2,1)=\\frac{\\partial f_2}{\\partial x_{1}}=k_{1}+2k_{2}(x_{2}-x_{1})$\n",
-    "\n",
-    "$J(2,2)=\\frac{\\partial f_2}{\\partial x_{2}}=-k_{1}-2k_{2}(x_{2}-x_{1})$\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
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-   "source": [
-    "m1=1; % kg \n",
-    "m2=2; % kg\n",
-    "k1=100; % N/m\n",
-    "k2=-10; % N/m^2"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 20,
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-    "function [f,J]=mass_spring(x)\n",
-    "    % Function to calculate function values f1 and f2 as well as Jacobian \n",
-    "    % for 2 masses and 2 identical nonlinear springs\n",
-    "    m1=1; % kg \n",
-    "    m2=2; % kg\n",
-    "    k1=100; % N/m\n",
-    "    k2=-10; % N/m^2\n",
-    "    g=9.81; % m/s^2\n",
-    "    x1=x(1);\n",
-    "    x2=x(2);\n",
-    "    J=[-k1-2*k2*(x2-x1)-k1-2*k2*x1,k1+2*k2*(x2-x1);\n",
-    "        k1+2*k2*(x2-x1),-k1-2*k2*(x2-x1)];\n",
-    "    f=[m1*g+k1*(x2-x1)+k2*(x2-x1).^2-k1*x1-k2*x1^2;\n",
-    "       m2*g-k1*(x2-x1)-k2*(x2-x1).^2];\n",
-    "end\n",
-    "    "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 21,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "f =\n",
-      "\n",
-      "  -190.19\n",
-      "   129.62\n",
-      "\n",
-      "J =\n",
-      "\n",
-      "  -200   120\n",
-      "   120  -120\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "[f,J]=mass_spring([1,0])"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 22,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "x1 =\n",
-      "\n",
-      "  -1.5142\n",
-      "  -1.4341\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   2.9812\n",
-      "   2.3946\n",
-      "\n",
-      "x2 =\n",
-      "\n",
-      "   0.049894\n",
-      "   0.248638\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   31.3492\n",
-      "    6.7678\n",
-      "\n",
-      "x3 =\n",
-      "\n",
-      "   0.29701\n",
-      "   0.49722\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   0.83201\n",
-      "   0.49995\n",
-      "\n",
-      "x =\n",
-      "\n",
-      "   0.29701\n",
-      "   0.49722\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   0.021392\n",
-      "   0.012890\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   1.4786e-05\n",
-      "   8.9091e-06\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   7.0642e-12\n",
-      "   4.2565e-12\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "x0=[3;2];\n",
-    "[f0,J0]=mass_spring(x0);\n",
-    "x1=x0-J0\\f0\n",
-    "ea=(x1-x0)./x1\n",
-    "[f1,J1]=mass_spring(x1);\n",
-    "x2=x1-J1\\f1\n",
-    "ea=(x2-x1)./x2\n",
-    "[f2,J2]=mass_spring(x2);\n",
-    "x3=x2-J2\\f2\n",
-    "ea=(x3-x2)./x3\n",
-    "x=x3\n",
-    "for i=1:3\n",
-    "    xold=x;\n",
-    "    [f,J]=mass_spring(x);\n",
-    "    x=x-J\\f;\n",
-    "    ea=(x-xold)./x\n",
-    "end"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 23,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "x =\n",
-      "\n",
-      "   0.30351\n",
-      "   0.50372\n",
-      "\n",
-      "X0 =\n",
-      "\n",
-      "   0.30351\n",
-      "   0.50372\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "x\n",
-    "X0=fsolve(@(x) mass_spring(x),[3;5])"
-   ]
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-       "</g>\n",
-       "</g>\n",
-       "</svg>"
-      ],
-      "text/plain": [
-       "<IPython.core.display.SVG object>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "[X,Y]=meshgrid(linspace(0,10,20),linspace(0,10,20));\n",
-    "[N,M]=size(X);\n",
-    "F=zeros(size(X));\n",
-    "for i=1:N\n",
-    "    for j=1:M\n",
-    "        [f,~]=mass_spring([X(i,j),Y(i,j)]);\n",
-    "        F1(i,j)=f(1);\n",
-    "        F2(i,j)=f(2);\n",
-    "    end\n",
-    "end\n",
-    "mesh(X,Y,F1)\n",
-    "xlabel('x_1')\n",
-    "ylabel('x_2')\n",
-    "colorbar()\n",
-    "figure()\n",
-    "mesh(X,Y,F2)\n",
-    "xlabel('x_1')\n",
-    "ylabel('x_2')\n",
-    "colorbar()"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": []
   }
  ],
  "metadata": {
diff --git a/12_matrix_condition/12_matrix_condition.ipynb b/12_matrix_condition/12_matrix_condition.ipynb
index cc25231..932c3cd 100644
--- a/12_matrix_condition/12_matrix_condition.ipynb
+++ b/12_matrix_condition/12_matrix_condition.ipynb
@@ -353,4888 +353,6 @@
     "\n",
     "no need to calculate the inv(K)"
    ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Iterative Methods\n",
-    "\n",
-    "## Gauss-Seidel method\n",
-    "\n",
-    "If we have an intial guess for each value of a vector $x$ that we are trying to solve, then it is easy enough to solve for one component given the others. \n",
-    "\n",
-    "Take a 3$\\times$3 matrix \n",
-    "\n",
-    "$Ax=b$\n",
-    "\n",
-    "$\\left[ \\begin{array}{ccc}\n",
-    "3 & -0.1 & -0.2  \\\\\n",
-    "0.1 & 7 & -0.3  \\\\\n",
-    "0.3 & -0.2 & 10 \\end{array} \\right]\n",
-    "\\left[ \\begin{array}{c}\n",
-    "x_{1} \\\\\n",
-    "x_{2} \\\\\n",
-    "x_{3} \\end{array} \\right]=\n",
-    "\\left[ \\begin{array}{c}\n",
-    "7.85 \\\\\n",
-    "-19.3 \\\\\n",
-    "71.4\\end{array} \\right]$\n",
-    "\n",
-    "$x_{1}=\\frac{7.85+0.1x_{2}+0.2x_{3}}{3}$\n",
-    "\n",
-    "$x_{2}=\\frac{-19.3-0.1x_{1}+0.3x_{3}}{7}$\n",
-    "\n",
-    "$x_{3}=\\frac{71.4+0.1x_{1}+0.2x_{2}}{10}$"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 12,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "A =\n",
-      "\n",
-      "    3.00000   -0.10000   -0.20000\n",
-      "    0.10000    7.00000   -0.30000\n",
-      "    0.30000   -0.20000   10.00000\n",
-      "\n",
-      "b =\n",
-      "\n",
-      "    7.8500\n",
-      "  -19.3000\n",
-      "   71.4000\n",
-      "\n",
-      "x =\n",
-      "\n",
-      "   3.0000\n",
-      "  -2.5000\n",
-      "   7.0000\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "A=[3 -0.1 -0.2;0.1 7 -0.3;0.3 -0.2 10]\n",
-    "b=[7.85;-19.3;71.4]\n",
-    "\n",
-    "x=A\\b"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "collapsed": true
-   },
-   "source": [
-    "### Gauss-Seidel Iterative approach\n",
-    "\n",
-    "As a first guess, we can use $x_{1}=x_{2}=x_{3}=0$\n",
-    "\n",
-    "$x_{1}=\\frac{7.85+0.1(0)+0.3(0)}{3}=2.6167$\n",
-    "\n",
-    "$x_{2}=\\frac{-19.3-0.1(2.6167)+0.3(0)}{7}=-2.7945$\n",
-    "\n",
-    "$x_{3}=\\frac{71.4+0.1(2.6167)+0.2(-2.7945)}{10}=7.0056$\n",
-    "\n",
-    "Then, we update the guess:\n",
-    "\n",
-    "$x_{1}=\\frac{7.85+0.1(-2.7945)+0.3(7.0056)}{3}=2.9906$\n",
-    "\n",
-    "$x_{2}=\\frac{-19.3-0.1(2.9906)+0.3(7.0056)}{7}=-2.4996$\n",
-    "\n",
-    "$x_{3}=\\frac{71.4+0.1(2.9906)+0.2(-2.4966)}{10}=7.00029$\n",
-    "\n",
-    "The results are conveerging to the solution we found with `\\` of $x_{1}=3,~x_{2}=-2.5,~x_{3}=7$\n",
-    "\n",
-    "We could also use an iterative method that solves for all of the x-components in one step:\n",
-    "\n",
-    "### Jacobi method\n",
-    "\n",
-    "$x_{1}^{i}=\\frac{7.85+0.1x_{2}^{i-1}+0.3x_{3}^{i-1}}{3}$\n",
-    "\n",
-    "$x_{2}^{i}=\\frac{-19.3-0.1x_{1}^{i-1}+0.3x_{3}^{i-1}}{7}$\n",
-    "\n",
-    "$x_{3}^{i}=\\frac{71.4+0.1x_{1}^{i-1}+0.2x_{2}^{i-1}}{10}$\n",
-    "\n",
-    "Here the solution is a matrix multiplication and vector addition\n",
-    "\n",
-    "$\\left[ \\begin{array}{c}\n",
-    "x_{1}^{i} \\\\\n",
-    "x_{2}^{i} \\\\\n",
-    "x_{3}^{i} \\end{array} \\right]=\n",
-    "\\left[ \\begin{array}{c}\n",
-    "7.85/3 \\\\\n",
-    "-19.3/7 \\\\\n",
-    "71.4/10\\end{array} \\right]-\n",
-    "\\left[ \\begin{array}{ccc}\n",
-    "0 & 0.1/3 & 0.2/3  \\\\\n",
-    "0.1/7 & 0 & -0.3/7  \\\\\n",
-    "0.3/10 & -0.2/10 & 0 \\end{array} \\right]\n",
-    "\\left[ \\begin{array}{c}\n",
-    "x_{1}^{i-1} \\\\\n",
-    "x_{2}^{i-1} \\\\\n",
-    "x_{3}^{i-1} \\end{array} \\right]$\n",
-    "\n",
-    "|x_{j}|Jacobi method |vs| Gauss-Seidel |\n",
-    "|--------|------------------------------|---|-------------------------------|\n",
-    "|$x_{1}^{i}=$ | $\\frac{7.85+0.1x_{2}^{i-1}+0.3x_{3}^{i-1}}{3}$ | | $\\frac{7.85+0.1x_{2}^{i-1}+0.3x_{3}^{i-1}}{3}$|\n",
-    "|$x_{2}^{i}=$ | $\\frac{-19.3-0.1x_{1}^{i-1}+0.3x_{3}^{i-1}}{7}$ | | $\\frac{-19.3-0.1x_{1}^{i}+0.3x_{3}^{i-1}}{7}$ |\n",
-    "|$x_{3}^{i}=$ | $\\frac{71.4+0.1x_{1}^{i-1}+0.2x_{2}^{i-1}}{10}$ | | $\\frac{71.4+0.1x_{1}^{i}+0.2x_{2}^{i}}{10}$|"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 14,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "ba =\n",
-      "\n",
-      "   2.6167\n",
-      "  -2.7571\n",
-      "   7.1400\n",
-      "\n",
-      "sA =\n",
-      "\n",
-      "   0.00000  -0.10000  -0.20000\n",
-      "   0.10000   0.00000  -0.30000\n",
-      "   0.30000  -0.20000   0.00000\n",
-      "\n",
-      "sA =\n",
-      "\n",
-      "   0.000000  -0.033333  -0.066667\n",
-      "   0.014286   0.000000  -0.042857\n",
-      "   0.030000  -0.020000   0.000000\n",
-      "\n",
-      "x1 =\n",
-      "\n",
-      "   2.6167\n",
-      "  -2.7571\n",
-      "   7.1400\n",
-      "\n",
-      "x2 =\n",
-      "\n",
-      "   3.0008\n",
-      "  -2.4885\n",
-      "   7.0064\n",
-      "\n",
-      "x3 =\n",
-      "\n",
-      "   3.0008\n",
-      "  -2.4997\n",
-      "   7.0002\n",
-      "\n",
-      "solution is converging to [3,-2.5,7]]\n"
-     ]
-    }
-   ],
-   "source": [
-    "ba=b./diag(A) % or ba=b./[A(1,1);A(2,2);A(3,3)]\n",
-    "sA=A-diag(diag(A)) % A with zeros on diagonal\n",
-    "sA(1,:)=sA(1,:)/A(1,1);\n",
-    "sA(2,:)=sA(2,:)/A(2,2);\n",
-    "sA(3,:)=sA(3,:)/A(3,3)\n",
-    "x0=[0;0;0];\n",
-    "x1=ba-sA*x0\n",
-    "x2=ba-sA*x1\n",
-    "x3=ba-sA*x2\n",
-    "fprintf('solution is converging to [3,-2.5,7]]\\n')"
-   ]
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-   "execution_count": 16,
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-   "outputs": [
-    {
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-     "output_type": "stream",
-     "text": [
-      "ans =\n",
-      "\n",
-      "    3\n",
-      "    7\n",
-      "   10\n",
-      "\n",
-      "ans =\n",
-      "\n",
-      "Diagonal Matrix\n",
-      "\n",
-      "    3    0    0\n",
-      "    0    7    0\n",
-      "    0    0   10\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "diag(A)\n",
-    "diag(diag(A))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "This method works if problem is diagonally dominant, \n",
-    "\n",
-    "$|a_{ii}|>\\sum_{j=1,j\\ne i}^{n}|a_{ij}|$\n",
-    "\n",
-    "If this condition is true, then Jacobi or Gauss-Seidel should converge\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 15,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "A =\n",
-      "\n",
-      "   0.10000   1.00000   3.00000\n",
-      "   1.00000   0.20000   3.00000\n",
-      "   5.00000   2.00000   0.30000\n",
-      "\n",
-      "b =\n",
-      "\n",
-      "   12\n",
-      "    2\n",
-      "    4\n",
-      "\n",
-      "ans =\n",
-      "\n",
-      "  -2.9393\n",
-      "   9.1933\n",
-      "   1.0336\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "A=[0.1,1,3;1,0.2,3;5,2,0.3]\n",
-    "b=[12;2;4]\n",
-    "A\\b"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 16,
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-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "ba =\n",
-      "\n",
-      "   120.000\n",
-      "    10.000\n",
-      "    13.333\n",
-      "\n",
-      "sA =\n",
-      "\n",
-      "   0   1   3\n",
-      "   1   0   3\n",
-      "   5   2   0\n",
-      "\n",
-      "sA =\n",
-      "\n",
-      "    0.00000   10.00000   30.00000\n",
-      "    5.00000    0.00000   15.00000\n",
-      "   16.66667    6.66667    0.00000\n",
-      "\n",
-      "x1 =\n",
-      "\n",
-      "   120.000\n",
-      "    10.000\n",
-      "    13.333\n",
-      "\n",
-      "x2 =\n",
-      "\n",
-      "   -380.00\n",
-      "   -790.00\n",
-      "  -2053.33\n",
-      "\n",
-      "x3 =\n",
-      "\n",
-      "   6.9620e+04\n",
-      "   3.2710e+04\n",
-      "   1.1613e+04\n",
-      "\n",
-      "solution is not converging to [-2.93,9.19,1.03]\n"
-     ]
-    }
-   ],
-   "source": [
-    "ba=b./diag(A) % or ba=b./[A(1,1);A(2,2);A(3,3)]\n",
-    "sA=A-diag(diag(A)) % A with zeros on diagonal\n",
-    "sA(1,:)=sA(1,:)/A(1,1);\n",
-    "sA(2,:)=sA(2,:)/A(2,2);\n",
-    "sA(3,:)=sA(3,:)/A(3,3)\n",
-    "x0=[0;0;0];\n",
-    "x1=ba-sA*x0\n",
-    "x2=ba-sA*x1\n",
-    "x3=ba-sA*x2\n",
-    "fprintf('solution is not converging to [-2.93,9.19,1.03]\\n')"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Gauss-Seidel with Relaxation\n",
-    "\n",
-    "In order to force the solution to converge faster, we can introduce a relaxation term $\\lambda$. \n",
-    "\n",
-    "where the new x values are weighted between the old and new:\n",
-    "\n",
-    "$x^{i}=\\lambda x^{i}+(1-\\lambda)x^{i-1}$\n",
-    "\n",
-    "after solving for x, lambda weights the current approximation with the previous approximation for the updated x\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 17,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
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-     "text": [
-      "A =\n",
-      "\n",
-      "    3.00000   -0.10000   -0.20000\n",
-      "    0.10000    7.00000   -0.30000\n",
-      "    0.30000   -0.20000   10.00000\n",
-      "\n",
-      "b =\n",
-      "\n",
-      "    7.8500\n",
-      "  -19.3000\n",
-      "   71.4000\n",
-      "\n"
-     ]
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-   "source": [
-    "% rearrange A and b\n",
-    "A=[3 -0.1 -0.2;0.1 7 -0.3;0.3 -0.2 10]\n",
-    "b=[7.85;-19.3;71.4]\n",
-    "\n",
-    "iters=zeros(100,1);\n",
-    "for i=1:100\n",
-    "    lambda=2/100*i;\n",
-    "    [x,ea,iters(i)]=Jacobi_rel(A,b,lambda);\n",
-    "end\n",
-    "plot([1:100]*2/100,iters)    "
-   ]
-  },
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-   "execution_count": 107,
-   "metadata": {
-    "collapsed": false
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-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "l =  0.99158\r\n"
-     ]
-    }
-   ],
-   "source": [
-    "l=fminbnd(@(l) lambda_fcn(A,b,l),0.5,1.5)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 108,
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-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "ans =\n",
-      "\n",
-      "   3.0000\n",
-      "  -2.5000\n",
-      "   7.0000\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "A\\b"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 109,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "x =\n",
-      "\n",
-      "   3.0000\n",
-      "  -2.5000\n",
-      "   7.0000\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   1.8289e-07\n",
-      "   2.1984e-08\n",
-      "   2.3864e-08\n",
-      "\n",
-      "iter =  8\n",
-      "x =\n",
-      "\n",
-      "   3.0000\n",
-      "  -2.5000\n",
-      "   7.0000\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   1.9130e-08\n",
-      "   7.6449e-08\n",
-      "   3.3378e-08\n",
-      "\n",
-      "iter =  8\n"
-     ]
-    }
-   ],
-   "source": [
-    "[x,ea,iter]=Jacobi_rel(A,b,l,0.000001)\n",
-    "[x,ea,iter]=Jacobi_rel(A,b,1,0.000001)\n"
-   ]
-  },
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-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Nonlinear Systems\n",
-    "\n",
-    "Consider two simultaneous nonlinear equations with two unknowns:\n",
-    "\n",
-    "$x_{1}^{2}+x_{1}x_{2}=10$\n",
-    "\n",
-    "$x_{2}+3x_{1}x_{2}^{2}=57$\n",
-    "\n",
-    "Graphically, we are looking for the solution:\n"
-   ]
-  },
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-   "cell_type": "code",
-   "execution_count": 19,
-   "metadata": {
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-   "source": [
-    "x11=linspace(0.5,3);\n",
-    "x12=(10-x11.^2)./x11;\n",
-    "\n",
-    "x22=linspace(2,8);\n",
-    "x21=(57-x22).*x22.^-2/3;\n",
-    "\n",
-    "plot(x11,x12,x21,x22)\n",
-    "% Solution at x_1=2, x_2=3\n",
-    "hold on;\n",
-    "plot(2,3,'o')\n",
-    "xlabel('x_1')\n",
-    "ylabel('x_2')"
-   ]
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-   "source": [
-    "## Newton-Raphson part II\n",
-    "\n",
-    "Remember the first order approximation for the next point in a function is:\n",
-    "\n",
-    "$f(x_{i+1})=f(x_{i})+(x_{i+1}-x_{i})f'(x_{i})$\n",
-    "\n",
-    "then, $f(x_{i+1})=0$ so we are left with:\n",
-    "\n",
-    "$x_{i+1}=x_{i}-\\frac{f(x_{i})}{f'(x_{i})}$\n",
-    "\n",
-    "We can use the same formula, but now we have multiple dimensions so we need to determine the Jacobian\n",
-    "\n",
-    "$[J]=\\left[ \\begin{array}{cccc}\n",
-    "\\frac{\\partial f_{1,i}}{\\partial x_{1}} & \\frac{\\partial f_{1,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{1,i}}{\\partial x_{n}}  \\\\\n",
-    "\\frac{\\partial f_{2,i}}{\\partial x_{1}} & \\frac{\\partial f_{2,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{2,i}}{\\partial x_{n}}  \\\\\n",
-    "\\vdots & \\vdots & & \\vdots \\\\\n",
-    "\\frac{\\partial f_{n,i}}{\\partial x_{1}} & \\frac{\\partial f_{n,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{n,i}}{\\partial x_{n}}  \\\\\n",
-    "\\end{array} \\right]$\n",
-    "\n",
-    "$\\left[ \\begin{array}{c}\n",
-    "f_{1,i+1} \\\\\n",
-    "f_{2,i+1} \\\\\n",
-    "\\vdots \\\\\n",
-    "f_{n,i+1}\\end{array} \\right]=\n",
-    "\\left[ \\begin{array}{c}\n",
-    "f_{1,i} \\\\\n",
-    "f_{2,i} \\\\\n",
-    "\\vdots \\\\\n",
-    "f_{n,i}\\end{array} \\right]+\n",
-    "\\left[ \\begin{array}{cccc}\n",
-    "\\frac{\\partial f_{1,i}}{\\partial x_{1}} & \\frac{\\partial f_{1,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{1,i}}{\\partial x_{n}}  \\\\\n",
-    "\\frac{\\partial f_{2,i}}{\\partial x_{1}} & \\frac{\\partial f_{2,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{2,i}}{\\partial x_{n}}  \\\\\n",
-    "\\vdots & \\vdots & & \\vdots \\\\\n",
-    "\\frac{\\partial f_{n,i}}{\\partial x_{1}} & \\frac{\\partial f_{n,i}}{\\partial x_{2}} & \n",
-    "\\cdots & \\frac{\\partial f_{n,i}}{\\partial x_{n}}  \\\\\n",
-    "\\end{array} \\right]\n",
-    "\\left( \\left[ \\begin{array}{c}\n",
-    "x_{i+1} \\\\\n",
-    "x_{i+1} \\\\\n",
-    "\\vdots \\\\\n",
-    "x_{i+1}\\end{array} \\right]-\n",
-    "\\left[ \\begin{array}{c}\n",
-    "x_{1,i} \\\\\n",
-    "x_{2,i} \\\\\n",
-    "\\vdots \\\\\n",
-    "x_{n,i}\\end{array} \\right]\\right)$\n",
-    "\n",
-    "### Solution is again in the form Ax=b\n",
-    "\n",
-    "$[J]([x_{i+1}]-[x_{i}])=-[f]$\n",
-    "\n",
-    "so\n",
-    "\n",
-    "$[x_{i+1}]= [x_{i}]-[J]^{-1}[f]$\n",
-    "\n",
-    "## Example of Jacobian calculation\n",
-    "\n",
-    "### Nonlinear springs supporting two masses in series\n",
-    "\n",
-    "Two springs are connected to two masses, with $m_1$=1 kg and $m_{2}$=2 kg. The springs are identical, but they have nonlinear spring constants, of $k_1$=100 N/m and $k_2$=-10 N/m\n",
-    "\n",
-    "We want to solve for the final position of the masses ($x_1$ and $x_2$)\n",
-    "\n",
-    "$m_{1}g+k_{1}(x_{2}-x_{1})+k_{2}(x_{2}-x_{1})^{2}+k_{1}x_{1}+k_{2}x_{1}^{2}=0$\n",
-    "\n",
-    "$m_{2}g-k_{1}(x_{2}-x_{1})-k_{2}(x_2-x_1)^{2}=0$\n",
-    "\n",
-    "$J(1,1)=\\frac{\\partial f_{1}}{\\partial x_{1}}=-k_{1}-2k_{2}(x_{2}-x_{1})+k_{1}+2k_{2}x_{1}$\n",
-    "\n",
-    "$J(1,2)=\\frac{\\partial f_1}{\\partial x_{2}}=k_{1}+2k_{2}(x_{2}-x_{1})$\n",
-    "\n",
-    "$J(2,1)=\\frac{\\partial f_2}{\\partial x_{1}}=k_{1}+2k_{2}(x_{2}-x_{1})$\n",
-    "\n",
-    "$J(2,2)=\\frac{\\partial f_2}{\\partial x_{2}}=-k_{1}-2k_{2}(x_{2}-x_{1})$\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
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-   "source": [
-    "m1=1; % kg \n",
-    "m2=2; % kg\n",
-    "k1=100; % N/m\n",
-    "k2=-10; % N/m^2"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 20,
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-    "function [f,J]=mass_spring(x)\n",
-    "    % Function to calculate function values f1 and f2 as well as Jacobian \n",
-    "    % for 2 masses and 2 identical nonlinear springs\n",
-    "    m1=1; % kg \n",
-    "    m2=2; % kg\n",
-    "    k1=100; % N/m\n",
-    "    k2=-10; % N/m^2\n",
-    "    g=9.81; % m/s^2\n",
-    "    x1=x(1);\n",
-    "    x2=x(2);\n",
-    "    J=[-k1-2*k2*(x2-x1)-k1-2*k2*x1,k1+2*k2*(x2-x1);\n",
-    "        k1+2*k2*(x2-x1),-k1-2*k2*(x2-x1)];\n",
-    "    f=[m1*g+k1*(x2-x1)+k2*(x2-x1).^2-k1*x1-k2*x1^2;\n",
-    "       m2*g-k1*(x2-x1)-k2*(x2-x1).^2];\n",
-    "end\n",
-    "    "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 21,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "f =\n",
-      "\n",
-      "  -190.19\n",
-      "   129.62\n",
-      "\n",
-      "J =\n",
-      "\n",
-      "  -200   120\n",
-      "   120  -120\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "[f,J]=mass_spring([1,0])"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 22,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "x1 =\n",
-      "\n",
-      "  -1.5142\n",
-      "  -1.4341\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   2.9812\n",
-      "   2.3946\n",
-      "\n",
-      "x2 =\n",
-      "\n",
-      "   0.049894\n",
-      "   0.248638\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   31.3492\n",
-      "    6.7678\n",
-      "\n",
-      "x3 =\n",
-      "\n",
-      "   0.29701\n",
-      "   0.49722\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   0.83201\n",
-      "   0.49995\n",
-      "\n",
-      "x =\n",
-      "\n",
-      "   0.29701\n",
-      "   0.49722\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   0.021392\n",
-      "   0.012890\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   1.4786e-05\n",
-      "   8.9091e-06\n",
-      "\n",
-      "ea =\n",
-      "\n",
-      "   7.0642e-12\n",
-      "   4.2565e-12\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "x0=[3;2];\n",
-    "[f0,J0]=mass_spring(x0);\n",
-    "x1=x0-J0\\f0\n",
-    "ea=(x1-x0)./x1\n",
-    "[f1,J1]=mass_spring(x1);\n",
-    "x2=x1-J1\\f1\n",
-    "ea=(x2-x1)./x2\n",
-    "[f2,J2]=mass_spring(x2);\n",
-    "x3=x2-J2\\f2\n",
-    "ea=(x3-x2)./x3\n",
-    "x=x3\n",
-    "for i=1:3\n",
-    "    xold=x;\n",
-    "    [f,J]=mass_spring(x);\n",
-    "    x=x-J\\f;\n",
-    "    ea=(x-xold)./x\n",
-    "end"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 23,
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-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "x =\n",
-      "\n",
-      "   0.30351\n",
-      "   0.50372\n",
-      "\n",
-      "X0 =\n",
-      "\n",
-      "   0.30351\n",
-      "   0.50372\n",
-      "\n"
-     ]
-    }
-   ],
-   "source": [
-    "x\n",
-    "X0=fsolve(@(x) mass_spring(x),[3;5])"
-   ]
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-       "</g>\n",
-       "\t<g id=\"gnuplot_plot_1b\"><title>gnuplot_plot_1b</title>\n",
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-       "\t</g>\n",
-       "</g>\n",
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-       "</g>\n",
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-       "\t<path d=\"M463.4,31.6 L463.4,373.8 L489.3,373.8 L489.3,31.6 L463.4,31.6 Z  \" stroke=\"black\"/></g>\n",
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-       "</g>\n",
-       "</g>\n",
-       "</svg>"
-      ],
-      "text/plain": [
-       "<IPython.core.display.SVG object>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "[X,Y]=meshgrid(linspace(0,10,20),linspace(0,10,20));\n",
-    "[N,M]=size(X);\n",
-    "F=zeros(size(X));\n",
-    "for i=1:N\n",
-    "    for j=1:M\n",
-    "        [f,~]=mass_spring([X(i,j),Y(i,j)]);\n",
-    "        F1(i,j)=f(1);\n",
-    "        F2(i,j)=f(2);\n",
-    "    end\n",
-    "end\n",
-    "mesh(X,Y,F1)\n",
-    "xlabel('x_1')\n",
-    "ylabel('x_2')\n",
-    "colorbar()\n",
-    "figure()\n",
-    "mesh(X,Y,F2)\n",
-    "xlabel('x_1')\n",
-    "ylabel('x_2')\n",
-    "colorbar()"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": []
   }
  ],
  "metadata": {
diff --git a/13_eigenvalues/.ipynb_checkpoints/13_eigenvalues-checkpoint.ipynb b/13_eigenvalues/.ipynb_checkpoints/13_eigenvalues-checkpoint.ipynb
index f6b2b95..13a92e8 100644
--- a/13_eigenvalues/.ipynb_checkpoints/13_eigenvalues-checkpoint.ipynb
+++ b/13_eigenvalues/.ipynb_checkpoints/13_eigenvalues-checkpoint.ipynb
@@ -4,7 +4,10 @@
    "cell_type": "code",
    "execution_count": 1,
    "metadata": {
-    "collapsed": true
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "skip"
+    }
    },
    "outputs": [],
    "source": [
@@ -15,7 +18,10 @@
    "cell_type": "code",
    "execution_count": 2,
    "metadata": {
-    "collapsed": true
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "skip"
+    }
    },
    "outputs": [],
    "source": [
@@ -24,22 +30,53 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
-    "# Eigenvalues\n",
-    "\n",
-    "Eigenvalues and eigen vectors are the solution to the set of equations where\n",
+    "# Eigenvalues"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "Eigenvalues and eigenvectors are the solution to the set of equations where\n",
     "\n",
     "$Ax=\\lambda x$\n",
     "\n",
     "or \n",
     "\n",
-    "$A-I \\lambda=0$\n",
-    "\n",
+    "$A-I \\lambda=0$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
     "Where A is the description of the system and I is the identity matrix with the same dimensions as A and $\\lambda$ is an eigenvalue of A. \n",
     "\n",
-    "These problems are seen in a number of engineering practices:\n",
-    "\n",
+    "These problems are seen in a number of engineering practices:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
     "1. Determining vibrational modes in structural devices\n",
     "\n",
     "2. Material science - vibrational modes of crystal lattices (phonons)\n",
@@ -48,8 +85,17 @@
     "\n",
     "4. Quantum mechanics - many solutions are eigenvalue problems\n",
     "\n",
-    "5. Solid mechanics, principle stresses and principle stress directions are eigenvalues and eigenvectors\n",
-    "\n",
+    "5. Solid mechanics, principle stresses and principle stress directions are eigenvalues and eigenvectors"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
     "One way of determining the eigenvalues is taking the determinant:\n",
     "\n",
     "$|A-\\lambda I|=0$\n",
@@ -59,7 +105,11 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "Take, A\n",
     "\n",
@@ -76,7 +126,11 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "$(2-\\lambda)(3-\\lambda)(4-\\lambda)-(4-\\lambda)-(2-\\lambda)=0$\n",
     "\n",
@@ -98,7 +152,10 @@
    "cell_type": "code",
    "execution_count": 3,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
    },
    "outputs": [
     {
@@ -122,7 +179,11 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "## Applications of Eigenvalue analysis\n",
     "\n",
@@ -136,8 +197,17 @@
     "\n",
     "$m_{1}\\frac{d^{2}x_{1}}{dt^{2}}=-kx_{1}+k(x_{2}-x_{1})$\n",
     "\n",
-    "$m_{2}\\frac{d^{2}x_{2}}{dt^{2}}=-k(x_{2}-x_{1})-kx_{2}$\n",
-    "\n",
+    "$m_{2}\\frac{d^{2}x_{2}}{dt^{2}}=-k(x_{2}-x_{1})-kx_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
     "we know that $x_{i}(t)\\propto sin(\\omega t)$ so we can substitute\n",
     "\n",
     "$x_{i}=X_{i}sin(\\omega t)$\n",
@@ -151,7 +221,11 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "now, \n",
     "\n",
@@ -174,7 +248,10 @@
    "cell_type": "code",
    "execution_count": 4,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
    },
    "outputs": [
     {
@@ -214,9 +291,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 4,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
    },
    "outputs": [
     {
@@ -256,7 +336,10 @@
    "cell_type": "code",
    "execution_count": 6,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
    },
    "outputs": [
     {
@@ -277,24 +360,52 @@
     }
    ],
    "source": [
-    "K*v(:,2)\n",
+    "K*v(:,2) % K*x=lambda*x\n",
     "e(2,2)*v(:,2)"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "### $m_{1}=m_{2}=$40 kg and $k_{1}=k_{2}=k_{3}=$200 N/m\n",
-    "![animation](./eig_200_40.gif)\n",
-    "\n",
+    "![animation](./eig_200_40.gif)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
     "### $m_{1}=$40 kg, $m_{2}=$50 kg, $k_{1}=$200 N/m, and $k_{2}=k_{3}=$100 N/m\n",
     "![animation](./eig_200_100_40_50.gif)"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Thanks"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "## Solid Mechanics\n",
     "\n",
@@ -567,6 +678,7 @@
   }
  ],
  "metadata": {
+  "celltoolbar": "Slideshow",
   "kernelspec": {
    "display_name": "Octave",
    "language": "octave",
diff --git a/13_eigenvalues/13_eigenvalues.ipynb b/13_eigenvalues/13_eigenvalues.ipynb
index f6b2b95..13a92e8 100644
--- a/13_eigenvalues/13_eigenvalues.ipynb
+++ b/13_eigenvalues/13_eigenvalues.ipynb
@@ -4,7 +4,10 @@
    "cell_type": "code",
    "execution_count": 1,
    "metadata": {
-    "collapsed": true
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "skip"
+    }
    },
    "outputs": [],
    "source": [
@@ -15,7 +18,10 @@
    "cell_type": "code",
    "execution_count": 2,
    "metadata": {
-    "collapsed": true
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "skip"
+    }
    },
    "outputs": [],
    "source": [
@@ -24,22 +30,53 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
-    "# Eigenvalues\n",
-    "\n",
-    "Eigenvalues and eigen vectors are the solution to the set of equations where\n",
+    "# Eigenvalues"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "Eigenvalues and eigenvectors are the solution to the set of equations where\n",
     "\n",
     "$Ax=\\lambda x$\n",
     "\n",
     "or \n",
     "\n",
-    "$A-I \\lambda=0$\n",
-    "\n",
+    "$A-I \\lambda=0$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
     "Where A is the description of the system and I is the identity matrix with the same dimensions as A and $\\lambda$ is an eigenvalue of A. \n",
     "\n",
-    "These problems are seen in a number of engineering practices:\n",
-    "\n",
+    "These problems are seen in a number of engineering practices:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
     "1. Determining vibrational modes in structural devices\n",
     "\n",
     "2. Material science - vibrational modes of crystal lattices (phonons)\n",
@@ -48,8 +85,17 @@
     "\n",
     "4. Quantum mechanics - many solutions are eigenvalue problems\n",
     "\n",
-    "5. Solid mechanics, principle stresses and principle stress directions are eigenvalues and eigenvectors\n",
-    "\n",
+    "5. Solid mechanics, principle stresses and principle stress directions are eigenvalues and eigenvectors"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
     "One way of determining the eigenvalues is taking the determinant:\n",
     "\n",
     "$|A-\\lambda I|=0$\n",
@@ -59,7 +105,11 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "Take, A\n",
     "\n",
@@ -76,7 +126,11 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "$(2-\\lambda)(3-\\lambda)(4-\\lambda)-(4-\\lambda)-(2-\\lambda)=0$\n",
     "\n",
@@ -98,7 +152,10 @@
    "cell_type": "code",
    "execution_count": 3,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
    },
    "outputs": [
     {
@@ -122,7 +179,11 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "## Applications of Eigenvalue analysis\n",
     "\n",
@@ -136,8 +197,17 @@
     "\n",
     "$m_{1}\\frac{d^{2}x_{1}}{dt^{2}}=-kx_{1}+k(x_{2}-x_{1})$\n",
     "\n",
-    "$m_{2}\\frac{d^{2}x_{2}}{dt^{2}}=-k(x_{2}-x_{1})-kx_{2}$\n",
-    "\n",
+    "$m_{2}\\frac{d^{2}x_{2}}{dt^{2}}=-k(x_{2}-x_{1})-kx_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
     "we know that $x_{i}(t)\\propto sin(\\omega t)$ so we can substitute\n",
     "\n",
     "$x_{i}=X_{i}sin(\\omega t)$\n",
@@ -151,7 +221,11 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "now, \n",
     "\n",
@@ -174,7 +248,10 @@
    "cell_type": "code",
    "execution_count": 4,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
    },
    "outputs": [
     {
@@ -214,9 +291,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 4,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
    },
    "outputs": [
     {
@@ -256,7 +336,10 @@
    "cell_type": "code",
    "execution_count": 6,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
    },
    "outputs": [
     {
@@ -277,24 +360,52 @@
     }
    ],
    "source": [
-    "K*v(:,2)\n",
+    "K*v(:,2) % K*x=lambda*x\n",
     "e(2,2)*v(:,2)"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "### $m_{1}=m_{2}=$40 kg and $k_{1}=k_{2}=k_{3}=$200 N/m\n",
-    "![animation](./eig_200_40.gif)\n",
-    "\n",
+    "![animation](./eig_200_40.gif)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
     "### $m_{1}=$40 kg, $m_{2}=$50 kg, $k_{1}=$200 N/m, and $k_{2}=k_{3}=$100 N/m\n",
     "![animation](./eig_200_100_40_50.gif)"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Thanks"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
    "source": [
     "## Solid Mechanics\n",
     "\n",
@@ -567,6 +678,7 @@
   }
  ],
  "metadata": {
+  "celltoolbar": "Slideshow",
   "kernelspec": {
    "display_name": "Octave",
    "language": "octave",
diff --git a/13_eigenvalues/octave-workspace b/13_eigenvalues/octave-workspace
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