From de553ce8a6cac89a8dd4847d2d600f0be073f6c0 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Thu, 16 Feb 2017 00:06:23 -0500 Subject: [PATCH 01/11] add lecture 09 --- .../lecture_09-checkpoint.ipynb | 6 + lecture_09/.lecture_09.md.swp | Bin 0 -> 16384 bytes lecture_09/even_odd_perm.svg | 331 +++++ lecture_09/lecture_09.ipynb | 1151 +++++++++++++++++ lecture_09/lecture_09.log | 28 + lecture_09/lecture_09.md | 802 ++++++++++++ lecture_09/lecture_09.pdf | Bin 0 -> 16189 bytes lecture_09/mass_springs.svg | 214 +++ 8 files changed, 2532 insertions(+) create mode 100644 lecture_09/.ipynb_checkpoints/lecture_09-checkpoint.ipynb create mode 100644 lecture_09/.lecture_09.md.swp create mode 100644 lecture_09/even_odd_perm.svg create mode 100644 lecture_09/lecture_09.ipynb create mode 100644 lecture_09/lecture_09.log create mode 100644 lecture_09/lecture_09.md create mode 100644 lecture_09/lecture_09.pdf create mode 100644 lecture_09/mass_springs.svg diff --git a/lecture_09/.ipynb_checkpoints/lecture_09-checkpoint.ipynb 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equations:\n", + "\n", + "1. $5x_{1}+3x_{2} =1$\n", + "\n", + "2. $x_{1}+2x_{2}+3x_{3} =2$\n", + "\n", + "3. $x_{1}+x_{2}+x_{3} =3$\n", + "\n", + "in matrix form:\n", + "\n", + "$\\left[ \\begin{array}{ccc}\n", + "5 & 3 & 0 \\\\\n", + "1 & 2 & 3 \\\\\n", + "1 & 1 & 1 \\end{array} \\right]\n", + "\\left[\\begin{array}{c} \n", + "x_{1} \\\\ \n", + "x_{2} \\\\\n", + "x_{3}\\end{array}\\right]=\\left[\\begin{array}{c} \n", + "1 \\\\\n", + "2 \\\\\n", + "3\\end{array}\\right]$\n", + "\n", + "$Ax=b$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Vectors \n", + "\n", + "column vector x (length of 3):\n", + "\n", + "$\\left[\\begin{array}{c} \n", + "x_{1} \\\\ \n", + "x_{2} \\\\\n", + "x_{3}\\end{array}\\right]$\n", + "\n", + "row vector y (length of 3):\n", + "\n", + "$\\left[\\begin{array}{ccc} \n", + "y_{1}~ \n", + "y_{2}~ \n", + "y_{3}\\end{array}\\right]$\n", + "\n", + "vector of length N:\n", + "\n", + "$\\left[\\begin{array}{c} \n", + "x_{1} \\\\ \n", + "x_{2} \\\\\n", + "\\vdots \\\\\n", + "x_{N}\\end{array}\\right]$\n", + "\n", + "The $i^{th}$ element of x is $x_{i}$" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "x =\n", + "\n", + " 1 2 3 4 5 6 7 8 9 10\n", + "\n" + ] + } + ], + "source": [ + "x=[1:10]" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 1\n", + " 2\n", + " 3\n", + " 4\n", + " 5\n", + " 6\n", + " 7\n", + " 8\n", + " 9\n", + " 10\n", + "\n" + ] + } + ], + "source": [ + "x'" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Matrices\n", + "\n", + "Matrix A is 3x3:\n", + "\n", + "$A=\\left[ \\begin{array}{ccc}\n", + "5 & 3 & 0 \\\\\n", + "1 & 2 & 3 \\\\\n", + "1 & 1 & 1 \\end{array} \\right]$\n", + "\n", + "elements in the matrix are denoted $A_{row~column}$, $A_{23}=3$\n", + "\n", + "In general, matrix, B, can be any size, $M \\times N$ (M-rows and N-columns):\n", + "\n", + "$B=\\left[ \\begin{array}{cccc}\n", + "B_{11} & B_{12} &...& B_{1N} \\\\\n", + "B_{21} & B_{22} &...& B_{2N} \\\\\n", + "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n", + "B_{M1} & B_{M2} &...& B_{MN}\\end{array} \\right]$" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A =\n", + "\n", + " 5 3 0\n", + " 1 2 3\n", + " 1 1 1\n", + "\n" + ] + } + ], + "source": [ + "A=[5,3,0;1,2,3;1,1,1]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Multiplication\n", + "\n", + "A column vector is a $1\\times N$ matrix and a row vector is a $M\\times 1$ matrix\n", + "\n", + "**Multiplying matrices is not commutative**\n", + "\n", + "$A B \\neq B A$\n", + "\n", + "Inner dimensions must agree, output is outer dimensions. \n", + "\n", + "A is $M1 \\times N1$ and B is $M2 \\times N2$\n", + "\n", + "C=AB\n", + "\n", + "Therefore N1=M2 and C is $M1 \\times N2$\n", + "\n", + "If $C'=BA$, then N2=M1 and C is $M2 \\times N1$\n", + "\n", + "\n", + "e.g. \n", + "$A=\\left[ \\begin{array}{cc}\n", + "5 & 3 & 0 \\\\\n", + "1 & 2 & 3 \\end{array} \\right]$\n", + "\n", + "$B=\\left[ \\begin{array}{cccc}\n", + "1 & 2 & 3 & 4 \\\\\n", + "5 & 6 & 7 & 8 \\\\\n", + "9 & 10 & 11 & 12 \\end{array} \\right]$\n", + "\n", + "C=AB\n", + "\n", + "$[2\\times 4] = [2 \\times 3][3 \\times 4]$\n", + "\n", + "The rule for multiplying matrices, A and B, is\n", + "\n", + "$C_{ij} = \\sum_{k=1}^{n}A_{ik}B_{kj}$\n", + "\n", + "In the previous example, \n", + "\n", + "$C_{11} = A_{11}B_{11}+A_{12}B_{21}+A_{13}B_{31} = 5*1+3*5+0*9=20$\n", + "\n", + "\n", + "Multiplication is associative:\n", + "\n", + "$(AB)C = A(BC)$\n", + "\n", + "and distributive:\n", + "\n", + "$A(B+C)=AB+AC$" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A =\n", + "\n", + " 5 3 0\n", + " 1 2 3\n", + "\n", + "B =\n", + "\n", + " 1 2 3 4\n", + " 5 6 7 8\n", + " 9 10 11 12\n", + "\n" + ] + } + ], + "source": [ + "A=[5,3,0;1,2,3] \n", + "B=[1,2,3,4;5,6,7,8;9,10,11,12]" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "C =\n", + "\n", + " 0 0 0 0\n", + " 0 0 0 0\n", + "\n", + "C =\n", + "\n", + " 20 28 36 44\n", + " 38 44 50 56\n", + "\n" + ] + } + ], + "source": [ + "C=zeros(2,4)\n", + "C(1,1)=A(1,1)*B(1,1)+A(1,2)*B(2,1)+A(1,3)*B(3,1);\n", + "C(1,2)=A(1,1)*B(1,2)+A(1,2)*B(2,2)+A(1,3)*B(3,2);\n", + "\n", + "C=A*B" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "error: operator *: nonconformant arguments (op1 is 3x4, op2 is 2x3)\r\n" + ] + } + ], + "source": [ + "Cp=B*A" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Representation of a problem with Matrices and Vectors\n", + "\n", + "If you have a set of known output, $y_{1},~y_{2},~...y_{N}$ and a set of equations that\n", + "relate unknown inputs, $x_{1},~x_{2},~...x_{N}$, then these can be written in a vector\n", + "matrix format as:\n", + "\n", + "$y=Ax=\\left[\\begin{array}{c} \n", + "y_{1} \\\\ \n", + "y_{2} \\\\\n", + "\\vdots \\\\\n", + "y_{N}\\end{array}\\right]\n", + "=\n", + "A\\left[\\begin{array}{c} \n", + "x_{1} \\\\ \n", + "x_{2} \\\\\n", + "\\vdots \\\\\n", + "x_{N}\\end{array}\\right]$\n", + "\n", + "$A=\n", + "\\left[\\begin{array}{cccc} \n", + "| & | & & | \\\\ \n", + "a_{1} & a_{2} & ... & a_{N} \\\\ \n", + "| & | & & | \\end{array}\\right]$\n", + "\n", + "or \n", + "\n", + "$y = a_{1}x_{1} + a_{2}x_{2} +...+a_{N}x_{N}$\n", + "\n", + "where each $a_{i}$ is a column vector and $x_{i}$ is a scalar taken from the $i^{th}$\n", + "element of x.\n", + "\n", + "Consider the following problem, 4 masses are connected in series to 4 springs with K=10 N/m. What are the final positions of the masses? \n", + "\n", + "![Springs-masses](mass_springs.svg)\n", + "\n", + "The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass:\n", + "\n", + "$m_{1}g+k(x_{2}-x_{1})-kx_{1}=0$\n", + "\n", + "$m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0$\n", + "\n", + "$m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0$\n", + "\n", + "$m_{4}g-k(x_{4}-x_{3})=0$\n", + "\n", + "in matrix form:\n", + "\n", + "$\\left[ \\begin{array}{cccc}\n", + "2k & -k & 0 & 0 \\\\\n", + "-k & 2k & -k & 0 \\\\\n", + "0 & -k & 2k & -k \\\\\n", + "0 & 0 & -k & k \\end{array} \\right]\n", + "\\left[ \\begin{array}{c}\n", + "x_{1} \\\\\n", + "x_{2} \\\\\n", + "x_{3} \\\\\n", + "x_{4} \\end{array} \\right]=\n", + "\\left[ \\begin{array}{c}\n", + "m_{1}g \\\\\n", + "m_{2}g \\\\\n", + "m_{3}g \\\\\n", + "m_{4}g \\end{array} \\right]$" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "K =\n", + "\n", + " 20 -10 0 0\n", + " -10 20 -10 0\n", + " 0 -10 20 -10\n", + " 0 0 -10 10\n", + "\n", + "y =\n", + "\n", + " 9.8100\n", + " 19.6200\n", + " 29.4300\n", + " 39.2400\n", + "\n", + "x =\n", + "\n", + " 9.8100\n", + " 18.6390\n", + " 25.5060\n", + " 29.4300\n", + "\n" + ] + } + ], + "source": [ + "k=10; % N/m\n", + "m1=1; % kg\n", + "m2=2;\n", + "m3=3;\n", + "m4=4;\n", + "g=9.81; % m/s^2\n", + "K=[2*k -k 0 0; -k 2*k -k 0; 0 -k 2*k -k; 0 0 -k k]\n", + "y=[m1*g;m2*g;m3*g;m4*g]\n", + "\n", + "x=K\\y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Matrix Operations\n", + "\n", + "Identity matrix `eye(M,N)` **Assume M=N unless specfied**\n", + "\n", + "$I_{ij} =1$ if $i=j$ \n", + "\n", + "$I_{ij} =0$ if $i\\neq j$ \n", + "\n", + "AI=A=IA\n", + "\n", + "Diagonal matrix, D, is similar to the identity matrix, but each diagonal element can be any\n", + "number\n", + "\n", + "$D_{ij} =d_{i}$ if $i=j$ \n", + "\n", + "$D_{ij} =0$ if $i\\neq j$ " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Transpose\n", + "\n", + "The transpose of a matrix changes the rows -> columns and columns-> rows\n", + "\n", + "$(A^{T})_{ij} = A_{ji}$\n", + "\n", + "for diagonal matrices, $D^{T}=D$\n", + "\n", + "in general, the transpose has the following qualities:\n", + " \n", + "1. $(A^{T})^{T}=A$\n", + "\n", + "2. $(AB)^{T} = B^{T}A^{T}$\n", + "\n", + "3. $(A+B)^{T} = A^{T} +B^{T}$ \n", + "\n", + "All matrices have a symmetric and antisymmetric part:\n", + "\n", + "$A= 1/2(A+A^{T}) +1/2(A-A^{T})$\n", + "\n", + "If $A=A^{T}$, then A is symmetric, if $A=-A^{T}$, then A is antisymmetric." + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 20 38\n", + " 28 44\n", + " 36 50\n", + " 44 56\n", + "\n", + "ans =\n", + "\n", + " 20 38\n", + " 28 44\n", + " 36 50\n", + " 44 56\n", + "\n", + "error: operator *: nonconformant arguments (op1 is 3x2, op2 is 4x3)\n" + ] + } + ], + "source": [ + "(A*B)'\n", + "\n", + "B'*A' % if this wasnt true, then inner dimensions wouldn;t match\n", + "\n", + "A'*B' == (A*B)'" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Square matrix operations\n", + "\n", + "### Trace\n", + "\n", + "The trace of a square matrix is the sum of the diagonal elements\n", + "\n", + "$tr~A=\\sum_{i=1}^{N}A_{ii}$\n", + "\n", + "The trace is invariant, meaning that if you change the basis through a rotation, then the\n", + "trace remains the same. \n", + "\n", + "It also has the following properties:\n", + "\n", + "1. $tr~A=tr~A^{T}$\n", + "\n", + "2. $tr~A+tr~B=tr(A+B)$\n", + "\n", + "3. $tr(kA)=k tr~A$\n", + "\n", + "4. $tr(AB)=tr(BA)$" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "id_m =\n", + "\n", + "Diagonal Matrix\n", + "\n", + " 1 0 0\n", + " 0 1 0\n", + " 0 0 1\n", + "\n", + "ans = 3\n" + ] + } + ], + "source": [ + "id_m=eye(3)\n", + "trace(id_m)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Inverse\n", + "\n", + "The inverse of a square matrix, $A^{-1}$ is defined such that\n", + "\n", + "$A^{-1}A=I=AA^{-1}$\n", + "\n", + "Not all square matrices have an inverse, they can be *singular* or *non-invertible*\n", + "\n", + "The inverse has the following properties:\n", + "\n", + "1. $(A^{-1})^{-1}=A$\n", + "\n", + "2. $(AB)^{-1}=B^{-1}A^{-1}$\n", + "\n", + "3. $(A^{-1})^{T}=(A^{T})^{-1}$" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A =\n", + "\n", + " 0.5762106 0.3533174 0.7172134\n", + " 0.7061664 0.4863733 0.9423436\n", + " 0.4255961 0.0016613 0.3561407\n", + "\n" + ] + } + ], + "source": [ + "A=rand(3,3)" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Ainv =\n", + "\n", + " 41.5613 -30.1783 -3.8467\n", + " 36.2130 -24.2201 -8.8415\n", + " -49.8356 36.1767 7.4460\n", + "\n" + ] + } + ], + "source": [ + "Ainv=inv(A)" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "B =\n", + "\n", + " 0.524529 0.470856 0.708116\n", + " 0.084491 0.730986 0.528292\n", + " 0.303545 0.782522 0.389282\n", + "\n" + ] + } + ], + "source": [ + "B=rand(3,3)" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " -182.185 125.738 40.598\n", + " -133.512 97.116 17.079\n", + " 282.422 -200.333 -46.861\n", + "\n", + "ans =\n", + "\n", + " -182.185 125.738 40.598\n", + " -133.512 97.116 17.079\n", + " 282.422 -200.333 -46.861\n", + "\n", + "ans =\n", + "\n", + " 41.5613 36.2130 -49.8356\n", + " -30.1783 -24.2201 36.1767\n", + " -3.8467 -8.8415 7.4460\n", + "\n", + "ans =\n", + "\n", + " 41.5613 36.2130 -49.8356\n", + " -30.1783 -24.2201 36.1767\n", + " -3.8467 -8.8415 7.4460\n", + "\n" + ] + } + ], + "source": [ + "inv(A*B)\n", + "inv(B)*inv(A)\n", + "\n", + "inv(A')\n", + "\n", + "inv(A)'" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Orthogonal Matrices\n", + "\n", + "Vectors are *orthogonal* if $x^{T}$ y=0, and a vector is *normalized* if $||x||_{2}=1$. A\n", + "square matrix is *orthogonal* if all its column vectors are orthogonal to each other and\n", + "normalized. The column vectors are then called *orthonormal* and the following results\n", + "\n", + "$U^{T}U=I=UU^{T}$\n", + "\n", + "and \n", + "\n", + "$||Ux||_{2}=||x||_{2}$\n", + "\n", + "### Determinant\n", + "\n", + "The **determinant** of a matrix has 3 properties\n", + "\n", + "1. The determinant of the identity matrix is one, $|I|=1$\n", + "\n", + "2. If you multiply a single row by scalar $t$ then the determinant is $t|A|$:\n", + "\n", + "$t|A|=\\left[ \\begin{array}{cccc}\n", + "tA_{11} & tA_{12} &...& tA_{1N} \\\\\n", + "A_{21} & A_{22} &...& A_{2N} \\\\\n", + "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n", + "A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$\n", + "\n", + "3. If you switch 2 rows, the determinant changes sign:\n", + "\n", + "\n", + "$-|A|=\\left[ \\begin{array}{cccc}\n", + "A_{21} & A_{22} &...& A_{2N} \\\\\n", + "A_{11} & A_{12} &...& A_{1N} \\\\\n", + "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n", + "A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$\n", + "\n", + "4. inverse of the determinant is the determinant of the inverse:\n", + "\n", + "$|A^{-1}|=\\frac{1}{|A|}=|A|^{-1}$\n", + "\n", + "For a $2\\times2$ matrix, \n", + "\n", + "$|A|=\\left|\\left[ \\begin{array}{cc}\n", + "A_{11} & A_{12} \\\\\n", + "A_{21} & A_{22} \\\\\n", + "\\end{array} \\right]\\right| = A_{11}A_{22}-A_{21}A_{12}$\n", + "\n", + "For a $3\\times3$ matrix,\n", + "\n", + "$|A|=\\left|\\left[ \\begin{array}{ccc}\n", + "A_{11} & A_{12} & A_{13} \\\\\n", + "A_{21} & A_{22} & A_{23} \\\\\n", + "A_{31} & A_{32} & A_{33}\\end{array} \\right]\\right|=$\n", + "\n", + "$A_{11}A_{22}A_{33}+A_{12}A_{23}A_{31}+A_{13}A_{21}A_{32}\n", + "-A_{31}A_{22}A_{13}-A_{32}A_{23}A_{11}-A_{33}A_{21}A_{12}$\n", + "\n", + "For larger matrices, the determinant is \n", + "\n", + "$|A|=\n", + "\\frac{1}{n!}\n", + "\\sum_{i_{r}}^{N}\n", + "\\sum_{j_{r}}^{N}\n", + "\\epsilon_{i_{1}...i_{N}}\n", + "\\epsilon_{j_{1}...j_{N}}\n", + "A_{i_{1}j_{1}}\n", + "A_{i_{2}j_{2}}\n", + "...\n", + "A_{i_{N}j_{N}}$\n", + "\n", + "Where the Levi-Cevita symbol is $\\epsilon_{i_{1}i_{2}...i_{N}} = 1$ if it is an even\n", + "permutation and $\\epsilon_{i_{1}i_{2}...i_{N}} = -1$ if it is an odd permutation and\n", + "$\\epsilon_{i_{1}i_{2}...i_{N}} = 0$ if $i_{n}=i_{m}$ e.g. $i_{1}=i_{7}$\n", + "\n", + "![Levi-Cevita rule for even and odd permutations for an $N\\times N$\n", + "determinant](even_odd_perm.svg)\n", + "\n", + "So a $4\\times4$ matrix determinant,\n", + "\n", + "$|A|=\\left|\\left[ \\begin{array}{cccc}\n", + "A_{11} & A_{12} & A_{13} & A_{14} \\\\\n", + "A_{21} & A_{22} & A_{23} & A_{24} \\\\\n", + "A_{31} & A_{32} & A_{33} & A_{34} \\\\\n", + "A_{41} & A_{42} & A_{43} & A_{44} \\end{array} \\right]\\right|$\n", + "\n", + "$=\\epsilon_{1234}\\epsilon_{1234}A_{11}A_{22}A_{33}A_{44}+\n", + "\\epsilon_{2341}\\epsilon_{1234}A_{21}A_{32}A_{43}A_{14}+\n", + "\\epsilon_{3412}\\epsilon_{1234}A_{31}A_{42}A_{13}A_{24}+\n", + "\\epsilon_{4123}\\epsilon_{1234}A_{41}A_{12}A_{23}A_{34}+...\n", + "\\epsilon_{4321}\\epsilon_{4321}A_{44}A_{33}A_{22}A_{11}$\n", + "\n", + "### Special Case for determinants\n", + "\n", + "The determinant of a diagonal matrix $|D|=D_{11}D_{22}D_{33}...D_{NN}$. \n", + "\n", + "Similarly, if a matrix is upper triangular (so all values of $A_{ij}=0$ when $j +Babel <3.9q> and hyphenation patterns for 81 language(s) loaded. +! You can't use `macro parameter character #' in vertical mode. +l.2 # + Linear Algebra (Review/Introduction) +? +! Interruption. + + +l.2 # + Linear Algebra (Review/Introduction) +? xx + +Here is how much of TeX's memory you used: + 6 strings out of 493030 + 147 string characters out of 6136260 + 53067 words of memory out of 5000000 + 3641 multiletter control sequences out of 15000+600000 + 3640 words of font info for 14 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 5i,0n,1p,57b,8s stack positions out of 5000i,500n,10000p,200000b,80000s +No pages of output. diff --git a/lecture_09/lecture_09.md b/lecture_09/lecture_09.md new file mode 100644 index 0000000..a1d53d1 --- /dev/null +++ b/lecture_09/lecture_09.md @@ -0,0 +1,802 @@ + +# Linear Algebra (Review/Introduction) + +Representation of linear equations: + +1. $5x_{1}+3x_{2} =1$ + +2. $x_{1}+2x_{2}+3x_{3} =2$ + +3. $x_{1}+x_{2}+x_{3} =3$ + +in matrix form: + +$\left[ \begin{array}{ccc} +5 & 3 & 0 \\ +1 & 2 & 3 \\ +1 & 1 & 1 \end{array} \right] +\left[\begin{array}{c} +x_{1} \\ +x_{2} \\ +x_{3}\end{array}\right]=\left[\begin{array}{c} +1 \\ +2 \\ +3\end{array}\right]$ + +$Ax=b$ + +### Vectors + +column vector x (length of 3): + +$\left[\begin{array}{c} +x_{1} \\ +x_{2} \\ +x_{3}\end{array}\right]$ + +row vector y (length of 3): + +$\left[\begin{array}{ccc} +y_{1}~ +y_{2}~ +y_{3}\end{array}\right]$ + +vector of length N: + +$\left[\begin{array}{c} +x_{1} \\ +x_{2} \\ +\vdots \\ +x_{N}\end{array}\right]$ + +The $i^{th}$ element of x is $x_{i}$ + + +```octave +x=[1:10] +``` + + x = + + 1 2 3 4 5 6 7 8 9 10 + + + + +```octave +x' +``` + + ans = + + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + + + +### Matrices + +Matrix A is 3x3: + +$A=\left[ \begin{array}{ccc} +5 & 3 & 0 \\ +1 & 2 & 3 \\ +1 & 1 & 1 \end{array} \right]$ + +elements in the matrix are denoted $A_{row~column}$, $A_{23}=3$ + +In general, matrix, B, can be any size, $M \times N$ (M-rows and N-columns): + +$B=\left[ \begin{array}{cccc} +B_{11} & B_{12} &...& B_{1N} \\ +B_{21} & B_{22} &...& B_{2N} \\ +\vdots & \vdots &\ddots& \vdots \\ +B_{M1} & B_{M2} &...& B_{MN}\end{array} \right]$ + + +```octave +A=[5,3,0;1,2,3;1,1,1] +``` + + A = + + 5 3 0 + 1 2 3 + 1 1 1 + + + +## Multiplication + +A column vector is a $1\times N$ matrix and a row vector is a $M\times 1$ matrix + +**Multiplying matrices is not commutative** + +$A B \neq B A$ + +Inner dimensions must agree, output is outer dimensions. + +A is $M1 \times N1$ and B is $M2 \times N2$ + +C=AB + +Therefore N1=M2 and C is $M1 \times N2$ + +If $C'=BA$, then N2=M1 and C is $M2 \times N1$ + + +e.g. +$A=\left[ \begin{array}{cc} +5 & 3 & 0 \\ +1 & 2 & 3 \end{array} \right]$ + +$B=\left[ \begin{array}{cccc} +1 & 2 & 3 & 4 \\ +5 & 6 & 7 & 8 \\ +9 & 10 & 11 & 12 \end{array} \right]$ + +C=AB + +$[2\times 4] = [2 \times 3][3 \times 4]$ + +The rule for multiplying matrices, A and B, is + +$C_{ij} = \sum_{k=1}^{n}A_{ik}B_{kj}$ + +In the previous example, + +$C_{11} = A_{11}B_{11}+A_{12}B_{21}+A_{13}B_{31} = 5*1+3*5+0*9=20$ + + +Multiplication is associative: + +$(AB)C = A(BC)$ + +and distributive: + +$A(B+C)=AB+AC$ + + +```octave +A=[5,3,0;1,2,3] +B=[1,2,3,4;5,6,7,8;9,10,11,12] +``` + + A = + + 5 3 0 + 1 2 3 + + B = + + 1 2 3 4 + 5 6 7 8 + 9 10 11 12 + + + + +```octave +C=zeros(2,4) +C(1,1)=A(1,1)*B(1,1)+A(1,2)*B(2,1)+A(1,3)*B(3,1); +C(1,2)=A(1,1)*B(1,2)+A(1,2)*B(2,2)+A(1,3)*B(3,2); + +C=A*B +``` + + C = + + 0 0 0 0 + 0 0 0 0 + + C = + + 20 28 36 44 + 38 44 50 56 + + + + +```octave +Cp=B*A +``` + + error: operator *: nonconformant arguments (op1 is 3x4, op2 is 2x3) + + +## Representation of a problem with Matrices and Vectors + +If you have a set of known output, $y_{1},~y_{2},~...y_{N}$ and a set of equations that +relate unknown inputs, $x_{1},~x_{2},~...x_{N}$, then these can be written in a vector +matrix format as: + +$y=Ax=\left[\begin{array}{c} +y_{1} \\ +y_{2} \\ +\vdots \\ +y_{N}\end{array}\right] += +A\left[\begin{array}{c} +x_{1} \\ +x_{2} \\ +\vdots \\ +x_{N}\end{array}\right]$ + +$A= +\left[\begin{array}{cccc} +| & | & & | \\ +a_{1} & a_{2} & ... & a_{N} \\ +| & | & & | \end{array}\right]$ + +or + +$y = a_{1}x_{1} + a_{2}x_{2} +...+a_{N}x_{N}$ + +where each $a_{i}$ is a column vector and $x_{i}$ is a scalar taken from the $i^{th}$ +element of x. + +Consider the following problem, 4 masses are connected in series to 4 springs with K=10 N/m. What are the final positions of the masses? + +![Springs-masses](mass_springs.svg) + +The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass: + +$m_{1}g+k(x_{2}-x_{1})-kx_{1}=0$ + +$m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0$ + +$m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0$ + +$m_{4}g-k(x_{4}-x_{3})=0$ + +in matrix form: + +$\left[ \begin{array}{cccc} +2k & -k & 0 & 0 \\ +-k & 2k & -k & 0 \\ +0 & -k & 2k & -k \\ +0 & 0 & -k & k \end{array} \right] +\left[ \begin{array}{c} +x_{1} \\ +x_{2} \\ +x_{3} \\ +x_{4} \end{array} \right]= +\left[ \begin{array}{c} +m_{1}g \\ +m_{2}g \\ +m_{3}g \\ +m_{4}g \end{array} \right]$ + + +```octave +k=10; % N/m +m1=1; % kg +m2=2; +m3=3; +m4=4; +g=9.81; % m/s^2 +K=[2*k -k 0 0; -k 2*k -k 0; 0 -k 2*k -k; 0 0 -k k] +y=[m1*g;m2*g;m3*g;m4*g] + +x=K\y +``` + + K = + + 20 -10 0 0 + -10 20 -10 0 + 0 -10 20 -10 + 0 0 -10 10 + + y = + + 9.8100 + 19.6200 + 29.4300 + 39.2400 + + x = + + 9.8100 + 18.6390 + 25.5060 + 29.4300 + + + +## Matrix Operations + +Identity matrix `eye(M,N)` **Assume M=N unless specfied** + +$I_{ij} =1$ if $i=j$ + +$I_{ij} =0$ if $i\neq j$ + +AI=A=IA + +Diagonal matrix, D, is similar to the identity matrix, but each diagonal element can be any +number + +$D_{ij} =d_{i}$ if $i=j$ + +$D_{ij} =0$ if $i\neq j$ + +### Transpose + +The transpose of a matrix changes the rows -> columns and columns-> rows + +$(A^{T})_{ij} = A_{ji}$ + +for diagonal matrices, $D^{T}=D$ + +in general, the transpose has the following qualities: + +1. $(A^{T})^{T}=A$ + +2. $(AB)^{T} = B^{T}A^{T}$ + +3. $(A+B)^{T} = A^{T} +B^{T}$ + +All matrices have a symmetric and antisymmetric part: + +$A= 1/2(A+A^{T}) +1/2(A-A^{T})$ + +If $A=A^{T}$, then A is symmetric, if $A=-A^{T}$, then A is antisymmetric. + + +```octave +(A*B)' + +B'*A' % if this wasnt true, then inner dimensions wouldn;t match + +A'*B' == (A*B)' +``` + + ans = + + 20 38 + 28 44 + 36 50 + 44 56 + + ans = + + 20 38 + 28 44 + 36 50 + 44 56 + + error: operator *: nonconformant arguments (op1 is 3x2, op2 is 4x3) + + +## Square matrix operations + +### Trace + +The trace of a square matrix is the sum of the diagonal elements + +$tr~A=\sum_{i=1}^{N}A_{ii}$ + +The trace is invariant, meaning that if you change the basis through a rotation, then the +trace remains the same. + +It also has the following properties: + +1. $tr~A=tr~A^{T}$ + +2. $tr~A+tr~B=tr(A+B)$ + +3. $tr(kA)=k tr~A$ + +4. $tr(AB)=tr(BA)$ + + +```octave +id_m=eye(3) +trace(id_m) +``` + + id_m = + + Diagonal Matrix + + 1 0 0 + 0 1 0 + 0 0 1 + + ans = 3 + + +### Inverse + +The inverse of a square matrix, $A^{-1}$ is defined such that + +$A^{-1}A=I=AA^{-1}$ + +Not all square matrices have an inverse, they can be *singular* or *non-invertible* + +The inverse has the following properties: + +1. $(A^{-1})^{-1}=A$ + +2. $(AB)^{-1}=B^{-1}A^{-1}$ + +3. $(A^{-1})^{T}=(A^{T})^{-1}$ + + +```octave +A=rand(3,3) +``` + + A = + + 0.5762106 0.3533174 0.7172134 + 0.7061664 0.4863733 0.9423436 + 0.4255961 0.0016613 0.3561407 + + + + +```octave +Ainv=inv(A) +``` + + Ainv = + + 41.5613 -30.1783 -3.8467 + 36.2130 -24.2201 -8.8415 + -49.8356 36.1767 7.4460 + + + + +```octave +B=rand(3,3) +``` + + B = + + 0.524529 0.470856 0.708116 + 0.084491 0.730986 0.528292 + 0.303545 0.782522 0.389282 + + + + +```octave +inv(A*B) +inv(B)*inv(A) + +inv(A') + +inv(A)' +``` + + ans = + + -182.185 125.738 40.598 + -133.512 97.116 17.079 + 282.422 -200.333 -46.861 + + ans = + + -182.185 125.738 40.598 + -133.512 97.116 17.079 + 282.422 -200.333 -46.861 + + ans = + + 41.5613 36.2130 -49.8356 + -30.1783 -24.2201 36.1767 + -3.8467 -8.8415 7.4460 + + ans = + + 41.5613 36.2130 -49.8356 + -30.1783 -24.2201 36.1767 + -3.8467 -8.8415 7.4460 + + + +### Orthogonal Matrices + +Vectors are *orthogonal* if $x^{T}$ y=0, and a vector is *normalized* if $||x||_{2}=1$. A +square matrix is *orthogonal* if all its column vectors are orthogonal to each other and +normalized. The column vectors are then called *orthonormal* and the following results + +$U^{T}U=I=UU^{T}$ + +and + +$||Ux||_{2}=||x||_{2}$ + +### Determinant + +The **determinant** of a matrix has 3 properties + +1. The determinant of the identity matrix is one, $|I|=1$ + +2. If you multiply a single row by scalar $t$ then the determinant is $t|A|$: + +$t|A|=\left[ \begin{array}{cccc} +tA_{11} & tA_{12} &...& tA_{1N} \\ +A_{21} & A_{22} &...& A_{2N} \\ +\vdots & \vdots &\ddots& \vdots \\ +A_{M1} & A_{M2} &...& A_{MN}\end{array} \right]$ + +3. If you switch 2 rows, the determinant changes sign: + + +$-|A|=\left[ \begin{array}{cccc} +A_{21} & A_{22} &...& A_{2N} \\ +A_{11} & A_{12} &...& A_{1N} \\ +\vdots & \vdots &\ddots& \vdots \\ +A_{M1} & A_{M2} &...& A_{MN}\end{array} \right]$ + +4. inverse of the determinant is the determinant of the inverse: + +$|A^{-1}|=\frac{1}{|A|}=|A|^{-1}$ + +For a $2\times2$ matrix, + +$|A|=\left|\left[ \begin{array}{cc} +A_{11} & A_{12} \\ +A_{21} & A_{22} \\ +\end{array} \right]\right| = A_{11}A_{22}-A_{21}A_{12}$ + +For a $3\times3$ matrix, + +$|A|=\left|\left[ \begin{array}{ccc} +A_{11} & A_{12} & A_{13} \\ +A_{21} & A_{22} & A_{23} \\ +A_{31} & A_{32} & A_{33}\end{array} \right]\right|=$ + +$A_{11}A_{22}A_{33}+A_{12}A_{23}A_{31}+A_{13}A_{21}A_{32} +-A_{31}A_{22}A_{13}-A_{32}A_{23}A_{11}-A_{33}A_{21}A_{12}$ + +For larger matrices, the determinant is + +$|A|= +\frac{1}{n!} +\sum_{i_{r}}^{N} +\sum_{j_{r}}^{N} +\epsilon_{i_{1}...i_{N}} +\epsilon_{j_{1}...j_{N}} +A_{i_{1}j_{1}} +A_{i_{2}j_{2}} +... +A_{i_{N}j_{N}}$ + +Where the Levi-Cevita symbol is $\epsilon_{i_{1}i_{2}...i_{N}} = 1$ if it is an even +permutation and $\epsilon_{i_{1}i_{2}...i_{N}} = -1$ if it is an odd permutation and +$\epsilon_{i_{1}i_{2}...i_{N}} = 0$ if $i_{n}=i_{m}$ e.g. $i_{1}=i_{7}$ + +![Levi-Cevita rule for even and odd permutations for an $N\times N$ +determinant](even_odd_perm.svg) + +So a $4\times4$ matrix determinant, + +$|A|=\left|\left[ \begin{array}{cccc} +A_{11} & A_{12} & A_{13} & A_{14} \\ +A_{21} & A_{22} & A_{23} & A_{24} \\ +A_{31} & A_{32} & A_{33} & A_{34} \\ +A_{41} & A_{42} & A_{43} & A_{44} \end{array} \right]\right|$ + +$=\epsilon_{1234}\epsilon_{1234}A_{11}A_{22}A_{33}A_{44}+ +\epsilon_{2341}\epsilon_{1234}A_{21}A_{32}A_{43}A_{14}+ +\epsilon_{3412}\epsilon_{1234}A_{31}A_{42}A_{13}A_{24}+ +\epsilon_{4123}\epsilon_{1234}A_{41}A_{12}A_{23}A_{34}+... +\epsilon_{4321}\epsilon_{4321}A_{44}A_{33}A_{22}A_{11}$ + +### Special Case for determinants + +The determinant of a diagonal matrix $|D|=D_{11}D_{22}D_{33}...D_{NN}$. + +Similarly, if a matrix is upper triangular (so all values of $A_{ij}=0$ when $j{i-l?Q^_J>s~ zzpCr1wMgZKMQIplS)fRd?yldU80ZP;3G56lp}4u}L@lhHO&sY&tqq(_giVa>j7{jI zO>E7a%?TLjIaqjkp`4r@O$=H*K-m5PWXc?i@l~$#Hfp{=(DDp-@HwumpDx z2^3IRY5CK}{2_hVb#tGOtkPMJIB4cKz}9v!3pJb3W@r2yi+A`CFa`Z~WYwwH8@U>m zGBy{;H$B7-)eWkzUV(}>JA}8Z@?hGwUL|f;c|?CLay#5jf(l$!C~4JC0;3T)G79W#2I0Uv=Qrq*o?LfplfBQb0yPj@c_0o$+`cwO*HA8K0SL zR-3jP{~}rMs;$kGYdU$1Qk$Iid^X_Po>HB>oChQUpY>s-Wfn)Y(PQx=54*`aHbrty z{9cw}hU;pE<9)zh$N z<5|Ju01C&XB=I*iT^v~iMtxW_0v{ZLP{UJPY|IJq2n7)s;tvL6TUtAgAVH&#ccR8P zp-IRYQdQRwm^}D8;o6CY2l8$}TQq@`_wCrYT~xL^qoXDe8RPL{0T%?H-j#D2N60LG z_95|5-%u}(y9O5|qa!3IBE#gg4a0i1S=Z8%t?K2l$ck$SYd(oee5XD^vkAQD2n|a( zdA2wnMAoOqwE_aCrt+{BF;eqR+Y13fqCYpKzXSldKUHCd1ndNgv~zrE?L~yD>zelv z@%Hl=8k~;kOD+4ZP#aX9tfnlI1 z;5!Xw&^HRgcMahzkmx6AguO7eBrry{(_SLRG#ZqaCl~izOb+#pBfCa;(9_TNy}i;5 zzTzl*@w>m!nC)#98?kUEV`&NQ&}I*Ech>U20oa*OLX` z)Y#>`d_|HPx3mqxM*?!3s%Z9i@cVA^NW}y~&eJ)wh9oxap<$8*7P(H##}@qbAZ9^? 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messages) diff --git a/lecture_10/lecture_10.ipynb b/lecture_10/lecture_10.ipynb new file mode 100644 index 0000000..78e529d --- /dev/null +++ b/lecture_10/lecture_10.ipynb @@ -0,0 +1,1462 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "%plot --format svg" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "setdefaults" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Gauss Elimination\n", + "### Solving sets of equations with matrix operations\n", + "\n", + "The number of dimensions of a matrix indicate the degrees of freedom of the system you are solving. \n", + "\n", + "If you have a set of known output, $y_{1},~y_{2},~...y_{N}$ and a set of equations that\n", + "relate unknown inputs, $x_{1},~x_{2},~...x_{N}$, then these can be written in a vector\n", + "matrix format as:\n", + "\n", + "$y=Ax$\n", + "\n", + "Consider a problem with 2 DOF:\n", + "\n", + "$x_{1}+3x_{2}=1$\n", + "\n", + "$2x_{1}+x_{2}=1$\n", + "\n", + "$\\left[ \\begin{array}{cc}\n", + "1 & 3 \\\\\n", + "2 & 1 \\end{array} \\right]\n", + "\\left[\\begin{array}{c} \n", + "x_{1} \\\\ \n", + "x_{2} \\end{array}\\right]=\n", + "\\left[\\begin{array}{c} \n", + "1 \\\\\n", + "1\\end{array}\\right]$\n", + "\n", + "The solution for $x_{1}$ and $x_{2}$ is the intersection of two lines:" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + 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+ "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x21=[-2:2];\n", + "x11=1-3*x21;\n", + "x21=[-2:2];\n", + "x22=1-2*x21;\n", + "plot(x11,x21,x21,x22)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "For a $3\\times3$ matrix, the solution is the intersection of the 3 planes.\n", + "\n", + "$10x_{1}+2x_{2}+x_{3}=1$\n", + "\n", + "$2x_{1}+x_{2}+x_{3}=1$\n", + "\n", + "$x_{1}+2x_{2}+10x_{3}=1$\n", + "\n", + "$\\left[ \\begin{array}{cc}\n", + "10 & 2 & 1\\\\\n", + "2 & 1 & 1 \\\\\n", + "1 & 2 & 10\\end{array} \\right]\n", + "\\left[\\begin{array}{c} \n", + "x_{1} \\\\ \n", + "x_{2} \\\\\n", + "x_{3} \\end{array}\\right]=\n", + "\\left[\\begin{array}{c} \n", + "1 \\\\\n", + "1 \\\\\n", + "1\\end{array}\\right]$" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "error: 'X22' undefined near line 1 column 16\n", + "error: 'X13' undefined near line 1 column 14\n", + "error: evaluating argument list element number 3\n" + ] + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-1.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-1.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-4\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t8\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x11=linspace(-2,2,5);\n", + "x12=linspace(-2,2,5);\n", + "[X11,X12]=meshgrid(x11,x12);\n", + "X13=1-10*X11-2*X22;\n", + "\n", + "x21=linspace(-2,2,5);\n", + "x22=linspace(-2,2,5);\n", + "[X21,X22]=meshgrid(x21,x22);\n", + "X23=1-2*X11-X22;\n", + "\n", + "x31=linspace(-2,2,5);\n", + "x32=linspace(-2,2,5);\n", + "[X31,X32]=meshgrid(x31,x32);\n", + "X33=1/10*(1-X31-2*X32);\n", + "\n", + "mesh(X11,X12,X13);\n", + "hold on;\n", + "mesh(X21,X22,X23)\n", + "mesh(X31,X32,X33)\n", + "x=[10,2, 1;2,1, 1; 1, 2, 10]\\[1;1;1];\n", + "plot3(x(1),x(2),x(3),'o')\n", + "view(45,45)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "After 3 DOF problems, the solutions are described as *hyperplane* intersections. Which are even harder to visualize" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Gauss elimination\n", + "### Solving sets of equations systematically\n", + "\n", + "$\\left[ \\begin{array}{ccc|c}\n", + " & A & & y \\\\\n", + "10 & 2 & 1 & 1\\\\\n", + "2 & 1 & 1 & 1 \\\\\n", + "1 & 2 & 10 & 1\\end{array} \n", + "\\right] $\n", + "\n", + "Ay(2,:)-Ay(1,:)/5 = ([2 1 1 1]-1/5[10 2 1 1])\n", + "\n", + "$\\left[ \\begin{array}{ccc|c}\n", + " & A & & y \\\\\n", + "10 & 2 & 1 & 1\\\\\n", + "0 & 3/5 & 4/5 & 4/5 \\\\\n", + "1 & 2 & 10 & 1\\end{array} \n", + "\\right] $\n", + "\n", + "Ay(3,:)-Ay(1,:)/10 = ([1 2 10 1]-1/10[10 2 1 1])\n", + "\n", + "$\\left[ \\begin{array}{ccc|c}\n", + " & A & & y \\\\\n", + "10 & 2 & 1 & 1\\\\\n", + "0 & 3/5 & 4/5 & 4/5 \\\\\n", + "0 & 1.8 & 9.9 & 0.9\\end{array} \n", + "\\right] $\n", + "\n", + "Ay(3,:)-1.8\\*5/3\\*Ay(2,:) = ([0 1.8 9.9 0.9]-3\\*[0 3/5 4/5 4/5])\n", + "\n", + "$\\left[ \\begin{array}{ccc|c}\n", + " & A & & y \\\\\n", + "10 & 2 & 1 & 1\\\\\n", + "0 & 3/5 & 4/5 & 4/5 \\\\\n", + "0 & 0 & 7.5 & -1.5\\end{array} \n", + "\\right] $\n", + "\n", + "now, $7.5x_{3}=-1.5$ so $x_{3}=-\\frac{1}{5}$\n", + "\n", + "then, $3/5x_{2}+4/5(-1/5)=1$ so $x_{2}=\\frac{8}{5}$\n", + "\n", + "finally, $10x_{1}+2(8/5)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "Consider the problem again from the intro to Linear Algebra, 4 masses are connected in series to 4 springs with K=10 N/m. What are the final positions of the masses? \n", + "\n", + "![Springs-masses](../lecture_09/mass_springs.svg)\n", + "\n", + "The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass:\n", + "\n", + "$m_{1}g+k(x_{2}-x_{1})-kx_{1}=0$\n", + "\n", + "$m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0$\n", + "\n", + "$m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0$\n", + "\n", + "$m_{4}g-k(x_{4}-x_{3})=0$\n", + "\n", + "in matrix form:\n", + "\n", + "$\\left[ \\begin{array}{cccc}\n", + "2k & -k & 0 & 0 \\\\\n", + "-k & 2k & -k & 0 \\\\\n", + "0 & -k & 2k & -k \\\\\n", + "0 & 0 & -k & k \\end{array} \\right]\n", + "\\left[ \\begin{array}{c}\n", + "x_{1} \\\\\n", + "x_{2} \\\\\n", + "x_{3} \\\\\n", + "x_{4} \\end{array} \\right]=\n", + "\\left[ \\begin{array}{c}\n", + "m_{1}g \\\\\n", + "m_{2}g \\\\\n", + "m_{3}g \\\\\n", + "m_{4}g \\end{array} \\right]$" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "K =\n", + "\n", + " 20 -10 0 0\n", + " -10 20 -10 0\n", + " 0 -10 20 -10\n", + " 0 0 -10 10\n", + "\n", + "y =\n", + "\n", + " 9.8100\n", + " 19.6200\n", + " 29.4300\n", + " 39.2400\n", + "\n" + ] + } + ], + "source": [ + "k=10; % N/m\n", + "m1=1; % kg\n", + "m2=2;\n", + "m3=3;\n", + "m4=4;\n", + "g=9.81; % m/s^2\n", + "K=[2*k -k 0 0; -k 2*k -k 0; 0 -k 2*k -k; 0 0 -k k]\n", + "y=[m1*g;m2*g;m3*g;m4*g]" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "K1 =\n", + "\n", + " 20.00000 -10.00000 0.00000 0.00000 9.81000\n", + " 0.00000 15.00000 -10.00000 0.00000 24.52500\n", + " 0.00000 -10.00000 20.00000 -10.00000 29.43000\n", + " 0.00000 0.00000 -10.00000 10.00000 39.24000\n", + "\n" + ] + } + ], + "source": [ + "K1=[K y];\n", + "K1(2,:)=K1(1,:)/2+K1(2,:)" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "K2 =\n", + "\n", + " 20.00000 -10.00000 0.00000 0.00000 9.81000\n", + " 0.00000 15.00000 -10.00000 0.00000 24.52500\n", + " 0.00000 0.00000 13.33333 -10.00000 45.78000\n", + " 0.00000 0.00000 -10.00000 10.00000 39.24000\n", + "\n" + ] + } + ], + "source": [ + "K2=K1;\n", + "K2(3,:)=K1(2,:)*2/3+K1(3,:)\n" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "K2 =\n", + "\n", + " 20.00000 -10.00000 0.00000 0.00000 9.81000\n", + " 0.00000 15.00000 -10.00000 0.00000 24.52500\n", + " 0.00000 0.00000 13.33333 -10.00000 45.78000\n", + " 0.00000 0.00000 0.00000 2.50000 73.57500\n", + "\n" + ] + } + ], + "source": [ + "K2(4,:)=-K2(3,:)*K2(4,3)/K2(3,3)+K2(4,:)" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "x4 = 29.430\n", + "x3 = 25.506\n", + "x2 = 18.639\n", + "x1 = 9.8100\n" + ] + } + ], + "source": [ + "yp=K2(:,5);\n", + "x4=yp(4)/K2(4,4)\n", + "x3=(yp(3)+10*x4)/K2(3,3)\n", + "x2=(yp(2)+10*x3)/K2(2,2)\n", + "x1=(yp(1)+10*x2)/K2(1,1)" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 9.8100\n", + " 18.6390\n", + " 25.5060\n", + " 29.4300\n", + "\n" + ] + } + ], + "source": [ + "K\\y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Automate Gauss Elimination\n", + "\n", + "We can automate Gauss elimination with a function whose input is A and y:\n", + "\n", + "`x=GaussNaive(A,y)`" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "x =\n", + "\n", + " 9.8100\n", + " 18.6390\n", + " 25.5060\n", + " 29.4300\n", + "\n" + ] + } + ], + "source": [ + "x=GaussNaive(K,y)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Problem (Diagonal element is zero)\n", + "\n", + "If a diagonal element is 0 or very small either:\n", + "\n", + "1. no solution found\n", + "2. errors are introduced \n", + "\n", + "Therefore, we would want to pivot before applying Gauss elimination\n", + "\n", + "Consider:\n", + "\n", + "(a) $\\left[ \\begin{array}{cccc}\n", + "0 & 2 & 3 \\\\\n", + "4 & 6 & 7 \\\\\n", + "2 & -3 & 6 \\end{array} \\right]\n", + "\\left[ \\begin{array}{c}\n", + "x_{1} \\\\\n", + "x_{2} \\\\\n", + "x_{3} \\end{array} \\right]=\n", + "\\left[ \\begin{array}{c}\n", + "8 \\\\\n", + "-3 \\\\\n", + "5\\end{array} \\right]$\n", + "\n", + "(b) $\\left[ \\begin{array}{cccc}\n", + "0.0003 & 3.0000 \\\\\n", + "1.0000 & 1.0000 \\end{array} \\right]\n", + "\\left[ \\begin{array}{c}\n", + "x_{1} \\\\\n", + "x_{2} \\end{array} \\right]=\n", + "\\left[ \\begin{array}{c}\n", + "2.0001 \\\\\n", + "1.0000 \\end{array} \\right]$" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "warning: division by zero\n", + "warning: called from\n", + " GaussNaive at line 16 column 12\n", + "warning: division by zero\n", + "warning: division by zero\n", + "ans =\n", + "\n", + " NaN\n", + " NaN\n", + " NaN\n", + "\n", + "ans =\n", + "\n", + " -5.423913\n", + " 0.021739\n", + " 2.652174\n", + "\n" + ] + } + ], + "source": [ + "format short\n", + "Aa=[0,2,3;4,6,7;2,-3,6]; ya=[8;-3;5];\n", + "GaussNaive(Aa,ya)\n", + "Aa\\ya" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "x =\n", + "\n", + " -5.423913\n", + " 0.021739\n", + " 2.652174\n", + "\n", + "Aug =\n", + "\n", + " 4.00000 6.00000 7.00000 -3.00000\n", + " 0.00000 -6.00000 2.50000 6.50000\n", + " 0.00000 0.00000 3.83333 10.16667\n", + "\n", + "npivots = 2\n" + ] + } + ], + "source": [ + "[x,Aug,npivots]=GaussPivot(Aa,ya)" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 0.325665420556713\n", + " 0.666666666666667\n", + "\n", + "ans =\n", + "\n", + " 0.333333333333333\n", + " 0.666666666666667\n", + "\n" + ] + } + ], + "source": [ + "format long\n", + "Ab=[0.3E-13,3.0000;1.0000,1.0000];yb=[2+0.1e-13;1.0000];\n", + "GaussNaive(Ab,yb)\n", + "Ab\\yb" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "x =\n", + "\n", + " 0.333333333333333\n", + " 0.666666666666667\n", + "\n", + "Aug =\n", + "\n", + " 1.000000000000000 1.000000000000000 1.000000000000000\n", + " 0.000000000000000 2.999999999999970 1.999999999999980\n", + "\n", + "npivots = 1\n", + "ans =\n", + "\n", + " 0.333333333333333\n", + " 0.666666666666667\n", + "\n" + ] + } + ], + "source": [ + "[x,Aug,npivots]=GaussPivot(Ab,yb)\n", + "Ab\\yb\n", + "format short" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Spring-Mass System again\n", + "Now, 4 masses are connected in series to 4 springs with $K_{1}$=10 N/m, $K_{2}$=5 N/m, \n", + "$K_{3}$=2 N/m \n", + "and $K_{4}$=1 N/m. What are the final positions of the masses? \n", + "\n", + "![Springs-masses](../lecture_09/mass_springs.svg)\n", + "\n", + "The masses have the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass:\n", + "\n", + "$m_{1}g+k_{2}(x_{2}-x_{1})-k_{1}x_{1}=0$\n", + "\n", + "$m_{2}g+k_{3}(x_{3}-x_{2})-k_{2}(x_{2}-x_{1})=0$\n", + "\n", + "$m_{3}g+k_{4}(x_{4}-x_{3})-k_{3}(x_{3}-x_{2})=0$\n", + "\n", + "$m_{4}g-k_{4}(x_{4}-x_{3})=0$\n", + "\n", + "in matrix form:\n", + "\n", + "$\\left[ \\begin{array}{cccc}\n", + "k_1+k_2 & -k_2 & 0 & 0 \\\\\n", + "-k_2 & k_2+k_3 & -k_3 & 0 \\\\\n", + "0 & -k_3 & k_3+k_4 & -k_4 \\\\\n", + "0 & 0 & -k_4 & k_4 \\end{array} \\right]\n", + "\\left[ \\begin{array}{c}\n", + "x_{1} \\\\\n", + "x_{2} \\\\\n", + "x_{3} \\\\\n", + "x_{4} \\end{array} \\right]=\n", + "\\left[ \\begin{array}{c}\n", + "m_{1}g \\\\\n", + "m_{2}g \\\\\n", + "m_{3}g \\\\\n", + "m_{4}g \\end{array} \\right]$" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "K =\n", + "\n", + " 15 -5 0 0\n", + " -5 7 -2 0\n", + " 0 -2 3 -1\n", + " 0 0 -1 1\n", + "\n", + "y =\n", + "\n", + " 9.8100\n", + " 19.6200\n", + " 29.4300\n", + " 39.2400\n", + "\n" + ] + } + ], + "source": [ + "k1=10; k2=5;k3=2;k4=1; % N/m\n", + "m1=1; % kg\n", + "m2=2;\n", + "m3=3;\n", + "m4=4;\n", + "g=9.81; % m/s^2\n", + "K=[k1+k2 -k2 0 0; -k2, k2+k3, -k3 0; 0 -k3, k3+k4, -k4; 0 0 -k4 k4]\n", + "y=[m1*g;m2*g;m3*g;m4*g]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Tridiagonal matrix\n", + "\n", + "This matrix, K, could be rewritten as 3 vectors e, f and g\n", + "\n", + "$e=\\left[ \\begin{array}{c}\n", + "0 \\\\\n", + "-5 \\\\\n", + "-2 \\\\\n", + "-1 \\end{array} \\right]$\n", + "\n", + "$f=\\left[ \\begin{array}{c}\n", + "15 \\\\\n", + "7 \\\\\n", + "3 \\\\\n", + "1 \\end{array} \\right]$\n", + "\n", + "$g=\\left[ \\begin{array}{c}\n", + "-5 \\\\\n", + "-2 \\\\\n", + "-1 \\\\\n", + "0 \\end{array} \\right]$\n", + "\n", + "Where all other components are 0 and the length of the vectors are n and the first component of e and the last component of g are zero\n", + "\n", + "`e(1)=0` \n", + "\n", + "`g(end)=0`\n", + "\n", + "No need to pivot and number of calculations reduced enormously.\n", + "\n", + "|method |Number of Floating point operations for n$\\times$n-matrix|\n", + "|----------------|---------|\n", + "| Naive Gauss | n-cubed |\n", + "| Tridiagonal | n |" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 9.8100 27.4680 61.8030 101.0430\n", + "\n" + ] + } + ], + "source": [ + "e=[0;-5;-2;-1];\n", + "g=[-5;-2;-1;0];\n", + "f=[15;7;3;1];\n", + "Tridiag(e,f,g,y)\n" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "% tic ... t=toc \n", + "% is Matlab timer used for debugging programs\n", + "t_GE = zeros(1,100);\n", + "t_GE_tridiag = zeros(1,100);\n", + "t_TD = zeros(1,100);\n", + "%for n = 1:200\n", + "for n=1:100\n", + " A = rand(n,n);\n", + " e = rand(n,1); e(1)=0;\n", + " f = rand(n,1);\n", + " g = rand(n,1); g(end)=0;\n", + " Atd=diag(f, 0) - diag(e(2:n), -1) - diag(g(1:n-1), 1);\n", + " b = rand(n,1);\n", + " tic;\n", + " x = GaussPivot(A,b);\n", + " t_GE(n) = toc;\n", + " tic;\n", + " x = GaussPivot(Atd,b);\n", + " t_GE_tridiag(n) = toc;\n", + " tic;\n", + " x = Tridiag(e,f,g,b);\n", + " t_TD(n) = toc;\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", 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Which are even harder to visualize + +## Gauss elimination +### Solving sets of equations systematically + +$\left[ \begin{array}{ccc|c} + & A & & y \\ +10 & 2 & 1 & 1\\ +2 & 1 & 1 & 1 \\ +1 & 2 & 10 & 1\end{array} +\right] $ + +Ay(2,:)-Ay(1,:)/5 = ([2 1 1 1]-1/5[10 2 1 1]) + +$\left[ \begin{array}{ccc|c} + & A & & y \\ +10 & 2 & 1 & 1\\ +0 & 3/5 & 4/5 & 4/5 \\ +1 & 2 & 10 & 1\end{array} +\right] $ + +Ay(3,:)-Ay(1,:)/10 = ([1 2 10 1]-1/10[10 2 1 1]) + +$\left[ \begin{array}{ccc|c} + & A & & y \\ +10 & 2 & 1 & 1\\ +0 & 3/5 & 4/5 & 4/5 \\ +0 & 1.8 & 9.9 & 0.9\end{array} +\right] $ + +Ay(3,:)-1.8\*5/3\*Ay(2,:) = ([0 1.8 9.9 0.9]-3\*[0 3/5 4/5 4/5]) + +$\left[ \begin{array}{ccc|c} + & A & & y \\ +10 & 2 & 1 & 1\\ +0 & 3/5 & 4/5 & 4/5 \\ +0 & 0 & 7.5 & -1.5\end{array} +\right] $ + +now, $7.5x_{3}=-1.5$ so $x_{3}=-\frac{1}{5}$ + +then, $3/5x_{2}+4/5(-1/5)=1$ so $x_{2}=\frac{8}{5}$ + +finally, $10x_{1}+2(8/5) + +Consider the problem again from the intro to Linear Algebra, 4 masses are connected in series to 4 springs with K=10 N/m. What are the final positions of the masses? + +![Springs-masses](../lecture_09/mass_springs.svg) + +The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass: + +$m_{1}g+k(x_{2}-x_{1})-kx_{1}=0$ + +$m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0$ + +$m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0$ + +$m_{4}g-k(x_{4}-x_{3})=0$ + +in matrix form: + +$\left[ \begin{array}{cccc} +2k & -k & 0 & 0 \\ +-k & 2k & -k & 0 \\ +0 & -k & 2k & -k \\ +0 & 0 & -k & k \end{array} \right] +\left[ \begin{array}{c} +x_{1} \\ +x_{2} \\ +x_{3} \\ +x_{4} \end{array} \right]= +\left[ \begin{array}{c} +m_{1}g \\ +m_{2}g \\ +m_{3}g \\ +m_{4}g \end{array} \right]$ + + +```octave +k=10; % N/m +m1=1; % kg +m2=2; +m3=3; +m4=4; +g=9.81; % m/s^2 +K=[2*k -k 0 0; -k 2*k -k 0; 0 -k 2*k -k; 0 0 -k k] +y=[m1*g;m2*g;m3*g;m4*g] +``` + + K = + + 20 -10 0 0 + -10 20 -10 0 + 0 -10 20 -10 + 0 0 -10 10 + + y = + + 9.8100 + 19.6200 + 29.4300 + 39.2400 + + + + +```octave +K1=[K y]; +K1(2,:)=K1(1,:)/2+K1(2,:) +``` + + K1 = + + 20.00000 -10.00000 0.00000 0.00000 9.81000 + 0.00000 15.00000 -10.00000 0.00000 24.52500 + 0.00000 -10.00000 20.00000 -10.00000 29.43000 + 0.00000 0.00000 -10.00000 10.00000 39.24000 + + + + +```octave +K2=K1; +K2(3,:)=K1(2,:)*2/3+K1(3,:) + +``` + + K2 = + + 20.00000 -10.00000 0.00000 0.00000 9.81000 + 0.00000 15.00000 -10.00000 0.00000 24.52500 + 0.00000 0.00000 13.33333 -10.00000 45.78000 + 0.00000 0.00000 -10.00000 10.00000 39.24000 + + + + +```octave +K2(4,:)=-K2(3,:)*K2(4,3)/K2(3,3)+K2(4,:) +``` + + K2 = + + 20.00000 -10.00000 0.00000 0.00000 9.81000 + 0.00000 15.00000 -10.00000 0.00000 24.52500 + 0.00000 0.00000 13.33333 -10.00000 45.78000 + 0.00000 0.00000 0.00000 2.50000 73.57500 + + + + +```octave +yp=K2(:,5); +x4=yp(4)/K2(4,4) +x3=(yp(3)+10*x4)/K2(3,3) +x2=(yp(2)+10*x3)/K2(2,2) +x1=(yp(1)+10*x2)/K2(1,1) +``` + + x4 = 29.430 + x3 = 25.506 + x2 = 18.639 + x1 = 9.8100 + + + +```octave +K\y +``` + + ans = + + 9.8100 + 18.6390 + 25.5060 + 29.4300 + + + +## Automate Gauss Elimination + +We can automate Gauss elimination with a function whose input is A and y: + +`x=GaussNaive(A,y)` + + +```octave +x=GaussNaive(K,y) +``` + + x = + + 9.8100 + 18.6390 + 25.5060 + 29.4300 + + + +## Problem (Diagonal element is zero) + +If a diagonal element is 0 or very small either: + +1. no solution found +2. errors are introduced + +Therefore, we would want to pivot before applying Gauss elimination + +Consider: + +(a) $\left[ \begin{array}{cccc} +0 & 2 & 3 \\ +4 & 6 & 7 \\ +2 & -3 & 6 \end{array} \right] +\left[ \begin{array}{c} +x_{1} \\ +x_{2} \\ +x_{3} \end{array} \right]= +\left[ \begin{array}{c} +8 \\ +-3 \\ +5\end{array} \right]$ + +(b) $\left[ \begin{array}{cccc} +0.0003 & 3.0000 \\ +1.0000 & 1.0000 \end{array} \right] +\left[ \begin{array}{c} +x_{1} \\ +x_{2} \end{array} \right]= +\left[ \begin{array}{c} +2.0001 \\ +1.0000 \end{array} \right]$ + + +```octave +format short +Aa=[0,2,3;4,6,7;2,-3,6]; ya=[8;-3;5]; +GaussNaive(Aa,ya) +Aa\ya +``` + + warning: division by zero + warning: called from + GaussNaive at line 16 column 12 + warning: division by zero + warning: division by zero + ans = + + NaN + NaN + NaN + + ans = + + -5.423913 + 0.021739 + 2.652174 + + + + +```octave +[x,Aug,npivots]=GaussPivot(Aa,ya) +``` + + x = + + -5.423913 + 0.021739 + 2.652174 + + Aug = + + 4.00000 6.00000 7.00000 -3.00000 + 0.00000 -6.00000 2.50000 6.50000 + 0.00000 0.00000 3.83333 10.16667 + + npivots = 2 + + + +```octave +format long +Ab=[0.3E-13,3.0000;1.0000,1.0000];yb=[2+0.1e-13;1.0000]; +GaussNaive(Ab,yb) +Ab\yb +``` + + ans = + + 0.325665420556713 + 0.666666666666667 + + ans = + + 0.333333333333333 + 0.666666666666667 + + + + +```octave +[x,Aug,npivots]=GaussPivot(Ab,yb) +Ab\yb +format short +``` + + x = + + 0.333333333333333 + 0.666666666666667 + + Aug = + + 1.000000000000000 1.000000000000000 1.000000000000000 + 0.000000000000000 2.999999999999970 1.999999999999980 + + npivots = 1 + ans = + + 0.333333333333333 + 0.666666666666667 + + + +### Spring-Mass System again +Now, 4 masses are connected in series to 4 springs with $K_{1}$=10 N/m, $K_{2}$=5 N/m, +$K_{3}$=2 N/m +and $K_{4}$=1 N/m. What are the final positions of the masses? + +![Springs-masses](../lecture_09/mass_springs.svg) + +The masses have the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass: + +$m_{1}g+k_{2}(x_{2}-x_{1})-k_{1}x_{1}=0$ + +$m_{2}g+k_{3}(x_{3}-x_{2})-k_{2}(x_{2}-x_{1})=0$ + +$m_{3}g+k_{4}(x_{4}-x_{3})-k_{3}(x_{3}-x_{2})=0$ + +$m_{4}g-k_{4}(x_{4}-x_{3})=0$ + +in matrix form: + +$\left[ \begin{array}{cccc} +k_1+k_2 & -k_2 & 0 & 0 \\ +-k_2 & k_2+k_3 & -k_3 & 0 \\ +0 & -k_3 & k_3+k_4 & -k_4 \\ +0 & 0 & -k_4 & k_4 \end{array} \right] +\left[ \begin{array}{c} +x_{1} \\ +x_{2} \\ +x_{3} \\ +x_{4} \end{array} \right]= +\left[ \begin{array}{c} +m_{1}g \\ +m_{2}g \\ +m_{3}g \\ +m_{4}g \end{array} \right]$ + + +```octave +k1=10; k2=5;k3=2;k4=1; % N/m +m1=1; % kg +m2=2; +m3=3; +m4=4; +g=9.81; % m/s^2 +K=[k1+k2 -k2 0 0; -k2, k2+k3, -k3 0; 0 -k3, k3+k4, -k4; 0 0 -k4 k4] +y=[m1*g;m2*g;m3*g;m4*g] +``` + + K = + + 15 -5 0 0 + -5 7 -2 0 + 0 -2 3 -1 + 0 0 -1 1 + + y = + + 9.8100 + 19.6200 + 29.4300 + 39.2400 + + + +## Tridiagonal matrix + +This matrix, K, could be rewritten as 3 vectors e, f and g + +$e=\left[ \begin{array}{c} +0 \\ +-5 \\ +-2 \\ +-1 \end{array} \right]$ + +$f=\left[ \begin{array}{c} +15 \\ +7 \\ +3 \\ +1 \end{array} \right]$ + +$g=\left[ \begin{array}{c} +-5 \\ +-2 \\ +-1 \\ +0 \end{array} \right]$ + +Where all other components are 0 and the length of the vectors are n and the first component of e and the last component of g are zero + +`e(1)=0` + +`g(end)=0` + +No need to pivot and number of calculations reduced enormously. + +|method |Number of Floating point operations for n$\times$n-matrix| +|----------------|---------| +| Naive Gauss | n-cubed | +| Tridiagonal | n | + + +```octave +e=[0;-5;-2;-1]; +g=[-5;-2;-1;0]; +f=[15;7;3;1]; +Tridiag(e,f,g,y) + +``` + + ans = + + 9.8100 27.4680 61.8030 101.0430 + + + + +```octave +% tic ... t=toc +% is Matlab timer used for debugging programs +t_GE = zeros(1,100); +t_GE_tridiag = zeros(1,100); +t_TD = zeros(1,100); +for n = 1:200 + A = rand(n,n); + e = rand(n,1); e(1)=0; + f = rand(n,1); + g = rand(n,1); g(end)=0; + Atd=diag(f, 0) - diag(e(2:n), -1) - diag(g(1:n-1), 1); + b = rand(n,1); + tic; + x = GaussPivot(A,b); + t_GE(n) = toc; + tic; + x = GaussPivot(Atd,b); + t_GE_tridiag(n) = toc; + tic; + x = Tridiag(e,f,g,b); + t_TD(n) = toc; +end +``` + + +```octave +n=1:200; +loglog(n,t_GE,n,t_TD,n,t_GE_tridiag) +xlabel('number of elements') +ylabel('time (s)') +``` + + +![svg](lecture_10_files/lecture_10_27_0.svg) + + + +```octave +plot(t_TD) +``` + + +![svg](lecture_10_files/lecture_10_28_0.svg) + + + +```octave + +``` diff --git a/lecture_10/lecture_10.out b/lecture_10/lecture_10.out new file mode 100644 index 0000000..cb5b517 --- /dev/null +++ b/lecture_10/lecture_10.out @@ -0,0 +1,8 @@ +\BOOKMARK [1][-]{section.1}{Gauss Elimination}{}% 1 +\BOOKMARK [2][-]{subsubsection.1.0.1}{Solving sets of equations with matrix operations}{section.1}% 2 +\BOOKMARK [2][-]{subsection.1.1}{Gauss elimination}{section.1}% 3 +\BOOKMARK [3][-]{subsubsection.1.1.1}{Solving sets of equations systematically}{subsection.1.1}% 4 +\BOOKMARK [2][-]{subsection.1.2}{Automate Gauss 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HOTqm=e8<+U literal 0 HcmV?d00001 diff --git a/lecture_10/lecture_10.tex b/lecture_10/lecture_10.tex new file mode 100644 index 0000000..1131881 --- /dev/null +++ b/lecture_10/lecture_10.tex @@ -0,0 +1,852 @@ + +% Default to the notebook output style + + + + +% Inherit from the specified cell style. + + + + + +\documentclass[11pt]{article} + + + + \usepackage[T1]{fontenc} + % Nicer default font (+ math font) than Computer Modern for most use cases + \usepackage{mathpazo} + + % Basic figure setup, for now with no caption control since it's done + % automatically by Pandoc (which extracts ![](path) syntax from Markdown). + \usepackage{graphicx} + % We will generate all images so they have a width \maxwidth. This means + % that they will get their normal width if they fit onto the page, but + % are scaled down if they would overflow the margins. + \makeatletter + \def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth + \else\Gin@nat@width\fi} + \makeatother + \let\Oldincludegraphics\includegraphics + % Set max figure width to be 80% of text width, for now hardcoded. + \renewcommand{\includegraphics}[1]{\Oldincludegraphics[width=.8\maxwidth]{#1}} + % Ensure that by default, figures have no caption (until we provide a + % proper Figure object with a Caption API and a way to capture that + % in the conversion process - todo). + \usepackage{caption} + \DeclareCaptionLabelFormat{nolabel}{} + \captionsetup{labelformat=nolabel} + + \usepackage{adjustbox} % Used to constrain images to a maximum size + \usepackage{xcolor} % Allow colors to be defined + \usepackage{enumerate} % Needed for markdown enumerations to work + \usepackage{geometry} % Used to adjust the document margins + \usepackage{amsmath} % Equations + \usepackage{amssymb} % Equations + \usepackage{textcomp} % defines textquotesingle + % Hack from http://tex.stackexchange.com/a/47451/13684: + \AtBeginDocument{% + \def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code + } + \usepackage{upquote} % Upright quotes for verbatim code + \usepackage{eurosym} % defines \euro + \usepackage[mathletters]{ucs} % Extended unicode (utf-8) support + \usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document + \usepackage{fancyvrb} % verbatim replacement that allows latex + \usepackage{grffile} % extends the file name processing of package graphics + % to support a larger range + % The hyperref package gives us a pdf with properly built + % internal navigation ('pdf bookmarks' for the table of contents, + % internal cross-reference links, web links for URLs, etc.) + \usepackage{hyperref} + \usepackage{longtable} % longtable support required by pandoc >1.10 + \usepackage{booktabs} % table support for pandoc > 1.12.2 + \usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment) + \usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout) + % normalem makes italics be italics, not underlines + + + + + % Colors for the hyperref package + \definecolor{urlcolor}{rgb}{0,.145,.698} + \definecolor{linkcolor}{rgb}{.71,0.21,0.01} + \definecolor{citecolor}{rgb}{.12,.54,.11} + + % ANSI colors + \definecolor{ansi-black}{HTML}{3E424D} + \definecolor{ansi-black-intense}{HTML}{282C36} + \definecolor{ansi-red}{HTML}{E75C58} + \definecolor{ansi-red-intense}{HTML}{B22B31} + \definecolor{ansi-green}{HTML}{00A250} + \definecolor{ansi-green-intense}{HTML}{007427} + \definecolor{ansi-yellow}{HTML}{DDB62B} + \definecolor{ansi-yellow-intense}{HTML}{B27D12} + \definecolor{ansi-blue}{HTML}{208FFB} + \definecolor{ansi-blue-intense}{HTML}{0065CA} + \definecolor{ansi-magenta}{HTML}{D160C4} + \definecolor{ansi-magenta-intense}{HTML}{A03196} + \definecolor{ansi-cyan}{HTML}{60C6C8} + \definecolor{ansi-cyan-intense}{HTML}{258F8F} + \definecolor{ansi-white}{HTML}{C5C1B4} + \definecolor{ansi-white-intense}{HTML}{A1A6B2} + + % commands and environments needed by pandoc snippets + % extracted from the output of `pandoc -s` + \providecommand{\tightlist}{% + \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} + \DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}} + % Add ',fontsize=\small' for more characters per line + \newenvironment{Shaded}{}{} + \newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}} + \newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}} + \newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}} + \newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}} + \newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}} + \newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}} + \newcommand{\RegionMarkerTok}[1]{{#1}} + \newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}} + \newcommand{\NormalTok}[1]{{#1}} + + % Additional commands for more recent versions of Pandoc + \newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}} + \newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}} + \newcommand{\ImportTok}[1]{{#1}} + \newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}} + \newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}} + \newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}} + \newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}} + \newcommand{\BuiltInTok}[1]{{#1}} + \newcommand{\ExtensionTok}[1]{{#1}} + \newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}} + \newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}} + \newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + + + % Define a nice break command that doesn't care if a line doesn't already + % exist. + \def\br{\hspace*{\fill} \\* } + % Math Jax compatability definitions + \def\gt{>} + \def\lt{<} + % Document 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PY@tok@k\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@se\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.13}{##1}}} +\expandafter\def\csname PY@tok@sd\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} + +\def\PYZbs{\char`\\} +\def\PYZus{\char`\_} +\def\PYZob{\char`\{} +\def\PYZcb{\char`\}} +\def\PYZca{\char`\^} +\def\PYZam{\char`\&} +\def\PYZlt{\char`\<} +\def\PYZgt{\char`\>} +\def\PYZsh{\char`\#} +\def\PYZpc{\char`\%} +\def\PYZdl{\char`\$} +\def\PYZhy{\char`\-} +\def\PYZsq{\char`\'} +\def\PYZdq{\char`\"} +\def\PYZti{\char`\~} +% for compatibility with earlier versions +\def\PYZat{@} +\def\PYZlb{[} +\def\PYZrb{]} +\makeatother + + + % Exact colors from NB + \definecolor{incolor}{rgb}{0.0, 0.0, 0.5} + \definecolor{outcolor}{rgb}{0.545, 0.0, 0.0} + + + + + % Prevent overflowing lines due to hard-to-break entities + \sloppy + % Setup hyperref package + \hypersetup{ + breaklinks=true, % so long urls are correctly broken across lines + colorlinks=true, + urlcolor=urlcolor, + linkcolor=linkcolor, + citecolor=citecolor, + } + % Slightly bigger margins than the latex defaults + + \geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in} + + + + \begin{document} + + + \maketitle + + + + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}14}]:} \PY{c}{\PYZpc{}plot \PYZhy{}\PYZhy{}format svg} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}15}]:} \PY{n}{setdefaults} +\end{Verbatim} + + \section{Gauss Elimination}\label{gauss-elimination} + +\subsubsection{Solving sets of equations with matrix +operations}\label{solving-sets-of-equations-with-matrix-operations} + +The number of dimensions of a matrix indicate the degrees of freedom of +the system you are solving. + +If you have a set of known output, \(y_{1},~y_{2},~...y_{N}\) and a set +of equations that relate unknown inputs, \(x_{1},~x_{2},~...x_{N}\), +then these can be written in a vector matrix format as: + +\(y=Ax\) + +Consider a problem with 2 DOF: + +\(x_{1}+3x_{2}=1\) + +\(2x_{1}+x_{2}=1\) + +\(\left[ \begin{array}{cc} 1 & 3 \\ 2 & 1 \end{array} \right] \left[\begin{array}{c} x_{1} \\ x_{2} \end{array}\right]= \left[\begin{array}{c} 1 \\ 1\end{array}\right]\) + +The solution for \(x_{1}\) and \(x_{2}\) is the intersection of two +lines: + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}16}]:} \PY{n}{x21}\PY{p}{=}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{:}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{;} + \PY{n}{x11}\PY{p}{=}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{o}{*}\PY{n}{x21}\PY{p}{;} + \PY{n}{x21}\PY{p}{=}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{:}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{;} + \PY{n}{x22}\PY{p}{=}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{x21}\PY{p}{;} + \PY{n+nb}{plot}\PY{p}{(}\PY{n}{x11}\PY{p}{,}\PY{n}{x21}\PY{p}{,}\PY{n}{x21}\PY{p}{,}\PY{n}{x22}\PY{p}{)} +\end{Verbatim} + + \begin{center} + \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{lecture_10_files/lecture_10_3_0.pdf} + \end{center} + { \hspace*{\fill} \\} + + For a $3\times3$ matrix, the solution is the intersection of the 3 +planes. + +\(10x_{1}+2x_{2}+x_{3}=1\) + +\(2x_{1}+x_{2}+x_{3}=1\) + +\(x_{1}+2x_{2}+10x_{3}=1\) + +$\left[ \begin{array}{ccc} 10 & 2 & 1 \\ 2 & 1 & 1 \\ 1 & 2 & 10\end{array} \right] + \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]= + \left[\begin{array}{c} 1 \\ 1 \\ 1\end{array}\right]$ + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}17}]:} \PY{n}{x11}\PY{p}{=}\PY{n+nb}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{)}\PY{p}{;} + 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\emph{hyperplane} +intersections. Which are even harder to visualize + + \subsection{Gauss elimination}\label{gauss-elimination} + +\subsubsection{Solving sets of equations +systematically}\label{solving-sets-of-equations-systematically} + +$\left[ +\begin{array}{ccc|c} + & A & & y \\ +10 & 2 & 1 & 1\\ +2 & 1 & 1 & 1 \\ +1 & 2 & 10 & 1\end{array} +\right] $ + +Ay(2,:)-Ay(1,:)/5 = ({[}2 1 1 1{]}-1/5{[}10 2 1 1{]}) + +$\left[ +\begin{array}{ccc|c} + & A & & y \\ +10 & 2 & 1 & 1\\ +0 & 3/5 & 4/5 & 4/5 \\ +1 & 2 & 10 & 1\end{array} +\right] $ + +Ay(3,:)-Ay(1,:)/10 = ({[}1 2 10 1{]}-1/10{[}10 2 1 1{]}) + +$\left[ +\begin{array}{ccc|c} + & A & & y \\ +10 & 2 & 1 & 1\\ +0 & 3/5 & 4/5 & 4/5 \\ +0 & 1.8 & 9.9 & 0.9\end{array} +\right] $ + +Ay(3,:)-1.8*5/3*Ay(2,:) = ({[}0 1.8 9.9 0.9{]}-3*{[}0 3/5 4/5 4/5{]}) + +$\left[ +\begin{array}{ccc|c} + & A & & y \\ +10 & 2 & 1 & 1\\ +0 & 3/5 & 4/5 & 4/5 \\ +0 & 0 & 7.5 & -1.5\end{array} +\right] $ + +now, \(7.5x_{3}=-1.5\) so \(x_{3}=-\frac{1}{5}\) + +then, \(3/5x_{2}+4/5(-1/5)=1\) so \(x_{2}=\frac{8}{5}\) + +finally, \$10x\_\{1\}+2(8/5) + + Consider the problem again from the intro to Linear Algebra, 4 masses +are connected in series to 4 springs with K=10 N/m. What are the final +positions of the masses? + +\begin{figure}[htbp] +\centering +\includegraphics{../lecture_09/mass_springs.svg} +\caption{Springs-masses} +\end{figure} + +The masses haves the following amounts, 1, 2, 3, and 4 kg for masses +1-4. Using a FBD for each mass: + +\(m_{1}g+k(x_{2}-x_{1})-kx_{1}=0\) + +\(m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0\) + +\(m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0\) + +\(m_{4}g-k(x_{4}-x_{3})=0\) + +in matrix form: + +\(\left[ \begin{array}{cccc} 2k & -k & 0 & 0 \\ -k & 2k & -k & 0 \\ 0 & -k & 2k & -k \\ 0 & 0 & -k & k \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array} \right]= \left[ \begin{array}{c} m_{1}g \\ m_{2}g \\ m_{3}g \\ m_{4}g \end{array} \right]\) + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}18}]:} \PY{n}{k}\PY{p}{=}\PY{l+m+mi}{10}\PY{p}{;} \PY{c}{\PYZpc{} N/m} + \PY{n}{m1}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{;} \PY{c}{\PYZpc{} kg} + \PY{n}{m2}\PY{p}{=}\PY{l+m+mi}{2}\PY{p}{;} + \PY{n}{m3}\PY{p}{=}\PY{l+m+mi}{3}\PY{p}{;} + \PY{n}{m4}\PY{p}{=}\PY{l+m+mi}{4}\PY{p}{;} + \PY{n}{g}\PY{p}{=}\PY{l+m+mf}{9.81}\PY{p}{;} \PY{c}{\PYZpc{} m/s\PYZca{}2} + \PY{n}{K}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{k} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{0} \PY{l+m+mi}{0}\PY{p}{;} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{2}\PY{o}{*}\PY{n}{k} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{0}\PY{p}{;} \PY{l+m+mi}{0} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{2}\PY{o}{*}\PY{n}{k} \PY{o}{\PYZhy{}}\PY{n}{k}\PY{p}{;} \PY{l+m+mi}{0} \PY{l+m+mi}{0} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{n}{k}\PY{p}{]} + \PY{n}{y}\PY{p}{=}\PY{p}{[}\PY{n}{m1}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m2}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m3}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m4}\PY{o}{*}\PY{n}{g}\PY{p}{]} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +K = + + 20 -10 0 0 + -10 20 -10 0 + 0 -10 20 -10 + 0 0 -10 10 + +y = + + 9.8100 + 19.6200 + 29.4300 + 39.2400 + + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}19}]:} \PY{n}{K1}\PY{p}{=}\PY{p}{[}\PY{n}{K} \PY{n}{y}\PY{p}{]}\PY{p}{;} + \PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{p}{=}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{+}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{:}\PY{p}{)} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +K1 = + + 20.00000 -10.00000 0.00000 0.00000 9.81000 + 0.00000 15.00000 -10.00000 0.00000 24.52500 + 0.00000 -10.00000 20.00000 -10.00000 29.43000 + 0.00000 0.00000 -10.00000 10.00000 39.24000 + + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}20}]:} \PY{n}{K2}\PY{p}{=}\PY{n}{K1}\PY{p}{;} + \PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{p}{=}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{o}{*}\PY{l+m+mi}{2}\PY{o}{/}\PY{l+m+mi}{3}\PY{o}{+}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{:}\PY{p}{)} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +K2 = + + 20.00000 -10.00000 0.00000 0.00000 9.81000 + 0.00000 15.00000 -10.00000 0.00000 24.52500 + 0.00000 0.00000 13.33333 -10.00000 45.78000 + 0.00000 0.00000 -10.00000 10.00000 39.24000 + + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}21}]:} \PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{p}{=}\PY{o}{\PYZhy{}}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{o}{*}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{+}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{:}\PY{p}{)} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +K2 = + + 20.00000 -10.00000 0.00000 0.00000 9.81000 + 0.00000 15.00000 -10.00000 0.00000 24.52500 + 0.00000 0.00000 13.33333 -10.00000 45.78000 + 0.00000 0.00000 0.00000 2.50000 73.57500 + + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}22}]:} \PY{n}{yp}\PY{p}{=}\PY{n}{K2}\PY{p}{(}\PY{p}{:}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{)}\PY{p}{;} + \PY{n}{x4}\PY{p}{=}\PY{n}{yp}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)} + \PY{n}{x3}\PY{p}{=}\PY{p}{(}\PY{n}{yp}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{10}\PY{o}{*}\PY{n}{x4}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)} + \PY{n}{x2}\PY{p}{=}\PY{p}{(}\PY{n}{yp}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{10}\PY{o}{*}\PY{n}{x3}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)} + \PY{n}{x1}\PY{p}{=}\PY{p}{(}\PY{n}{yp}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{10}\PY{o}{*}\PY{n}{x2}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +x4 = 29.430 +x3 = 25.506 +x2 = 18.639 +x1 = 9.8100 + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}23}]:} \PY{n}{K}\PY{o}{\PYZbs{}}\PY{n}{y} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +ans = + + 9.8100 + 18.6390 + 25.5060 + 29.4300 + + + \end{Verbatim} + + \subsection{Automate Gauss +Elimination}\label{automate-gauss-elimination} + +We can automate Gauss elimination with a function whose input is A and +y: + +\texttt{x=GaussNaive(A,y)} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}24}]:} \PY{n}{x}\PY{p}{=}\PY{n}{GaussNaive}\PY{p}{(}\PY{n}{K}\PY{p}{,}\PY{n}{y}\PY{p}{)} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +x = + + 9.8100 + 18.6390 + 25.5060 + 29.4300 + + + \end{Verbatim} + + \subsection{Problem (Diagonal element is +zero)}\label{problem-diagonal-element-is-zero} + +If a diagonal element is 0 or very small either: + +\begin{enumerate} +\def\labelenumi{\arabic{enumi}.} +\tightlist +\item + no solution found +\item + errors are introduced +\end{enumerate} + +Therefore, we would want to pivot before applying Gauss elimination + +Consider: + +\begin{enumerate} +\def\labelenumi{(\alph{enumi})} +\item + \(\left[ \begin{array}{cccc} 0 & 2 & 3 \\ 4 & 6 & 7 \\ 2 & -3 & 6 \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array} \right]= \left[ \begin{array}{c} 8 \\ -3 \\ 5\end{array} \right]\) +\item + \(\left[ \begin{array}{cccc} 0.0003 & 3.0000 \\ 1.0000 & 1.0000 \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \end{array} \right]= \left[ \begin{array}{c} 2.0001 \\ 1.0000 \end{array} \right]\) +\end{enumerate} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}25}]:} \PY{n}{format} \PY{n}{short} + \PY{n}{Aa}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{;}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{,}\PY{l+m+mi}{7}\PY{p}{;}\PY{l+m+mi}{2}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{]}\PY{p}{;} \PY{n}{ya}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{8}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{p}{;}\PY{l+m+mi}{5}\PY{p}{]}\PY{p}{;} + \PY{n}{GaussNaive}\PY{p}{(}\PY{n}{Aa}\PY{p}{,}\PY{n}{ya}\PY{p}{)} + \PY{n}{Aa}\PY{o}{\PYZbs{}}\PY{n}{ya} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +warning: division by zero +warning: called from + GaussNaive at line 16 column 12 +warning: division by zero +warning: division by zero +ans = + + NaN + NaN + NaN + +ans = + + -5.423913 + 0.021739 + 2.652174 + + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}26}]:} \PY{p}{[}\PY{n}{x}\PY{p}{,}\PY{n}{Aug}\PY{p}{,}\PY{n}{npivots}\PY{p}{]}\PY{p}{=}\PY{n}{GaussPivot}\PY{p}{(}\PY{n}{Aa}\PY{p}{,}\PY{n}{ya}\PY{p}{)} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +x = + + -5.423913 + 0.021739 + 2.652174 + +Aug = + + 4.00000 6.00000 7.00000 -3.00000 + 0.00000 -6.00000 2.50000 6.50000 + 0.00000 0.00000 3.83333 10.16667 + +npivots = 2 + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}27}]:} \PY{n}{format} \PY{n}{long} + \PY{n}{Ab}\PY{p}{=}\PY{p}{[}\PY{l+m+mf}{0.3E\PYZhy{}13}\PY{p}{,}\PY{l+m+mf}{3.0000}\PY{p}{;}\PY{l+m+mf}{1.0000}\PY{p}{,}\PY{l+m+mf}{1.0000}\PY{p}{]}\PY{p}{;}\PY{n}{yb}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{2}\PY{o}{+}\PY{l+m+mf}{0.1e\PYZhy{}13}\PY{p}{;}\PY{l+m+mf}{1.0000}\PY{p}{]}\PY{p}{;} + \PY{n}{GaussNaive}\PY{p}{(}\PY{n}{Ab}\PY{p}{,}\PY{n}{yb}\PY{p}{)} + \PY{n}{Ab}\PY{o}{\PYZbs{}}\PY{n}{yb} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +ans = + + 0.325665420556713 + 0.666666666666667 + +ans = + + 0.333333333333333 + 0.666666666666667 + + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}28}]:} \PY{p}{[}\PY{n}{x}\PY{p}{,}\PY{n}{Aug}\PY{p}{,}\PY{n}{npivots}\PY{p}{]}\PY{p}{=}\PY{n}{GaussPivot}\PY{p}{(}\PY{n}{Ab}\PY{p}{,}\PY{n}{yb}\PY{p}{)} + \PY{n}{Ab}\PY{o}{\PYZbs{}}\PY{n}{yb} + \PY{n}{format} \PY{n}{short} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +x = + + 0.333333333333333 + 0.666666666666667 + +Aug = + + 1.000000000000000 1.000000000000000 1.000000000000000 + 0.000000000000000 2.999999999999970 1.999999999999980 + +npivots = 1 +ans = + + 0.333333333333333 + 0.666666666666667 + + + \end{Verbatim} + + \subsubsection{Spring-Mass System again}\label{spring-mass-system-again} + +Now, 4 masses are connected in series to 4 springs with \(K_{1}\)=10 +N/m, \(K_{2}\)=5 N/m, \(K_{3}\)=2 N/m and \(K_{4}\)=1 N/m. What are the +final positions of the masses? + +\begin{figure}[htbp] +\centering +\includegraphics{../lecture_09/mass_springs.svg} +\caption{Springs-masses} +\end{figure} + +The masses have the following amounts, 1, 2, 3, and 4 kg for masses 1-4. +Using a FBD for each mass: + +\(m_{1}g+k_{2}(x_{2}-x_{1})-k_{1}x_{1}=0\) + +\(m_{2}g+k_{3}(x_{3}-x_{2})-k_{2}(x_{2}-x_{1})=0\) + +\(m_{3}g+k_{4}(x_{4}-x_{3})-k_{3}(x_{3}-x_{2})=0\) + +\(m_{4}g-k_{4}(x_{4}-x_{3})=0\) + +in matrix form: + +\(\left[ \begin{array}{cccc} k_1+k_2 & -k_2 & 0 & 0 \\ -k_2 & k_2+k_3 & -k_3 & 0 \\ 0 & -k_3 & k_3+k_4 & -k_4 \\ 0 & 0 & -k_4 & k_4 \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array} \right]= \left[ \begin{array}{c} m_{1}g \\ m_{2}g \\ m_{3}g \\ m_{4}g \end{array} \right]\) + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}29}]:} \PY{n}{k1}\PY{p}{=}\PY{l+m+mi}{10}\PY{p}{;} \PY{n}{k2}\PY{p}{=}\PY{l+m+mi}{5}\PY{p}{;}\PY{n}{k3}\PY{p}{=}\PY{l+m+mi}{2}\PY{p}{;}\PY{n}{k4}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{;} \PY{c}{\PYZpc{} N/m} + \PY{n}{m1}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{;} \PY{c}{\PYZpc{} kg} + \PY{n}{m2}\PY{p}{=}\PY{l+m+mi}{2}\PY{p}{;} + \PY{n}{m3}\PY{p}{=}\PY{l+m+mi}{3}\PY{p}{;} + \PY{n}{m4}\PY{p}{=}\PY{l+m+mi}{4}\PY{p}{;} + \PY{n}{g}\PY{p}{=}\PY{l+m+mf}{9.81}\PY{p}{;} \PY{c}{\PYZpc{} m/s\PYZca{}2} + \PY{n}{K}\PY{p}{=}\PY{p}{[}\PY{n}{k1}\PY{o}{+}\PY{n}{k2} \PY{o}{\PYZhy{}}\PY{n}{k2} \PY{l+m+mi}{0} \PY{l+m+mi}{0}\PY{p}{;} \PY{o}{\PYZhy{}}\PY{n}{k2}\PY{p}{,} \PY{n}{k2}\PY{o}{+}\PY{n}{k3}\PY{p}{,} \PY{o}{\PYZhy{}}\PY{n}{k3} \PY{l+m+mi}{0}\PY{p}{;} \PY{l+m+mi}{0} \PY{o}{\PYZhy{}}\PY{n}{k3}\PY{p}{,} \PY{n}{k3}\PY{o}{+}\PY{n}{k4}\PY{p}{,} \PY{o}{\PYZhy{}}\PY{n}{k4}\PY{p}{;} \PY{l+m+mi}{0} \PY{l+m+mi}{0} \PY{o}{\PYZhy{}}\PY{n}{k4} \PY{n}{k4}\PY{p}{]} + \PY{n}{y}\PY{p}{=}\PY{p}{[}\PY{n}{m1}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m2}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m3}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m4}\PY{o}{*}\PY{n}{g}\PY{p}{]} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +K = + + 15 -5 0 0 + -5 7 -2 0 + 0 -2 3 -1 + 0 0 -1 1 + +y = + + 9.8100 + 19.6200 + 29.4300 + 39.2400 + + + \end{Verbatim} + + \subsection{Tridiagonal matrix}\label{tridiagonal-matrix} + +This matrix, K, could be rewritten as 3 vectors e, f and g + +\(e=\left[ \begin{array}{c} 0 \\ -5 \\ -2 \\ -1 \end{array} \right]\) + +\(f=\left[ \begin{array}{c} 15 \\ 7 \\ 3 \\ 1 \end{array} \right]\) + +\(g=\left[ \begin{array}{c} -5 \\ -2 \\ -1 \\ 0 \end{array} \right]\) + +Where all other components are 0 and the length of the vectors are n and +the first component of e and the last component of g are zero + +\texttt{e(1)=0} + +\texttt{g(end)=0} + +No need to pivot and number of calculations reduced enormously. + +\begin{longtable}[c]{@{}ll@{}} +\toprule +method & Number of Floating point operations for +n\(\times\)n-matrix\tabularnewline +\midrule +\endhead +Naive Gauss & n-cubed\tabularnewline +Tridiagonal & n\tabularnewline +\bottomrule +\end{longtable} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}30}]:} \PY{n+nb}{e}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{5}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} + \PY{n}{g}\PY{p}{=}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{5}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{;}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{;} + \PY{n}{f}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{15}\PY{p}{;}\PY{l+m+mi}{7}\PY{p}{;}\PY{l+m+mi}{3}\PY{p}{;}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} + \PY{n}{Tridiag}\PY{p}{(}\PY{n+nb}{e}\PY{p}{,}\PY{n}{f}\PY{p}{,}\PY{n}{g}\PY{p}{,}\PY{n}{y}\PY{p}{)} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +ans = + + 9.8100 27.4680 61.8030 101.0430 + + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}12}]:} \PY{c}{\PYZpc{} tic ... t=toc } + \PY{c}{\PYZpc{} is Matlab timer used for debugging programs} + \PY{n}{t\PYZus{}GE} \PY{p}{=} \PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{100}\PY{p}{)}\PY{p}{;} + \PY{n}{t\PYZus{}GE\PYZus{}tridiag} \PY{p}{=} \PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{100}\PY{p}{)}\PY{p}{;} + \PY{n}{t\PYZus{}TD} \PY{p}{=} \PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{100}\PY{p}{)}\PY{p}{;} + \PY{c}{\PYZpc{}for n = 1:200} + \PY{k}{for} \PY{n}{n}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{100} + \PY{n}{A} \PY{p}{=} \PY{n+nb}{rand}\PY{p}{(}\PY{n}{n}\PY{p}{,}\PY{n}{n}\PY{p}{)}\PY{p}{;} + \PY{n+nb}{e} \PY{p}{=} \PY{n+nb}{rand}\PY{p}{(}\PY{n}{n}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} \PY{n+nb}{e}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{=}\PY{l+m+mi}{0}\PY{p}{;} + \PY{n}{f} \PY{p}{=} 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Cooper" Date: Tue, 21 Feb 2017 11:47:09 -0500 Subject: [PATCH 03/11] update lecture 10 --- lecture_10/lecture_10.aux | 14 +- lecture_10/lecture_10.ipynb | 497 +++++++++++++----- lecture_10/lecture_10.log | 83 ++- lecture_10/lecture_10.pdf | Bin 162693 -> 190161 bytes lecture_10/lecture_10.tex | 199 ++++--- .../lecture_10_files/lecture_10_29_0.pdf | Bin 0 -> 10972 bytes .../lecture_10_files/lecture_10_29_0.svg | 148 ++++++ .../lecture_10_files/lecture_10_6_0.pdf | Bin 0 -> 30054 bytes .../lecture_10_files/lecture_10_6_0.svg | 397 ++++++++++++++ lecture_10/octave-workspace | Bin 86061 -> 134485 bytes 10 files changed, 1093 insertions(+), 245 deletions(-) create mode 100644 lecture_10/lecture_10_files/lecture_10_29_0.pdf create mode 100644 lecture_10/lecture_10_files/lecture_10_29_0.svg create mode 100644 lecture_10/lecture_10_files/lecture_10_6_0.pdf create mode 100644 lecture_10/lecture_10_files/lecture_10_6_0.svg diff --git a/lecture_10/lecture_10.aux b/lecture_10/lecture_10.aux index 14a6f0d..d8d4c62 100644 --- a/lecture_10/lecture_10.aux +++ b/lecture_10/lecture_10.aux @@ -28,13 +28,13 @@ \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Springs-masses\relax }}{4}{figure.caption.1}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Automate Gauss Elimination}{6}{subsection.1.2}} \newlabel{automate-gauss-elimination}{{1.2}{6}{Automate Gauss Elimination}{subsection.1.2}{}} -\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Problem (Diagonal element is zero)}{6}{subsection.1.3}} -\newlabel{problem-diagonal-element-is-zero}{{1.3}{6}{Problem (Diagonal element is zero)}{subsection.1.3}{}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.3.1}Spring-Mass System again}{8}{subsubsection.1.3.1}} -\newlabel{spring-mass-system-again}{{1.3.1}{8}{Spring-Mass System again}{subsubsection.1.3.1}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Problem (Diagonal element is zero)}{7}{subsection.1.3}} +\newlabel{problem-diagonal-element-is-zero}{{1.3}{7}{Problem (Diagonal element is zero)}{subsection.1.3}{}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.3.1}Spring-Mass System again}{9}{subsubsection.1.3.1}} +\newlabel{spring-mass-system-again}{{1.3.1}{9}{Spring-Mass System again}{subsubsection.1.3.1}{}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Springs-masses\relax }}{9}{figure.caption.2}} -\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}Tridiagonal matrix}{9}{subsection.1.4}} -\newlabel{tridiagonal-matrix}{{1.4}{9}{Tridiagonal matrix}{subsection.1.4}{}} \gdef \LT@i {\LT@entry - {2}{67.64778pt}\LT@entry + {2}{61.69104pt}\LT@entry {1}{261.39424pt}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}Tridiagonal matrix}{10}{subsection.1.4}} +\newlabel{tridiagonal-matrix}{{1.4}{10}{Tridiagonal matrix}{subsection.1.4}{}} diff --git a/lecture_10/lecture_10.ipynb b/lecture_10/lecture_10.ipynb index 78e529d..46d4f00 100644 --- a/lecture_10/lecture_10.ipynb +++ b/lecture_10/lecture_10.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 14, + "execution_count": 1, "metadata": { "collapsed": true }, @@ -13,7 +13,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 2, "metadata": { "collapsed": true }, @@ -58,7 +58,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -223,6 +223,30 @@ "plot(x11,x21,x21,x22)" ] }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 0.40000\n", + " 0.20000\n", + "\n" + ] + } + ], + "source": [ + "A=[1,3;2,1]; y=[1;1];\n", + "A\\y % matlab's Ax=y solution for x" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -251,20 +275,11 @@ }, { "cell_type": "code", - 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"\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n", + "\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n", "\t\n", "\tgnuplot_plot_3a\n", - "\n", + "\n", + "\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", "\n", "\n", - "\t\n", + "\t\n", "\n", "\t\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\n", + "\t\t\n", + "\t\tx1\n", + "\t\n", + "\n", "\n", - "\t\n", + "\t\n", + "\t\tx2\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tx3\n", + "\t\n", + "\n", "\n", - "\t\n", + "\n", "\n", "" ], @@ -600,18 +690,19 @@ } ], "source": [ - "x11=linspace(-2,2,5);\n", - "x12=linspace(-2,2,5);\n", + "N=25;\n", + "x11=linspace(-2,2,N);\n", + "x12=linspace(-2,2,N);\n", "[X11,X12]=meshgrid(x11,x12);\n", - "X13=1-10*X11-2*X22;\n", + "X13=1-10*X11-2*X12;\n", "\n", - "x21=linspace(-2,2,5);\n", - "x22=linspace(-2,2,5);\n", + "x21=linspace(-2,2,N);\n", + "x22=linspace(-2,2,N);\n", "[X21,X22]=meshgrid(x21,x22);\n", "X23=1-2*X11-X22;\n", "\n", - "x31=linspace(-2,2,5);\n", - "x32=linspace(-2,2,5);\n", + "x31=linspace(-2,2,N);\n", + "x32=linspace(-2,2,N);\n", "[X31,X32]=meshgrid(x31,x32);\n", "X33=1/10*(1-X31-2*X32);\n", "\n", @@ -621,7 +712,10 @@ "mesh(X31,X32,X33)\n", "x=[10,2, 1;2,1, 1; 1, 2, 10]\\[1;1;1];\n", "plot3(x(1),x(2),x(3),'o')\n", - "view(45,45)" + "xlabel('x1')\n", + "ylabel('x2')\n", + "zlabel('x3')\n", + "view(10,45)" ] }, { @@ -676,7 +770,7 @@ "\n", "then, $3/5x_{2}+4/5(-1/5)=1$ so $x_{2}=\\frac{8}{5}$\n", "\n", - "finally, $10x_{1}+2(8/5)" + "finally, $10x_{1}+2(8/5)+1(-\\frac{1}{5})=1$" ] }, { @@ -720,7 +814,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 10, "metadata": { "collapsed": false }, @@ -759,7 +853,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -785,7 +879,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -811,7 +905,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -836,7 +930,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -862,7 +956,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -898,7 +992,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -962,7 +1056,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -1000,7 +1094,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 32, "metadata": { "collapsed": false }, @@ -1031,7 +1125,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 33, "metadata": { "collapsed": false }, @@ -1062,7 +1156,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 34, "metadata": { "collapsed": false }, @@ -1096,6 +1190,28 @@ "format short" ] }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = -3.0000\n", + "ans = 3.0000\n" + ] + } + ], + "source": [ + "% determinant is (-1)^(number_of_pivots)*diagonal_elements\n", + "det(Ab)\n", + "Aug(1,1)*Aug(2,2)" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -1138,7 +1254,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 24, "metadata": { "collapsed": false }, @@ -1211,13 +1327,13 @@ "\n", "|method |Number of Floating point operations for n$\\times$n-matrix|\n", "|----------------|---------|\n", - "| Naive Gauss | n-cubed |\n", + "| Gauss | n-cubed |\n", "| Tridiagonal | n |" ] }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -1229,6 +1345,13 @@ "ans =\n", "\n", " 9.8100 27.4680 61.8030 101.0430\n", + "\n", + "ans =\n", + "\n", + " 9.8100\n", + " 27.4680\n", + " 61.8030\n", + " 101.0430\n", "\n" ] } @@ -1237,12 +1360,13 @@ "e=[0;-5;-2;-1];\n", "g=[-5;-2;-1;0];\n", "f=[15;7;3;1];\n", - "Tridiag(e,f,g,y)\n" + "Tridiag(e,f,g,y)\n", + "K\\y\n" ] }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 26, "metadata": { "collapsed": true }, @@ -1265,7 +1389,7 @@ " x = GaussPivot(A,b);\n", " t_GE(n) = toc;\n", " tic;\n", - " x = GaussPivot(Atd,b);\n", + " x = A\\b;\n", " t_GE_tridiag(n) = toc;\n", " tic;\n", " x = Tridiag(e,f,g,b);\n", @@ -1275,7 +1399,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 28, "metadata": { "collapsed": false }, @@ -1329,26 +1453,31 @@ "\n", "\n", "\t\t\n", + "\t\t10-5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", "\t\t10-4\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t10-3\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t10-2\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t10-1\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t100\n", "\t\n", "\n", @@ -1358,21 +1487,16 @@ "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t101\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t102\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t103\n", - "\t\n", - "\n", - "\n", "\n", "\n", "\t\n", @@ -1388,19 +1512,38 @@ "\n", "\n", "\n", - "\tgnuplot_plot_1a\n", - "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tGauss elim\n", + "\n", "\n", "\n", - "\t\n", + "\t\n", + "\t\tGauss elim\n", "\t\n", - "\tgnuplot_plot_2a\n", + "\n", "\n", - "\t\n", + "\t\n", "\t\n", - "\tgnuplot_plot_3a\n", + "\tMatlab \\\n", + "\n", + "\t\n", + "\t\tMatlab \n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tTriDiag\n", "\n", - "\t\n", + "\t\n", + "\t\tTriDiag\n", + "\t\n", + "\n", + "\n", + "\t\n", "\t\n", "\n", "\n", @@ -1422,12 +1565,92 @@ } ], "source": [ - "n=1:200;\n", + "n=1:100;\n", "loglog(n,t_GE,n,t_TD,n,t_GE_tridiag)\n", + "legend('Gauss elim','Matlab \\','TriDiag')\n", "xlabel('number of elements')\n", "ylabel('time (s)')" ] }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "x =\n", + "\n", + " 9.8100\n", + " 27.4680\n", + " 61.8030\n", + " 101.0430\n", + "\n", + "Aug =\n", + "\n", + " 15.00000 -5.00000 0.00000 0.00000 9.81000\n", + " 0.00000 5.33333 -2.00000 0.00000 22.89000\n", + " 0.00000 0.00000 2.25000 -1.00000 38.01375\n", + " 0.00000 0.00000 0.00000 0.55556 56.13500\n", + "\n", + "npivots = 0\n" + ] + } + ], + "source": [ + "[x,Aug,npivots]=GaussPivot(K,y)" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A =\n", + "\n", + " 15.00000 -5.00000 0.00000 0.00000\n", + " 0.00000 5.33333 -2.00000 0.00000\n", + " 0.00000 0.00000 2.25000 -1.00000\n", + " 0.00000 0.00000 0.00000 0.55556\n", + "\n" + ] + } + ], + "source": [ + "A=Aug(1:4,1:4)" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 100.00\n", + "detA = 100.00\n" + ] + } + ], + "source": [ + "det(A)\n", + "detA=A(1,1)*A(2,2)*A(3,3)*A(4,4)" + ] + }, { "cell_type": "code", "execution_count": null, diff --git a/lecture_10/lecture_10.log b/lecture_10/lecture_10.log index 088d911..9e3113d 100644 --- a/lecture_10/lecture_10.log +++ b/lecture_10/lecture_10.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex 2017.1.11) 20 FEB 2017 17:21 +This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex 2017.1.11) 21 FEB 2017 11:46 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -772,28 +772,27 @@ Underfull \hbox (badness 10000) in paragraph at lines 323--324 [] -[2 <./lecture_10_files/lecture_10_3_0.pdf>] -LaTeX Font Info: Try loading font information for TS1+cmtt on input line 359 +LaTeX Font Info: Try loading font information for TS1+cmtt on input line 327 . - (/usr/share/texlive/texmf-dist/tex/latex/base/ts1cmtt.fd File: ts1cmtt.fd 2014/09/29 v2.5h Standard LaTeX font definitions -) - -File: lecture_10_files/lecture_10_5_1.pdf Graphic file (type pdf) +) [2 <./lecture_10_files/lecture_10_3_0.pdf>] + +File: lecture_10_files/lecture_10_6_0.pdf Graphic file (type pdf) - -Package pdftex.def Info: lecture_10_files/lecture_10_5_1.pdf used on input line - 371. + +Package pdftex.def Info: lecture_10_files/lecture_10_6_0.pdf used on input line + 382. (pdftex.def) Requested size: 449.6789pt x 337.25917pt. -Underfull \hbox (badness 10000) in paragraph at lines 373--374 +Underfull \hbox (badness 10000) in paragraph at lines 384--385 [] +[3 <./lecture_10_files/lecture_10_6_0.pdf>] LaTeX Font Info: Font shape `T1/ppl/bx/n' in size <12> not available -(Font) Font shape `T1/ppl/b/n' tried instead on input line 378. -[3 <./lecture_10_files/lecture_10_5_1.pdf>] +(Font) Font shape `T1/ppl/b/n' tried instead on input line 389. + ! LaTeX Error: Unknown graphics extension: .svg. @@ -801,10 +800,10 @@ See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... -l.433 ...egraphics{../lecture_09/mass_springs.svg} +l.444 ...egraphics{../lecture_09/mass_springs.svg} ? -[4] [5] [6] [7] +[4] [5] [6] [7] [8] ! LaTeX Error: Unknown graphics extension: .svg. @@ -812,45 +811,45 @@ See the LaTeX manual or LaTeX Companion for explanation. 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1131881..633c945 100644 --- a/lecture_10/lecture_10.tex +++ b/lecture_10/lecture_10.tex @@ -277,11 +277,11 @@ \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}14}]:} \PY{c}{\PYZpc{}plot \PYZhy{}\PYZhy{}format svg} +{\color{incolor}In [{\color{incolor}1}]:} \PY{c}{\PYZpc{}plot \PYZhy{}\PYZhy{}format svg} \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}15}]:} \PY{n}{setdefaults} +{\color{incolor}In [{\color{incolor}2}]:} \PY{n}{setdefaults} \end{Verbatim} \section{Gauss Elimination}\label{gauss-elimination} @@ -310,11 +310,11 @@ \subsubsection{Solving sets of equations with matrix lines: \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}16}]:} \PY{n}{x21}\PY{p}{=}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{:}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{;} - \PY{n}{x11}\PY{p}{=}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{o}{*}\PY{n}{x21}\PY{p}{;} - 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$3\times3$ matrix, the solution is the intersection of the 3 + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}4}]:} \PY{n}{A}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{;}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} \PY{n}{y}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{;}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} + \PY{n}{A}\PY{o}{\PYZbs{}}\PY{n}{y} \PY{c}{\PYZpc{} matlab\PYZsq{}s Ax=y solution for x} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +ans = + + 0.40000 + 0.20000 + + + \end{Verbatim} + + For a \(3\times3\) matrix, the solution is the intersection of the 3 planes. \(10x_{1}+2x_{2}+x_{3}=1\) @@ -331,44 +345,41 @@ \subsubsection{Solving sets of equations with matrix \(x_{1}+2x_{2}+10x_{3}=1\) -$\left[ \begin{array}{ccc} 10 & 2 & 1 \\ 2 & 1 & 1 \\ 1 & 2 & 10\end{array} \right] +$\left[ \begin{array}{ccc} 10 & 2 & 1\\ 2 & 1 & 1 \\ 1 & 2 & 10\end{array} \right] \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]= 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\begin{Verbatim}[commandchars=\\\{\}] -error: 'X22' undefined near line 1 column 16 -error: 'X13' undefined near line 1 column 14 -error: evaluating argument list element number 3 - - \end{Verbatim} - \begin{center} - \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{lecture_10_files/lecture_10_5_1.pdf} + \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{lecture_10_files/lecture_10_6_0.pdf} \end{center} { \hspace*{\fill} \\} @@ -416,13 +427,13 @@ \subsubsection{Solving sets of equations 10 & 2 & 1 & 1\\ 0 & 3/5 & 4/5 & 4/5 \\ 0 & 0 & 7.5 & -1.5\end{array} -\right] $ +\right]$ now, \(7.5x_{3}=-1.5\) so \(x_{3}=-\frac{1}{5}\) then, \(3/5x_{2}+4/5(-1/5)=1\) so \(x_{2}=\frac{8}{5}\) -finally, \$10x\_\{1\}+2(8/5) +finally, \(10x_{1}+2(8/5)+1(-\frac{1}{5})=1\) Consider the problem again from the intro to Linear Algebra, 4 masses are connected in series to 4 springs with K=10 N/m. What are the final @@ -450,7 +461,7 @@ \subsubsection{Solving sets of equations \(\left[ \begin{array}{cccc} 2k & -k & 0 & 0 \\ -k & 2k & -k & 0 \\ 0 & -k & 2k & -k \\ 0 & 0 & -k & k \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array} \right]= \left[ \begin{array}{c} m_{1}g \\ m_{2}g \\ m_{3}g \\ m_{4}g \end{array} \right]\) \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}18}]:} \PY{n}{k}\PY{p}{=}\PY{l+m+mi}{10}\PY{p}{;} \PY{c}{\PYZpc{} N/m} +{\color{incolor}In [{\color{incolor}10}]:} \PY{n}{k}\PY{p}{=}\PY{l+m+mi}{10}\PY{p}{;} \PY{c}{\PYZpc{} N/m} \PY{n}{m1}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{;} \PY{c}{\PYZpc{} kg} \PY{n}{m2}\PY{p}{=}\PY{l+m+mi}{2}\PY{p}{;} \PY{n}{m3}\PY{p}{=}\PY{l+m+mi}{3}\PY{p}{;} @@ -479,7 +490,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}19}]:} \PY{n}{K1}\PY{p}{=}\PY{p}{[}\PY{n}{K} \PY{n}{y}\PY{p}{]}\PY{p}{;} +{\color{incolor}In [{\color{incolor}11}]:} \PY{n}{K1}\PY{p}{=}\PY{p}{[}\PY{n}{K} \PY{n}{y}\PY{p}{]}\PY{p}{;} \PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{p}{=}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{+}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{:}\PY{p}{)} \end{Verbatim} @@ -495,7 +506,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}20}]:} \PY{n}{K2}\PY{p}{=}\PY{n}{K1}\PY{p}{;} +{\color{incolor}In [{\color{incolor}12}]:} \PY{n}{K2}\PY{p}{=}\PY{n}{K1}\PY{p}{;} \PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{p}{=}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{o}{*}\PY{l+m+mi}{2}\PY{o}{/}\PY{l+m+mi}{3}\PY{o}{+}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{:}\PY{p}{)} \end{Verbatim} @@ -511,7 +522,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}21}]:} \PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{p}{=}\PY{o}{\PYZhy{}}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{o}{*}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{+}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{:}\PY{p}{)} +{\color{incolor}In [{\color{incolor}13}]:} \PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{p}{=}\PY{o}{\PYZhy{}}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{o}{*}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{+}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{:}\PY{p}{)} \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] @@ -526,7 +537,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}22}]:} \PY{n}{yp}\PY{p}{=}\PY{n}{K2}\PY{p}{(}\PY{p}{:}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{)}\PY{p}{;} +{\color{incolor}In [{\color{incolor}14}]:} \PY{n}{yp}\PY{p}{=}\PY{n}{K2}\PY{p}{(}\PY{p}{:}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{)}\PY{p}{;} \PY{n}{x4}\PY{p}{=}\PY{n}{yp}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)} \PY{n}{x3}\PY{p}{=}\PY{p}{(}\PY{n}{yp}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{10}\PY{o}{*}\PY{n}{x4}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)} \PY{n}{x2}\PY{p}{=}\PY{p}{(}\PY{n}{yp}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{10}\PY{o}{*}\PY{n}{x3}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)} @@ -542,7 +553,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}23}]:} \PY{n}{K}\PY{o}{\PYZbs{}}\PY{n}{y} +{\color{incolor}In [{\color{incolor}15}]:} \PY{n}{K}\PY{o}{\PYZbs{}}\PY{n}{y} \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] @@ -565,7 +576,7 @@ \subsubsection{Solving sets of equations \texttt{x=GaussNaive(A,y)} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}24}]:} \PY{n}{x}\PY{p}{=}\PY{n}{GaussNaive}\PY{p}{(}\PY{n}{K}\PY{p}{,}\PY{n}{y}\PY{p}{)} +{\color{incolor}In [{\color{incolor}16}]:} \PY{n}{x}\PY{p}{=}\PY{n}{GaussNaive}\PY{p}{(}\PY{n}{K}\PY{p}{,}\PY{n}{y}\PY{p}{)} \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] @@ -606,7 +617,7 @@ \subsubsection{Solving sets of equations \end{enumerate} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}25}]:} \PY{n}{format} \PY{n}{short} +{\color{incolor}In [{\color{incolor}17}]:} \PY{n}{format} \PY{n}{short} \PY{n}{Aa}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{;}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{,}\PY{l+m+mi}{7}\PY{p}{;}\PY{l+m+mi}{2}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{]}\PY{p}{;} \PY{n}{ya}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{8}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{p}{;}\PY{l+m+mi}{5}\PY{p}{]}\PY{p}{;} \PY{n}{GaussNaive}\PY{p}{(}\PY{n}{Aa}\PY{p}{,}\PY{n}{ya}\PY{p}{)} \PY{n}{Aa}\PY{o}{\PYZbs{}}\PY{n}{ya} @@ -634,7 +645,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}26}]:} \PY{p}{[}\PY{n}{x}\PY{p}{,}\PY{n}{Aug}\PY{p}{,}\PY{n}{npivots}\PY{p}{]}\PY{p}{=}\PY{n}{GaussPivot}\PY{p}{(}\PY{n}{Aa}\PY{p}{,}\PY{n}{ya}\PY{p}{)} +{\color{incolor}In [{\color{incolor}32}]:} \PY{p}{[}\PY{n}{x}\PY{p}{,}\PY{n}{Aug}\PY{p}{,}\PY{n}{npivots}\PY{p}{]}\PY{p}{=}\PY{n}{GaussPivot}\PY{p}{(}\PY{n}{Aa}\PY{p}{,}\PY{n}{ya}\PY{p}{)} \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] @@ -655,7 +666,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}27}]:} \PY{n}{format} \PY{n}{long} +{\color{incolor}In [{\color{incolor}33}]:} \PY{n}{format} \PY{n}{long} \PY{n}{Ab}\PY{p}{=}\PY{p}{[}\PY{l+m+mf}{0.3E\PYZhy{}13}\PY{p}{,}\PY{l+m+mf}{3.0000}\PY{p}{;}\PY{l+m+mf}{1.0000}\PY{p}{,}\PY{l+m+mf}{1.0000}\PY{p}{]}\PY{p}{;}\PY{n}{yb}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{2}\PY{o}{+}\PY{l+m+mf}{0.1e\PYZhy{}13}\PY{p}{;}\PY{l+m+mf}{1.0000}\PY{p}{]}\PY{p}{;} \PY{n}{GaussNaive}\PY{p}{(}\PY{n}{Ab}\PY{p}{,}\PY{n}{yb}\PY{p}{)} \PY{n}{Ab}\PY{o}{\PYZbs{}}\PY{n}{yb} @@ -676,7 +687,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}28}]:} \PY{p}{[}\PY{n}{x}\PY{p}{,}\PY{n}{Aug}\PY{p}{,}\PY{n}{npivots}\PY{p}{]}\PY{p}{=}\PY{n}{GaussPivot}\PY{p}{(}\PY{n}{Ab}\PY{p}{,}\PY{n}{yb}\PY{p}{)} +{\color{incolor}In [{\color{incolor}34}]:} \PY{p}{[}\PY{n}{x}\PY{p}{,}\PY{n}{Aug}\PY{p}{,}\PY{n}{npivots}\PY{p}{]}\PY{p}{=}\PY{n}{GaussPivot}\PY{p}{(}\PY{n}{Ab}\PY{p}{,}\PY{n}{yb}\PY{p}{)} \PY{n}{Ab}\PY{o}{\PYZbs{}}\PY{n}{yb} \PY{n}{format} \PY{n}{short} \end{Verbatim} @@ -699,6 +710,18 @@ \subsubsection{Solving sets of equations 0.666666666666667 + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}36}]:} \PY{c}{\PYZpc{} determinant is (\PYZhy{}1)\PYZca{}(number\PYZus{}of\PYZus{}pivots)*diagonal\PYZus{}elements} + \PY{n+nb}{det}\PY{p}{(}\PY{n}{Ab}\PY{p}{)} + \PY{n}{Aug}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{n}{Aug}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +ans = -3.0000 +ans = 3.0000 + \end{Verbatim} \subsubsection{Spring-Mass System again}\label{spring-mass-system-again} @@ -729,7 +752,7 @@ \subsubsection{Solving sets of equations \(\left[ \begin{array}{cccc} k_1+k_2 & -k_2 & 0 & 0 \\ -k_2 & k_2+k_3 & -k_3 & 0 \\ 0 & -k_3 & k_3+k_4 & -k_4 \\ 0 & 0 & -k_4 & k_4 \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array} \right]= \left[ \begin{array}{c} m_{1}g \\ m_{2}g \\ m_{3}g \\ m_{4}g \end{array} \right]\) \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}29}]:} \PY{n}{k1}\PY{p}{=}\PY{l+m+mi}{10}\PY{p}{;} \PY{n}{k2}\PY{p}{=}\PY{l+m+mi}{5}\PY{p}{;}\PY{n}{k3}\PY{p}{=}\PY{l+m+mi}{2}\PY{p}{;}\PY{n}{k4}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{;} \PY{c}{\PYZpc{} N/m} +{\color{incolor}In [{\color{incolor}24}]:} \PY{n}{k1}\PY{p}{=}\PY{l+m+mi}{10}\PY{p}{;} \PY{n}{k2}\PY{p}{=}\PY{l+m+mi}{5}\PY{p}{;}\PY{n}{k3}\PY{p}{=}\PY{l+m+mi}{2}\PY{p}{;}\PY{n}{k4}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{;} \PY{c}{\PYZpc{} N/m} \PY{n}{m1}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{;} \PY{c}{\PYZpc{} kg} \PY{n}{m2}\PY{p}{=}\PY{l+m+mi}{2}\PY{p}{;} \PY{n}{m3}\PY{p}{=}\PY{l+m+mi}{3}\PY{p}{;} @@ -782,16 +805,17 @@ \subsubsection{Solving sets of equations n\(\times\)n-matrix\tabularnewline \midrule \endhead -Naive Gauss & n-cubed\tabularnewline +Gauss & n-cubed\tabularnewline Tridiagonal & n\tabularnewline \bottomrule \end{longtable} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}30}]:} \PY{n+nb}{e}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{5}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} +{\color{incolor}In [{\color{incolor}25}]:} \PY{n+nb}{e}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{5}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} \PY{n}{g}\PY{p}{=}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{5}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{;}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{;} \PY{n}{f}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{15}\PY{p}{;}\PY{l+m+mi}{7}\PY{p}{;}\PY{l+m+mi}{3}\PY{p}{;}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} \PY{n}{Tridiag}\PY{p}{(}\PY{n+nb}{e}\PY{p}{,}\PY{n}{f}\PY{p}{,}\PY{n}{g}\PY{p}{,}\PY{n}{y}\PY{p}{)} + \PY{n}{K}\PY{o}{\PYZbs{}}\PY{n}{y} \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] @@ -799,11 +823,18 @@ \subsubsection{Solving sets of equations 9.8100 27.4680 61.8030 101.0430 +ans = + + 9.8100 + 27.4680 + 61.8030 + 101.0430 + \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}12}]:} \PY{c}{\PYZpc{} tic ... t=toc } +{\color{incolor}In [{\color{incolor}26}]:} \PY{c}{\PYZpc{} tic ... t=toc } \PY{c}{\PYZpc{} is Matlab timer used for debugging programs} \PY{n}{t\PYZus{}GE} \PY{p}{=} \PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{100}\PY{p}{)}\PY{p}{;} \PY{n}{t\PYZus{}GE\PYZus{}tridiag} \PY{p}{=} \PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{100}\PY{p}{)}\PY{p}{;} @@ -820,7 +851,7 @@ \subsubsection{Solving sets of equations \PY{n}{x} \PY{p}{=} \PY{n}{GaussPivot}\PY{p}{(}\PY{n}{A}\PY{p}{,}\PY{n}{b}\PY{p}{)}\PY{p}{;} \PY{n}{t\PYZus{}GE}\PY{p}{(}\PY{n}{n}\PY{p}{)} \PY{p}{=} \PY{n+nb}{toc}\PY{p}{;} \PY{n+nb}{tic}\PY{p}{;} - \PY{n}{x} \PY{p}{=} \PY{n}{GaussPivot}\PY{p}{(}\PY{n}{Atd}\PY{p}{,}\PY{n}{b}\PY{p}{)}\PY{p}{;} + \PY{n}{x} \PY{p}{=} \PY{n}{A}\PY{o}{\PYZbs{}}\PY{n}{b}\PY{p}{;} \PY{n}{t\PYZus{}GE\PYZus{}tridiag}\PY{p}{(}\PY{n}{n}\PY{p}{)} \PY{p}{=} \PY{n+nb}{toc}\PY{p}{;} \PY{n+nb}{tic}\PY{p}{;} \PY{n}{x} \PY{p}{=} \PY{n}{Tridiag}\PY{p}{(}\PY{n+nb}{e}\PY{p}{,}\PY{n}{f}\PY{p}{,}\PY{n}{g}\PY{p}{,}\PY{n}{b}\PY{p}{)}\PY{p}{;} @@ -829,17 +860,67 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}13}]:} \PY{n}{n}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{200}\PY{p}{;} +{\color{incolor}In [{\color{incolor}28}]:} \PY{n}{n}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{100}\PY{p}{;} 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zhsr@?wtn>ez8+E|cDCN}DJ-^5vgU>P4u8U>P4(PwN4p>}D9Fdu=LJmIFl(Ff%@!Cq zI&k)MOgXrCoMzv)9)Lxjzb(C?_7kT){*J{8|5o@CZMR+aQ455uy5MTo#wFA9H{EWk z>xPAmF9M`hIgq6yudX`0M{4jn!Kd?cH`x*-KDJw;5m-{y{GUzwU{~Q6(+xil?@?Y2 z`^e^1kYZY)Qmce^fcxr*-Fw%LWh*ZW+5h1}9T_kv^Zd`DHsIK?igSF6VIU|{(PcJ^ zEjbvncfrdza{LBw#V4KO?C88LWUQq!2TthDP&8t*)4%kF*=%3T;SW}rerwg_@T@Hx zE-&4}Cc9Hz$4IW_Y`OIN<`*07IHH=TY8I?G$@Z?9tMKg`hjXyj@7&a<&)FRkuji|h zd7RnXF9oC|yk$eO_bpMELQY;N``pn3D>*M$Yzv(6D4Bi8{HGPaH5+qwA8VMYw0IBa zQIh#K(b;R+EUjo(U6d|oT;A!x-`sRL%bxX>Dy~}0Zhth*qsVd+XJcxGzNeTxr)K7@ z7xOce*<*vu(+nT~#xA%l{5-&CE9Z&jf{5QuW^p#hItSnLaAjMLEpv<*TFWVUSY=bw x?#zi)D_prPZ3nwJXTs_|Yff`+ja4Z>{b>ukH0Mfg;J(#t!#Mr0%wiAD{{UU+ll1@q From 3780f0c36732e6e6d24aca6beb2e1b1dbcfe90a1 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Feb 2017 12:51:57 -0500 Subject: [PATCH 04/11] added HW4 --- HW4/README.md | 37 ++++++ HW4/collar_mass.svg | 270 ++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 307 insertions(+) create mode 100644 HW4/README.md create mode 100644 HW4/collar_mass.svg diff --git a/HW4/README.md b/HW4/README.md new file mode 100644 index 0000000..f9c6f19 --- /dev/null +++ b/HW4/README.md @@ -0,0 +1,37 @@ +# Homework #3 +## due 3/1/17 by 11:59pm + + +1. Use your repository 'roots_and_optimization'. Document all the HW4 work under the +heading `# Homework #4` in your `README.md` file + + a. Create a function called 'collar_potential_energy' that computes the total + potential energy of a collar connected to a spring and sliding on a rod. As shown in + the figure given a position, xc, and angle, theta: + + ![Collar-mass on an inclined rod](collar_mass.svg) + + The spring is unstretched when $x_{C}=0.5$. The potential energy due to gravity is: + + $PE_{g}=m x_{C}g\sin\theta$ + + where m=0.5 kg, and g is the acceleration due to gravity, + + and the potential energy due to the spring is: + + $PE_{s}=1/2K (\Delta L)^{2}$ + + where $\Delta L = 0.5 - \sqrt{0.5^{2}+(0.5-x_{C})^{2}}$ + + b. Use the `goldmin.m` function to solve for the minimum potential energy at xc when + theta=0. *create an anonymous function with `@(x) collar_potential_energy(x,theta)` in + the input for goldmin. Be sure to include the script that solves for xc* + + c. Create a for-loop that solves for the minimum potential energy position, xc, at a + given angle, theta, for theta = 0..90 degrees. + + d. Include a plot of xc vs theta. `plot(theta,xc)` with + + `![Steady-state position of collar on rod at angle theta](plot.png)` + +3. Commit your changes to your repository. Sync your local repository with github. diff --git a/HW4/collar_mass.svg b/HW4/collar_mass.svg new file mode 100644 index 0000000..cc18c3d --- /dev/null +++ b/HW4/collar_mass.svg @@ -0,0 +1,270 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + xc + + + + + + + + 𝝷 + + + L0=0.5m + + From 4a4ee8b39b30e3407210e9f7cdacdf81591c2d3e Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Feb 2017 12:54:00 -0500 Subject: [PATCH 05/11] update HW4 --- HW4/README.md | 2 +- HW4/collar_mass.png | Bin 0 -> 6794 bytes 2 files changed, 1 insertion(+), 1 deletion(-) create mode 100644 HW4/collar_mass.png diff --git a/HW4/README.md b/HW4/README.md index f9c6f19..3cacedc 100644 --- a/HW4/README.md +++ b/HW4/README.md @@ -9,7 +9,7 @@ heading `# Homework #4` in your `README.md` file potential energy of a collar connected to a spring and sliding on a rod. As shown in the figure given a position, xc, and angle, theta: - ![Collar-mass on an inclined rod](collar_mass.svg) + ![Collar-mass on an inclined rod](collar_mass.png) The spring is unstretched when $x_{C}=0.5$. 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is unstretched when $x_{C}=0.5$. The potential energy due to gravity is: + The spring is unstretched when x_C=0.5 m. The potential energy due to gravity is: - $PE_{g}=m x_{C}g\sin\theta$ + PE_g=m x_C*g*sin(theta) where m=0.5 kg, and g is the acceleration due to gravity, and the potential energy due to the spring is: - $PE_{s}=1/2K (\Delta L)^{2}$ + PE_s=1/2*K *(DL)^2$ - where $\Delta L = 0.5 - \sqrt{0.5^{2}+(0.5-x_{C})^{2}}$ + where DL = 0.5 - sqrt(0.5^2+(0.5-x_C)^2) b. Use the `goldmin.m` function to solve for the minimum potential energy at xc when theta=0. *create an anonymous function with `@(x) collar_potential_energy(x,theta)` in From f29d2cd48cf2333e32379a56402ef6b5a19f580e Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Feb 2017 12:58:13 -0500 Subject: [PATCH 07/11] update HW4 --- HW4/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/HW4/README.md b/HW4/README.md index ed5ca61..0b8a5f8 100644 --- a/HW4/README.md +++ b/HW4/README.md @@ -13,13 +13,13 @@ heading `# Homework #4` in your `README.md` file The spring is unstretched when x_C=0.5 m. The potential energy due to gravity is: - PE_g=m x_C*g*sin(theta) + PE_g=m x_C\*g\*sin(theta) where m=0.5 kg, and g is the acceleration due to gravity, and the potential energy due to the spring is: - PE_s=1/2*K *(DL)^2$ + PE_s=1/2\*K \*(DL)^2 where DL = 0.5 - sqrt(0.5^2+(0.5-x_C)^2) From 72ca990cf5c532001f3317577e52f1f54ab1a1b9 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Feb 2017 13:34:12 -0500 Subject: [PATCH 08/11] added suggested gauss elimination --- linear_algebra/gauss_suggested.md | 71 +++++++++++++++++++++++++++++++ 1 file changed, 71 insertions(+) create mode 100644 linear_algebra/gauss_suggested.md diff --git a/linear_algebra/gauss_suggested.md b/linear_algebra/gauss_suggested.md new file mode 100644 index 0000000..6b1f6b6 --- /dev/null +++ b/linear_algebra/gauss_suggested.md @@ -0,0 +1,71 @@ +# Linear Algebra Review +## (Gauss Elimination) Suggested problems +### No due date + +1. Solve for x when Ax=b for the following problems: + + a. $A=\left[ \begin{array}{cc} + 1 & 3 \\ + 2 & 1 \end{array} \right] + b= + \left[\begin{array}{c} + 1 \\ + 1\end{array}\right]$ + + a. $A=\left[ \begin{array}{cc} + 1 & 1 \\ + 2 & 3 \end{array} \right] + b= + \left[\begin{array}{c} + 3 \\ + 4\end{array}\right]$ + + a. $A=\left[ \begin{array}{cc} + 1 & 1 \\ + 2 & -2 \end{array} \right] + b= + \left[\begin{array}{c} + 4 \\ + 2\end{array}\right]$ + + b. $A=\left[ \begin{array}{ccc} + 1 & 3 & 1 \\ + -4 & -9 & 2 \\ + 0 & 3 & 6\end{array} \right] + b= + \left[\begin{array}{c} + 0 \\ + 0 \\ + 0\end{array}\right]$ + + c. $A=\left[ \begin{array}{ccc} + 1 & 3 & 1 \\ + -4 & -9 & 2 \\ + 0 & 3 & 6\end{array} \right] + b= + \left[\begin{array}{c} + 1 \\ + -1 \\ + -3\end{array}\right]$ + + d. $A=\left[ \begin{array}{ccc} + 1 & 3 & -5 \\ + 1 & 4 & -8 \\ + -3 & -7 & 9\end{array} \right] + b= + \left[\begin{array}{c} + 1 \\ + -1 \\ + -3\end{array}\right]$ + + d. $A=\left[ \begin{array}{ccc} + 1 & 2 & -1 \\ + 2 & 2 & 2 \\ + 1 & -1 & 2\end{array} \right] + b= + \left[\begin{array}{c} + 2 \\ + 12 \\ + 5\end{array}\right]$ + +2. Calculate the determinant of A from 1a-g. From 543fb4befaf120a39c88cde7fce8a17b07c6798d Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Feb 2017 13:34:49 -0500 Subject: [PATCH 09/11] added suggested gauss elimination --- linear_algebra/gauss_suggested.pdf | Bin 0 -> 87205 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 linear_algebra/gauss_suggested.pdf diff --git a/linear_algebra/gauss_suggested.pdf b/linear_algebra/gauss_suggested.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a78b2778173adb57f91a5eccafa5b843897da538 GIT binary patch literal 87205 zcmaf(Gn6n&6r{(tZQHhO+qP}n_WWbpwr$(C_w89E8*C+A9dy+93#!f~RS*%QWu#+; zBAs6zS%+d|Ctx73H?o4_;en!;F|{*yu^`~!WFz=rf}$6*v~e+YBA^$uF?2B%F*UX~ zF@@segK~CpGBvb?^4L7ln6leqKS$taLi(`uc{qSBLqrA%Ib85Sv&6F$g9OEmEnBF2|1!!f zsZs>e8dd+`$*xv%e`5C5F_ACcJF=$(Ay%8zg}@@+MzYO}c916h$+BCsX20Xo=b=fv zFjJ&?fOUqqZ+-G?-8E5m|1wvVdAv5fDV`cfs%ozcGH#XX*yo4JF(k;dYkCwKUUEt9 zu#{OTWt7WU^{WjZ9j8c48D$s!u8=2IqBo32r-=?Pm20%p{G}zbj zN*1ug(jUrZcxhk&z89*Wq&^o|QNR!bGD?X-3QPsK4Ftsm8=P@K$tI?>TNPlX3}dHI 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HcmV?d00001 From 35dbdeb9cc03d9f912cee00b2dccea4182c531cf Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Wed, 22 Feb 2017 11:42:29 -0500 Subject: [PATCH 10/11] add spring constant --- HW4/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/HW4/README.md b/HW4/README.md index 0b8a5f8..7f04a9a 100644 --- a/HW4/README.md +++ b/HW4/README.md @@ -21,7 +21,7 @@ heading `# Homework #4` in your `README.md` file PE_s=1/2\*K \*(DL)^2 - where DL = 0.5 - sqrt(0.5^2+(0.5-x_C)^2) + where DL = 0.5 - sqrt(0.5^2+(0.5-x_C)^2) and K=30 N/m. b. Use the `goldmin.m` function to solve for the minimum potential energy at xc when theta=0. *create an anonymous function with `@(x) collar_potential_energy(x,theta)` in From 46f626edf1d24a7dc9a189eb6c4ba024123f8242 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Wed, 22 Feb 2017 22:01:53 -0500 Subject: [PATCH 11/11] added lecture 11 --- lecture_11/LU_naive.m | 27 + lecture_11/LU_pivot.m | 36 + lecture_11/lecture_11.aux | 28 + lecture_11/lecture_11.bbl | 0 lecture_11/lecture_11.blg | 48 + lecture_11/lecture_11.ipynb | 685 +++++++++++++++ lecture_11/lecture_11.log | 826 ++++++++++++++++++ lecture_11/lecture_11.out | 4 + lecture_11/lecture_11.pdf | Bin 0 -> 154124 bytes lecture_11/lecture_11.tex | 656 ++++++++++++++ .../lecture_11_files/lecture_11_6_0.pdf | Bin 0 -> 9079 bytes .../lecture_11_files/lecture_11_6_0.svg | 149 ++++ lecture_11/mass_springs.png | Bin 0 -> 5031 bytes lecture_11/mass_springs.svg | 214 +++++ lecture_11/nohup.out | 2 + lecture_11/octave-workspace | Bin 0 -> 178581 bytes 16 files changed, 2675 insertions(+) create mode 100644 lecture_11/LU_naive.m create mode 100644 lecture_11/LU_pivot.m create mode 100644 lecture_11/lecture_11.aux create mode 100644 lecture_11/lecture_11.bbl create mode 100644 lecture_11/lecture_11.blg create mode 100644 lecture_11/lecture_11.ipynb create mode 100644 lecture_11/lecture_11.log create mode 100644 lecture_11/lecture_11.out create mode 100644 lecture_11/lecture_11.pdf create mode 100644 lecture_11/lecture_11.tex create mode 100644 lecture_11/lecture_11_files/lecture_11_6_0.pdf create mode 100644 lecture_11/lecture_11_files/lecture_11_6_0.svg create mode 100644 lecture_11/mass_springs.png create mode 100644 lecture_11/mass_springs.svg create mode 100644 lecture_11/nohup.out create mode 100644 lecture_11/octave-workspace diff --git a/lecture_11/LU_naive.m b/lecture_11/LU_naive.m new file mode 100644 index 0000000..92efde6 --- /dev/null +++ b/lecture_11/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 diff --git a/lecture_11/LU_pivot.m b/lecture_11/LU_pivot.m new file mode 100644 index 0000000..37abb26 --- /dev/null +++ b/lecture_11/LU_pivot.m @@ -0,0 +1,36 @@ +function [L,U,P] = LU_pivot(A) +% GaussPivot: Gauss elimination pivoting +% x = GaussPivot(A,b): Gauss elimination with pivoting. +% input: +% A = coefficient matrix +% b = 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; +P=diag(ones(n,1)); +% forward elimination +for k = 1:n-1 + % partial pivoting + [big,i]=max(abs(U(k:n,k))); + ipr=i+k-1; + if ipr~=k + P([k,ipr],:)=P([ipr,k],:); % if the max is not the current index ipr, pivot count + %L([k,ipr],:)=L([ipr,k],:); + %U([k,ipr],:)=U([ipr,k],:); + end + 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 diff --git a/lecture_11/lecture_11.aux b/lecture_11/lecture_11.aux new file mode 100644 index 0000000..cf3c530 --- /dev/null +++ b/lecture_11/lecture_11.aux @@ -0,0 +1,28 @@ +\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}}} 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with 494 characters, +and the built_in function-call counts, 0 in all, are: += -- 0 +> -- 0 +< -- 0 ++ -- 0 +- -- 0 +* -- 0 +:= -- 0 +add.period$ -- 0 +call.type$ -- 0 +change.case$ -- 0 +chr.to.int$ -- 0 +cite$ -- 0 +duplicate$ -- 0 +empty$ -- 0 +format.name$ -- 0 +if$ -- 0 +int.to.chr$ -- 0 +int.to.str$ -- 0 +missing$ -- 0 +newline$ -- 0 +num.names$ -- 0 +pop$ -- 0 +preamble$ -- 0 +purify$ -- 0 +quote$ -- 0 +skip$ -- 0 +stack$ -- 0 +substring$ -- 0 +swap$ -- 0 +text.length$ -- 0 +text.prefix$ -- 0 +top$ -- 0 +type$ -- 0 +warning$ -- 0 +while$ -- 0 +width$ -- 0 +write$ -- 0 +(There were 3 error messages) diff --git a/lecture_11/lecture_11.ipynb b/lecture_11/lecture_11.ipynb new file mode 100644 index 0000000..c290a47 --- /dev/null +++ b/lecture_11/lecture_11.ipynb @@ -0,0 +1,685 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "%plot --format svg" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "setdefaults" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# LU Decomposition\n", + "### efficient storage of matrices for solutions\n", + "\n", + "Considering the same solution set:\n", + "\n", + "$y=Ax$\n", + "\n", + "Assume that we can perform Gauss elimination and achieve this formula:\n", + "\n", + "$Ux=d$ \n", + "\n", + "Where, $U$ is an upper triangular matrix that we derived from Gauss elimination and $d$ is the set of dependent variables after Gauss elimination. \n", + "\n", + "Assume there is a lower triangular matrix, $L$, with ones on the diagonal and same dimensions of $U$ and the following is true:\n", + "\n", + "$L(Ux-d)=Ax-y=0$\n", + "\n", + "Now, $Ax=LUx$, so $A=LU$, and $y=Ld$.\n", + "\n", + "$2x_{1}+x_{2}=1$\n", + "\n", + "$x_{1}+3x_{2}=1$\n", + "\n", + "\n", + "$\\left[ \\begin{array}{cc}\n", + "2 & 1 \\\\\n", + "1 & 3 \\end{array} \\right]\n", + "\\left[\\begin{array}{c} \n", + "x_{1} \\\\ \n", + "x_{2} \\end{array}\\right]=\n", + "\\left[\\begin{array}{c} \n", + "1 \\\\\n", + "1\\end{array}\\right]$\n", + "\n", + "f21=0.5\n", + "\n", + "A(2,1)=1-1 = 0 \n", + "\n", + "A(2,2)=3-0.5=2.5\n", + "\n", + "y(2)=1-0.5=0.5\n", + "\n", + "$L(Ux-d)=\n", + "\\left[ \\begin{array}{cc}\n", + "1 & 0 \\\\\n", + "0.5 & 1 \\end{array} \\right]\n", + "\\left(\\left[ \\begin{array}{cc}\n", + "2 & 1 \\\\\n", + "0 & 2.5 \\end{array} \\right]\n", + "\\left[\\begin{array}{c} \n", + "x_{1} \\\\ \n", + "x_{2} \\end{array}\\right]-\n", + "\\left[\\begin{array}{c} \n", + "1 \\\\\n", + "0.5\\end{array}\\right]\\right)=0$\n" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A =\n", + "\n", + " 2 1\n", + " 1 3\n", + "\n", + "L =\n", + "\n", + " 1.00000 0.00000\n", + " 0.50000 1.00000\n", + "\n", + "U =\n", + "\n", + " 2.00000 1.00000\n", + " 0.00000 2.50000\n", + "\n", + "ans =\n", + "\n", + " 2 1\n", + " 1 3\n", + "\n", + "d =\n", + "\n", + " 1.00000\n", + " 0.50000\n", + "\n", + "y =\n", + "\n", + " 1\n", + " 1\n", + "\n" + ] + } + ], + "source": [ + "A=[2,1;1,3]\n", + "L=[1,0;0.5,1]\n", + "U=[2,1;0,2.5]\n", + "L*U\n", + "\n", + "d=[1;0.5]\n", + "y=L*d" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Pivoting for LU factorization\n", + "\n", + "LU factorization uses the same method as Gauss elimination so it is also necessary to perform partial pivoting when creating the lower and upper triangular matrices. \n", + "\n", + "Matlab and Octave use pivoting in the command \n", + "\n", + "`[L,U,P]=lu(A)`\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "'lu' is a built-in function from the file libinterp/corefcn/lu.cc\n", + "\n", + " -- Built-in Function: [L, U] = lu (A)\n", + " -- Built-in Function: [L, U, P] = lu (A)\n", + " -- Built-in Function: [L, U, P, Q] = lu (S)\n", + " -- Built-in Function: [L, U, P, Q, R] = lu (S)\n", + " -- Built-in Function: [...] = lu (S, THRES)\n", + " -- Built-in Function: Y = lu (...)\n", + " -- Built-in Function: [...] = lu (..., \"vector\")\n", + " Compute the LU decomposition of A.\n", + "\n", + " If A is full subroutines from LAPACK are used and if A is sparse\n", + " then UMFPACK is used.\n", + "\n", + " The result is returned in a permuted form, according to the\n", + " optional return value P. For example, given the matrix 'a = [1, 2;\n", + " 3, 4]',\n", + "\n", + " [l, u, p] = lu (A)\n", + "\n", + " returns\n", + "\n", + " l =\n", + "\n", + " 1.00000 0.00000\n", + " 0.33333 1.00000\n", + "\n", + " u =\n", + "\n", + " 3.00000 4.00000\n", + " 0.00000 0.66667\n", + "\n", + " p =\n", + "\n", + " 0 1\n", + " 1 0\n", + "\n", + " The matrix is not required to be square.\n", + "\n", + " When called with two or three output arguments and a spare input\n", + " matrix, 'lu' does not attempt to perform sparsity preserving column\n", + " permutations. Called with a fourth output argument, the sparsity\n", + " preserving column transformation Q is returned, such that 'P * A *\n", + " Q = L * U'.\n", + "\n", + " Called with a fifth output argument and a sparse input matrix, 'lu'\n", + " attempts to use a scaling factor R on the input matrix such that 'P\n", + " * (R \\ A) * Q = L * U'. This typically leads to a sparser and more\n", + " stable factorization.\n", + "\n", + " An additional input argument THRES, that defines the pivoting\n", + " threshold can be given. THRES can be a scalar, in which case it\n", + " defines the UMFPACK pivoting tolerance for both symmetric and\n", + " unsymmetric cases. If THRES is a 2-element vector, then the first\n", + " element defines the pivoting tolerance for the unsymmetric UMFPACK\n", + " pivoting strategy and the second for the symmetric strategy. By\n", + " default, the values defined by 'spparms' are used ([0.1, 0.001]).\n", + "\n", + " Given the string argument \"vector\", 'lu' returns the values of P\n", + " and Q as vector values, such that for full matrix, 'A (P,:) = L *\n", + " U', and 'R(P,:) * A (:, Q) = L * U'.\n", + "\n", + " With two output arguments, returns the permuted forms of the upper\n", + " and lower triangular matrices, such that 'A = L * U'. With one\n", + " output argument Y, then the matrix returned by the LAPACK routines\n", + " is returned. If the input matrix is sparse then the matrix L is\n", + " embedded into U to give a return value similar to the full case.\n", + " For both full and sparse matrices, 'lu' loses the permutation\n", + " information.\n", + "\n", + " See also: luupdate, ilu, chol, hess, qr, qz, schur, svd.\n", + "\n", + "Additional help for built-in functions and operators is\n", + "available in the online version of the manual. Use the command\n", + "'doc ' to search the manual index.\n", + "\n", + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" + ] + } + ], + "source": [ + "help lu" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5e-05\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0001\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.00015\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0002\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.00025\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0003\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tLU decomp\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\tLU decomp\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tOctave \\\\\n", + "\n", + "\t\n", + "\t\tOctave \\\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% time LU solution vs backslash\n", + "t_lu=zeros(100,1);\n", + "t_bs=zeros(100,1);\n", + "for N=1:100\n", + " A=rand(N,N);\n", + " y=rand(N,1);\n", + " [L,U,P]=lu(A);\n", + "\n", + " tic; d=L\\y; x=U\\d; t_lu(N)=toc;\n", + "\n", + " tic; x=A\\y; t_bs(N)=toc;\n", + "end\n", + "plot([1:100],t_lu,[1:100],t_bs) \n", + "legend('LU decomp','Octave \\\\')" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "Consider the problem again from the intro to Linear Algebra, 4 masses are connected in series to 4 springs with K=10 N/m. What are the final positions of the masses? \n", + "\n", + "![Springs-masses](../lecture_09/mass_springs.svg)\n", + "\n", + "The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass:\n", + "\n", + "$m_{1}g+k(x_{2}-x_{1})-kx_{1}=0$\n", + "\n", + "$m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0$\n", + "\n", + "$m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0$\n", + "\n", + "$m_{4}g-k(x_{4}-x_{3})=0$\n", + "\n", + "in matrix form:\n", + "\n", + "$\\left[ \\begin{array}{cccc}\n", + "2k & -k & 0 & 0 \\\\\n", + "-k & 2k & -k & 0 \\\\\n", + "0 & -k & 2k & -k \\\\\n", + "0 & 0 & -k & k \\end{array} \\right]\n", + "\\left[ \\begin{array}{c}\n", + "x_{1} \\\\\n", + "x_{2} \\\\\n", + "x_{3} \\\\\n", + "x_{4} \\end{array} \\right]=\n", + "\\left[ \\begin{array}{c}\n", + "m_{1}g \\\\\n", + "m_{2}g \\\\\n", + "m_{3}g \\\\\n", + "m_{4}g \\end{array} \\right]$" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "K =\n", + "\n", + " 20 -10 0 0\n", + " -10 20 -10 0\n", + " 0 -10 20 -10\n", + " 0 0 -10 10\n", + "\n", + "y =\n", + "\n", + " 9.8100\n", + " 19.6200\n", + " 29.4300\n", + " 39.2400\n", + "\n" + ] + } + ], + "source": [ + "k=10; % N/m\n", + "m1=1; % kg\n", + "m2=2;\n", + "m3=3;\n", + "m4=4;\n", + "g=9.81; % m/s^2\n", + "K=[2*k -k 0 0; -k 2*k -k 0; 0 -k 2*k -k; 0 0 -k k]\n", + "y=[m1*g;m2*g;m3*g;m4*g]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This matrix, K, is symmetric. \n", + "\n", + "`K(i,j)==K(j,i)`\n", + "\n", + "Now we can use,\n", + "\n", + "## Cholesky Factorization\n", + "\n", + "We can decompose the matrix, K into two matrices, $U$ and $U^{T}$, where\n", + "\n", + "$K=U^{T}U$\n", + "\n", + "each of the components of U can be calculated with the following equations:\n", + "\n", + "$u_{ii}=\\sqrt{a_{ii}-\\sum_{k=1}^{i-1}u_{ki}^{2}}$\n", + "\n", + "$u_{ij}=\\frac{a_{ij}-\\sum_{k=1}^{i-1}u_{ki}u_{kj}}{u_{ii}}$\n", + "\n", + "so for K" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "K =\n", + "\n", + " 20 -10 0 0\n", + " -10 20 -10 0\n", + " 0 -10 20 -10\n", + " 0 0 -10 10\n", + "\n" + ] + } + ], + "source": [ + "K" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "u11 = 4.4721\n", + "u12 = -2.2361\n", + "u13 = 0\n", + "u14 = 0\n", + "u22 = 3.8730\n", + "u23 = -2.5820\n", + "u24 = 0\n", + "u33 = 3.6515\n", + "u34 = -2.7386\n", + "u44 = 1.5811\n", + "U =\n", + "\n", + " 4.47214 -2.23607 0.00000 0.00000\n", + " 0.00000 3.87298 -2.58199 0.00000\n", + " 0.00000 0.00000 3.65148 -2.73861\n", + " 0.00000 0.00000 0.00000 1.58114\n", + "\n" + ] + } + ], + "source": [ + "u11=sqrt(K(1,1))\n", + "u12=(K(1,2))/u11\n", + "u13=(K(1,3))/u11\n", + "u14=(K(1,4))/u11\n", + "u22=sqrt(K(2,2)-u12^2)\n", + "u23=(K(2,3)-u12*u13)/u22\n", + "u24=(K(2,4)-u12*u14)/u22\n", + "u33=sqrt(K(3,3)-u13^2-u23^2)\n", + "u34=(K(3,4)-u13*u14-u23*u24)/u33\n", + "u44=sqrt(K(4,4)-u14^2-u24^2-u34^2)\n", + "U=[u11,u12,u13,u14;0,u22,u23,u24;0,0,u33,u34;0,0,0,u44]" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 20.00000 -10.00000 0.00000 0.00000\n", + " -10.00000 20.00000 -10.00000 0.00000\n", + " 0.00000 -10.00000 20.00000 -10.00000\n", + " 0.00000 0.00000 -10.00000 10.00000\n", + "\n" + ] + } + ], + "source": [ + "U'*U" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "average time spent for Cholesky factored solution = 1.623154e-05+/-1.166726e-05\n", + "average time spent for backslash solution = 1.675844e-05+/-1.187234e-05\n" + ] + } + ], + "source": [ + "% time solution for Cholesky vs backslash\n", + "t_chol=zeros(1000,1);\n", + "t_bs=zeros(1000,1);\n", + "for i=1:1000\n", + " tic; d=U'*y; x=U\\d; t_chol(i)=toc;\n", + " tic; x=K\\y; t_bs(i)=toc;\n", + "end\n", + "fprintf('average time spent for Cholesky factored solution = %e+/-%e',mean(t_chol),std(t_chol))\n", + "\n", + "fprintf('average time spent for backslash solution = %e+/-%e',mean(t_bs),std(t_bs))" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Octave", + "language": "octave", + "name": "octave" + }, + "language_info": { + "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": "octave", + "version": "0.19.14" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lecture_11/lecture_11.log b/lecture_11/lecture_11.log new file mode 100644 index 0000000..10423a7 --- /dev/null +++ b/lecture_11/lecture_11.log @@ -0,0 +1,826 @@ +This is pdfTeX, 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hardcoded. + \renewcommand{\includegraphics}[1]{\Oldincludegraphics[width=.8\maxwidth]{#1}} + % Ensure that by default, figures have no caption (until we provide a + % proper Figure object with a Caption API and a way to capture that + % in the conversion process - todo). + \usepackage{caption} + \DeclareCaptionLabelFormat{nolabel}{} + \captionsetup{labelformat=nolabel} + + \usepackage{adjustbox} % Used to constrain images to a maximum size + \usepackage{xcolor} % Allow colors to be defined + \usepackage{enumerate} % Needed for markdown enumerations to work + \usepackage{geometry} % Used to adjust the document margins + \usepackage{amsmath} % Equations + \usepackage{amssymb} % Equations + \usepackage{textcomp} % defines textquotesingle + % Hack from http://tex.stackexchange.com/a/47451/13684: + \AtBeginDocument{% + \def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code + } + \usepackage{upquote} % Upright quotes for verbatim code + \usepackage{eurosym} % defines \euro + \usepackage[mathletters]{ucs} % Extended unicode (utf-8) support + \usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document + \usepackage{fancyvrb} % verbatim replacement that allows latex + \usepackage{grffile} % extends the file name processing of package graphics + % to support a larger range + % The hyperref package gives us a pdf with properly built + % internal navigation ('pdf bookmarks' for the table of contents, + % internal cross-reference links, web links for URLs, etc.) + \usepackage{hyperref} + \usepackage{longtable} % longtable support required by pandoc >1.10 + \usepackage{booktabs} % table support for pandoc > 1.12.2 + \usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment) + \usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout) + % normalem makes italics be italics, not underlines + + + + + % Colors for the hyperref package + \definecolor{urlcolor}{rgb}{0,.145,.698} + \definecolor{linkcolor}{rgb}{.71,0.21,0.01} + \definecolor{citecolor}{rgb}{.12,.54,.11} + + % ANSI colors + \definecolor{ansi-black}{HTML}{3E424D} + \definecolor{ansi-black-intense}{HTML}{282C36} + \definecolor{ansi-red}{HTML}{E75C58} + \definecolor{ansi-red-intense}{HTML}{B22B31} + \definecolor{ansi-green}{HTML}{00A250} + \definecolor{ansi-green-intense}{HTML}{007427} + \definecolor{ansi-yellow}{HTML}{DDB62B} + \definecolor{ansi-yellow-intense}{HTML}{B27D12} + \definecolor{ansi-blue}{HTML}{208FFB} + \definecolor{ansi-blue-intense}{HTML}{0065CA} + \definecolor{ansi-magenta}{HTML}{D160C4} + \definecolor{ansi-magenta-intense}{HTML}{A03196} + \definecolor{ansi-cyan}{HTML}{60C6C8} + \definecolor{ansi-cyan-intense}{HTML}{258F8F} + \definecolor{ansi-white}{HTML}{C5C1B4} + \definecolor{ansi-white-intense}{HTML}{A1A6B2} + + % commands and environments needed by pandoc snippets + % extracted from the output of `pandoc -s` + \providecommand{\tightlist}{% + \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} + \DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}} + % Add ',fontsize=\small' for more characters per line + \newenvironment{Shaded}{}{} + \newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}} + \newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}} + \newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}} + \newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}} + \newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}} + \newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}} + \newcommand{\RegionMarkerTok}[1]{{#1}} + \newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}} + \newcommand{\NormalTok}[1]{{#1}} + + % Additional commands for more recent versions of Pandoc + \newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}} + \newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}} + \newcommand{\ImportTok}[1]{{#1}} + \newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}} + \newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}} + \newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}} + \newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}} + \newcommand{\BuiltInTok}[1]{{#1}} + \newcommand{\ExtensionTok}[1]{{#1}} + \newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}} + \newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}} + \newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + + + % Define a nice break command that doesn't care if a line doesn't already + % exist. + \def\br{\hspace*{\fill} \\* } + % Math Jax compatability definitions + \def\gt{>} + \def\lt{<} + % Document parameters + \title{lecture\_11} + + + + + % Pygments definitions + +\makeatletter +\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax% + \let\PY@ul=\relax \let\PY@tc=\relax% + \let\PY@bc=\relax \let\PY@ff=\relax} +\def\PY@tok#1{\csname PY@tok@#1\endcsname} +\def\PY@toks#1+{\ifx\relax#1\empty\else% + \PY@tok{#1}\expandafter\PY@toks\fi} +\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{% + \PY@it{\PY@bf{\PY@ff{#1}}}}}}} +\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}} + +\expandafter\def\csname PY@tok@gd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.00,0.00}{##1}}} +\expandafter\def\csname PY@tok@gu\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.50,0.00,0.50}{##1}}} +\expandafter\def\csname PY@tok@gt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.27,0.87}{##1}}} +\expandafter\def\csname PY@tok@gs\endcsname{\let\PY@bf=\textbf} +\expandafter\def\csname PY@tok@gr\endcsname{\def\PY@tc##1{\textcolor[rgb]{1.00,0.00,0.00}{##1}}} +\expandafter\def\csname PY@tok@cm\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@vg\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@vi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname 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+\def\PYZca{\char`\^} +\def\PYZam{\char`\&} +\def\PYZlt{\char`\<} +\def\PYZgt{\char`\>} +\def\PYZsh{\char`\#} +\def\PYZpc{\char`\%} +\def\PYZdl{\char`\$} +\def\PYZhy{\char`\-} +\def\PYZsq{\char`\'} +\def\PYZdq{\char`\"} +\def\PYZti{\char`\~} +% for compatibility with earlier versions +\def\PYZat{@} +\def\PYZlb{[} +\def\PYZrb{]} +\makeatother + + + % Exact colors from NB + \definecolor{incolor}{rgb}{0.0, 0.0, 0.5} + \definecolor{outcolor}{rgb}{0.545, 0.0, 0.0} + + + + + % Prevent overflowing lines due to hard-to-break entities + \sloppy + % Setup hyperref package + \hypersetup{ + breaklinks=true, % so long urls are correctly broken across lines + colorlinks=true, + urlcolor=urlcolor, + linkcolor=linkcolor, + citecolor=citecolor, + } + % Slightly bigger margins than the latex defaults + + \geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in} + + + + \begin{document} + + + \maketitle + + + + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}18}]:} \PY{c}{\PYZpc{}plot \PYZhy{}\PYZhy{}format svg} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}19}]:} \PY{n}{setdefaults} +\end{Verbatim} + + \section{LU Decomposition}\label{lu-decomposition} + +\subsubsection{efficient storage of matrices for +solutions}\label{efficient-storage-of-matrices-for-solutions} + +Considering the same solution set: + +\(y=Ax\) + +Assume that we can perform Gauss elimination and achieve this formula: + +\(Ux=d\) + +Where, \(U\) is an upper triangular matrix that we derived from Gauss +elimination and \(d\) is the set of dependent variables after Gauss +elimination. + +Assume there is a lower triangular matrix, \(L\), with ones on the +diagonal and same dimensions of \(U\) and the following is true: + +\(L(Ux-d)=Ax-y=0\) + +Now, \(Ax=LUx\), so \(A=LU\), and \(y=Ld\). + +\(2x_{1}+x_{2}=1\) + +\(x_{1}+3x_{2}=1\) + +\(\left[ \begin{array}{cc} 2 & 1 \\ 1 & 3 \end{array} \right] \left[\begin{array}{c} x_{1} \\ x_{2} \end{array}\right]= \left[\begin{array}{c} 1 \\ 1\end{array}\right]\) + +f21=0.5 + +A(2,1)=1-1 = 0 + +A(2,2)=3-0.5=2.5 + +y(2)=1-0.5=0.5 + +\(L(Ux-d)= \left[ \begin{array}{cc} 1 & 0 \\ 0.5 & 1 \end{array} \right] \left(\left[ \begin{array}{cc} 2 & 1 \\ 0 & 2.5 \end{array} \right] \left[\begin{array}{c} x_{1} \\ x_{2} \end{array}\right]- \left[\begin{array}{c} 1 \\ 0.5\end{array}\right]\right)=0\) + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}3}]:} \PY{n}{A}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{;}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{]} + \PY{n}{L}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{;}\PY{l+m+mf}{0.5}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]} + \PY{n}{U}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{;}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mf}{2.5}\PY{p}{]} + \PY{n}{L}\PY{o}{*}\PY{n}{U} + + \PY{n}{d}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{;}\PY{l+m+mf}{0.5}\PY{p}{]} + \PY{n}{y}\PY{p}{=}\PY{n}{L}\PY{o}{*}\PY{n}{d} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +A = + + 2 1 + 1 3 + +L = + + 1.00000 0.00000 + 0.50000 1.00000 + +U = + + 2.00000 1.00000 + 0.00000 2.50000 + +ans = + + 2 1 + 1 3 + +d = + + 1.00000 + 0.50000 + +y = + + 1 + 1 + + + \end{Verbatim} + + \subsection{Pivoting for LU +factorization}\label{pivoting-for-lu-factorization} + +LU factorization uses the same method as Gauss elimination so it is also +necessary to perform partial pivoting when creating the lower and upper +triangular matrices. + +Matlab and Octave use pivoting in the command + +\texttt{{[}L,U,P{]}=lu(A)} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}4}]:} \PY{n}{help} \PY{n+nb}{lu} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +'lu' is a built-in function from the file libinterp/corefcn/lu.cc + + -- Built-in Function: [L, U] = lu (A) + -- Built-in Function: [L, U, P] = lu (A) + -- Built-in Function: [L, U, P, Q] = lu (S) + -- Built-in Function: [L, U, P, Q, R] = lu (S) + -- Built-in Function: [{\ldots}] = lu (S, THRES) + -- Built-in Function: Y = lu ({\ldots}) + -- Built-in Function: [{\ldots}] = lu ({\ldots}, "vector") + Compute the LU decomposition of A. + + If A is full subroutines from LAPACK are used and if A is sparse + then UMFPACK is used. + + The result is returned in a permuted form, according to the + optional return value P. For example, given the matrix 'a = [1, 2; + 3, 4]', + + [l, u, p] = lu (A) + + returns + + l = + + 1.00000 0.00000 + 0.33333 1.00000 + + u = + + 3.00000 4.00000 + 0.00000 0.66667 + + p = + + 0 1 + 1 0 + + The matrix is not required to be square. + + When called with two or three output arguments and a spare input + matrix, 'lu' does not attempt to perform sparsity preserving column + permutations. Called with a fourth output argument, the sparsity + preserving column transformation Q is returned, such that 'P * A * + Q = L * U'. + + Called with a fifth output argument and a sparse input matrix, 'lu' + attempts to use a scaling factor R on the input matrix such that 'P + * (R \textbackslash{} A) * Q = L * U'. This typically leads to a sparser and more + stable factorization. + + An additional input argument THRES, that defines the pivoting + threshold can be given. THRES can be a scalar, in which case it + defines the UMFPACK pivoting tolerance for both symmetric and + unsymmetric cases. If THRES is a 2-element vector, then the first + element defines the pivoting tolerance for the unsymmetric UMFPACK + pivoting strategy and the second for the symmetric strategy. By + default, the values defined by 'spparms' are used ([0.1, 0.001]). + + Given the string argument "vector", 'lu' returns the values of P + and Q as vector values, such that for full matrix, 'A (P,:) = L * + U', and 'R(P,:) * A (:, Q) = L * U'. + + With two output arguments, returns the permuted forms of the upper + and lower triangular matrices, such that 'A = L * U'. With one + output argument Y, then the matrix returned by the LAPACK routines + is returned. If the input matrix is sparse then the matrix L is + embedded into U to give a return value similar to the full case. + For both full and sparse matrices, 'lu' loses the permutation + information. + + See also: luupdate, ilu, chol, hess, qr, qz, schur, svd. + +Additional help for built-in functions and operators is +available in the online version of the manual. Use the command +'doc ' to search the manual index. + +Help and information about Octave is also available on the WWW +at http://www.octave.org and via the help@octave.org +mailing list. + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}22}]:} \PY{c}{\PYZpc{} time LU solution vs backslash} + \PY{n}{t\PYZus{}lu}\PY{p}{=}\PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{100}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} + \PY{n}{t\PYZus{}bs}\PY{p}{=}\PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{100}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} + \PY{k}{for} \PY{n}{N}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{100} + \PY{n}{A}\PY{p}{=}\PY{n+nb}{rand}\PY{p}{(}\PY{n}{N}\PY{p}{,}\PY{n}{N}\PY{p}{)}\PY{p}{;} + \PY{n}{y}\PY{p}{=}\PY{n+nb}{rand}\PY{p}{(}\PY{n}{N}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} + \PY{p}{[}\PY{n}{L}\PY{p}{,}\PY{n}{U}\PY{p}{,}\PY{n}{P}\PY{p}{]}\PY{p}{=}\PY{n+nb}{lu}\PY{p}{(}\PY{n}{A}\PY{p}{)}\PY{p}{;} + + \PY{n+nb}{tic}\PY{p}{;} \PY{n}{d}\PY{p}{=}\PY{n}{L}\PY{o}{\PYZbs{}}\PY{n}{y}\PY{p}{;} \PY{n}{x}\PY{p}{=}\PY{n}{U}\PY{o}{\PYZbs{}}\PY{n}{d}\PY{p}{;} \PY{n}{t\PYZus{}lu}\PY{p}{(}\PY{n}{N}\PY{p}{)}\PY{p}{=}\PY{n+nb}{toc}\PY{p}{;} + + \PY{n+nb}{tic}\PY{p}{;} \PY{n}{x}\PY{p}{=}\PY{n}{A}\PY{o}{\PYZbs{}}\PY{n}{y}\PY{p}{;} \PY{n}{t\PYZus{}bs}\PY{p}{(}\PY{n}{N}\PY{p}{)}\PY{p}{=}\PY{n+nb}{toc}\PY{p}{;} + \PY{k}{end} + \PY{n+nb}{plot}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{100}\PY{p}{]}\PY{p}{,}\PY{n}{t\PYZus{}lu}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{100}\PY{p}{]}\PY{p}{,}\PY{n}{t\PYZus{}bs}\PY{p}{)} + \PY{n+nb}{legend}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{LU decomp\PYZsq{}}\PY{p}{,}\PY{l+s}{\PYZsq{}}\PY{l+s}{Octave \PYZbs{}\PYZbs{}\PYZsq{}}\PY{p}{)} +\end{Verbatim} + + \begin{center} + \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{lecture_11_files/lecture_11_6_0.pdf} + \end{center} + { \hspace*{\fill} \\} + + Consider the problem again from the intro to Linear Algebra, 4 masses +are connected in series to 4 springs with K=10 N/m. What are the final +positions of the masses? + +\begin{figure}[htbp] +\centering +\includegraphics{mass_springs.png} +\caption{Springs-masses} +\end{figure} + +The masses haves the following amounts, 1, 2, 3, and 4 kg for masses +1-4. Using a FBD for each mass: + +\(m_{1}g+k(x_{2}-x_{1})-kx_{1}=0\) + +\(m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0\) + +\(m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0\) + +\(m_{4}g-k(x_{4}-x_{3})=0\) + +in matrix form: + +\(\left[ \begin{array}{cccc} 2k & -k & 0 & 0 \\ -k & 2k & -k & 0 \\ 0 & -k & 2k & -k \\ 0 & 0 & -k & k \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array} \right]= \left[ \begin{array}{c} m_{1}g \\ m_{2}g \\ m_{3}g \\ m_{4}g \end{array} \right]\) + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}24}]:} \PY{n}{k}\PY{p}{=}\PY{l+m+mi}{10}\PY{p}{;} \PY{c}{\PYZpc{} N/m} + \PY{n}{m1}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{;} \PY{c}{\PYZpc{} kg} + \PY{n}{m2}\PY{p}{=}\PY{l+m+mi}{2}\PY{p}{;} + \PY{n}{m3}\PY{p}{=}\PY{l+m+mi}{3}\PY{p}{;} + \PY{n}{m4}\PY{p}{=}\PY{l+m+mi}{4}\PY{p}{;} + \PY{n}{g}\PY{p}{=}\PY{l+m+mf}{9.81}\PY{p}{;} \PY{c}{\PYZpc{} m/s\PYZca{}2} + \PY{n}{K}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{k} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{0} \PY{l+m+mi}{0}\PY{p}{;} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{2}\PY{o}{*}\PY{n}{k} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{0}\PY{p}{;} \PY{l+m+mi}{0} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{2}\PY{o}{*}\PY{n}{k} \PY{o}{\PYZhy{}}\PY{n}{k}\PY{p}{;} \PY{l+m+mi}{0} \PY{l+m+mi}{0} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{n}{k}\PY{p}{]} + \PY{n}{y}\PY{p}{=}\PY{p}{[}\PY{n}{m1}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m2}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m3}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m4}\PY{o}{*}\PY{n}{g}\PY{p}{]} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +K = + + 20 -10 0 0 + -10 20 -10 0 + 0 -10 20 -10 + 0 0 -10 10 + +y = + + 9.8100 + 19.6200 + 29.4300 + 39.2400 + + + \end{Verbatim} + + This matrix, K, is symmetric. + +\texttt{K(i,j)==K(j,i)} + +Now we can use, + +\subsection{Cholesky Factorization}\label{cholesky-factorization} + +We can decompose the matrix, K into two matrices, \(U\) and \(U^{T}\), +where + +\(K=U^{T}U\) + +each of the components of U can be calculated with the following +equations: + +\(u_{ii}=\sqrt{a_{ii}-\sum_{k=1}^{i-1}u_{ki}^{2}}\) + +\(u_{ij}=\frac{a_{ij}-\sum_{k=1}^{i-1}u_{ki}u_{kj}}{u_{ii}}\) + +so for K + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}25}]:} \PY{n}{K} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +K = + + 20 -10 0 0 + -10 20 -10 0 + 0 -10 20 -10 + 0 0 -10 10 + + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}26}]:} \PY{n}{u11}\PY{p}{=}\PY{n+nb}{sqrt}\PY{p}{(}\PY{n}{K}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)} + \PY{n}{u12}\PY{p}{=}\PY{p}{(}\PY{n}{K}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)}\PY{p}{)}\PY{o}{/}\PY{n}{u11} + \PY{n}{u13}\PY{p}{=}\PY{p}{(}\PY{n}{K}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{p}{)}\PY{o}{/}\PY{n}{u11} + \PY{n}{u14}\PY{p}{=}\PY{p}{(}\PY{n}{K}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)}\PY{p}{)}\PY{o}{/}\PY{n}{u11} + \PY{n}{u22}\PY{p}{=}\PY{n+nb}{sqrt}\PY{p}{(}\PY{n}{K}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{u12}\PYZca{}\PY{l+m+mi}{2}\PY{p}{)} + \PY{n}{u23}\PY{p}{=}\PY{p}{(}\PY{n}{K}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{u12}\PY{o}{*}\PY{n}{u13}\PY{p}{)}\PY{o}{/}\PY{n}{u22} + \PY{n}{u24}\PY{p}{=}\PY{p}{(}\PY{n}{K}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{u12}\PY{o}{*}\PY{n}{u14}\PY{p}{)}\PY{o}{/}\PY{n}{u22} + \PY{n}{u33}\PY{p}{=}\PY{n+nb}{sqrt}\PY{p}{(}\PY{n}{K}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{u13}\PYZca{}\PY{l+m+mi}{2}\PY{o}{\PYZhy{}}\PY{n}{u23}\PYZca{}\PY{l+m+mi}{2}\PY{p}{)} + \PY{n}{u34}\PY{p}{=}\PY{p}{(}\PY{n}{K}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{u13}\PY{o}{*}\PY{n}{u14}\PY{o}{\PYZhy{}}\PY{n}{u23}\PY{o}{*}\PY{n}{u24}\PY{p}{)}\PY{o}{/}\PY{n}{u33} + \PY{n}{u44}\PY{p}{=}\PY{n+nb}{sqrt}\PY{p}{(}\PY{n}{K}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{u14}\PYZca{}\PY{l+m+mi}{2}\PY{o}{\PYZhy{}}\PY{n}{u24}\PYZca{}\PY{l+m+mi}{2}\PY{o}{\PYZhy{}}\PY{n}{u34}\PYZca{}\PY{l+m+mi}{2}\PY{p}{)} + \PY{n}{U}\PY{p}{=}\PY{p}{[}\PY{n}{u11}\PY{p}{,}\PY{n}{u12}\PY{p}{,}\PY{n}{u13}\PY{p}{,}\PY{n}{u14}\PY{p}{;}\PY{l+m+mi}{0}\PY{p}{,}\PY{n}{u22}\PY{p}{,}\PY{n}{u23}\PY{p}{,}\PY{n}{u24}\PY{p}{;}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{n}{u33}\PY{p}{,}\PY{n}{u34}\PY{p}{;}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{n}{u44}\PY{p}{]} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +u11 = 4.4721 +u12 = -2.2361 +u13 = 0 +u14 = 0 +u22 = 3.8730 +u23 = -2.5820 +u24 = 0 +u33 = 3.6515 +u34 = -2.7386 +u44 = 1.5811 +U = + + 4.47214 -2.23607 0.00000 0.00000 + 0.00000 3.87298 -2.58199 0.00000 + 0.00000 0.00000 3.65148 -2.73861 + 0.00000 0.00000 0.00000 1.58114 + + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}27}]:} \PY{n}{U}\PY{o}{\PYZsq{}}\PY{o}{*}\PY{n}{U} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +ans = + + 20.00000 -10.00000 0.00000 0.00000 + -10.00000 20.00000 -10.00000 0.00000 + 0.00000 -10.00000 20.00000 -10.00000 + 0.00000 0.00000 -10.00000 10.00000 + + + \end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}37}]:} \PY{c}{\PYZpc{} time solution for Cholesky vs backslash} + \PY{n}{t\PYZus{}chol}\PY{p}{=}\PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{1000}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} + \PY{n}{t\PYZus{}bs}\PY{p}{=}\PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{1000}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} + \PY{k}{for} \PY{n}{i}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{1000} + \PY{n+nb}{tic}\PY{p}{;} \PY{n}{d}\PY{p}{=}\PY{n}{U}\PY{o}{\PYZsq{}}\PY{o}{*}\PY{n}{y}\PY{p}{;} \PY{n}{x}\PY{p}{=}\PY{n}{U}\PY{o}{\PYZbs{}}\PY{n}{d}\PY{p}{;} \PY{n}{t\PYZus{}chol}\PY{p}{(}\PY{n}{i}\PY{p}{)}\PY{p}{=}\PY{n+nb}{toc}\PY{p}{;} + \PY{n+nb}{tic}\PY{p}{;} \PY{n}{x}\PY{p}{=}\PY{n}{K}\PY{o}{\PYZbs{}}\PY{n}{y}\PY{p}{;} \PY{n}{t\PYZus{}bs}\PY{p}{(}\PY{n}{i}\PY{p}{)}\PY{p}{=}\PY{n+nb}{toc}\PY{p}{;} + \PY{k}{end} + \PY{n+nb}{fprintf}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{average time spent for Cholesky factored solution = \PYZpc{}e+/\PYZhy{}\PYZpc{}e\PYZsq{}}\PY{p}{,}\PY{n+nb}{mean}\PY{p}{(}\PY{n}{t\PYZus{}chol}\PY{p}{)}\PY{p}{,}\PY{n+nb}{std}\PY{p}{(}\PY{n}{t\PYZus{}chol}\PY{p}{)}\PY{p}{)} + + \PY{n+nb}{fprintf}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{average time spent for backslash solution = \PYZpc{}e+/\PYZhy{}\PYZpc{}e\PYZsq{}}\PY{p}{,}\PY{n+nb}{mean}\PY{p}{(}\PY{n}{t\PYZus{}bs}\PY{p}{)}\PY{p}{,}\PY{n+nb}{std}\PY{p}{(}\PY{n}{t\PYZus{}bs}\PY{p}{)}\PY{p}{)} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +average time spent for Cholesky factored solution = 1.623154e-05+/-1.166726e-05 +average time spent for backslash solution = 1.675844e-05+/-1.187234e-05 + + \end{Verbatim} + + + % Add a bibliography block to the postdoc + + + + \end{document} diff --git a/lecture_11/lecture_11_files/lecture_11_6_0.pdf b/lecture_11/lecture_11_files/lecture_11_6_0.pdf new file mode 100644 index 0000000000000000000000000000000000000000..40e3f285bc346d547a3c8211b26c74614a6db81b GIT binary patch literal 9079 zcma)C2UJr_x29K7nt*^1=|~A7B!u33?_CKHn$!@KssyArL68p8LKT(Xq)1VEZz9q~ zP*9NGd;#yh|Gn@2Z>@Ju+I)Lv_RQXMvS)pJICT}31weu#WSljd6(?k1fDpjl#+gh~ z5&+ahxH+I40U}sPp9}y10F|9wPzWUU+Xaq7C?ag#?GR+r(qvvJBm(YA=98Ihs^xAZ zN;!98@)2R$>2td2T3AZ$Y&3%5vql2|xAN%BlUwJHs?XkP9J4)6l-RF{AL`Fs&H!G$ zdVH|AynX5llRn>c-nkXa-t|SJJqL2~Y0UGc%Fc`3<*%FD?V%y(XZvGR0n#mt%jBoO zdfxbjpgo!&938G62KSAY-#bpwm-;j}(53KGsN3^F^;p{)`@0qjhcC0!-`{D>-B>*L zSOy#q#-D0zNk3!%y!?4k>X*mO3edxk0UjocGXvZQ%L904B$cKf$>a3GQG`*00SnmoL!(a~UD+q4ZHqlw4`SFcAz+5(A5Uwo#PH+iqv%$Io zmL^2RN_U0ou@L4`P)SGMeQR5O)8a(VL+3N-$gnnh(g86-l%~kk5Ya&}aPYm+9F8dh 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