diff --git a/HW5/README.html b/HW5/README.html
new file mode 100644
index 0000000..96595a9
--- /dev/null
+++ b/HW5/README.html
@@ -0,0 +1,53 @@
+
+
+
+ Homework #5
+due 3/28/17 by 11:59pm
+Include all work as either an m-file script, m-file function, or example code included with ``` and document your code in the README.md file
+
+Create a new github repository called ‘linear_algebra’.
+
+Add rcc02007 and pez16103 as collaborators.
Clone the repository to your computer.
Create an LU-decomposition function called lu_tridiag.m that takes 3 vectors as inputs and calculates the LU-decomposition of a tridiagonal matrix. The output should be 3 vectors, the diagonal of the Upper matrix, and the two off-diagonal vectors of the Lower and Upper matrices.
+[ud,uo,lo]=lu_tridiag(e,f,g);
Use the output from lu_tridiag.m to create a forward substitution and back-substitution function called solve_tridiag.m that provides the solution of Ax=b given the vectors from the output of [ud,uo,lo]=lu_tridiag(e,f,g). Note: do not use the backslash solver \, create an algebraic solution
+x=solve_tridiag(ud,uo,lo,b);
Test your function on the matrices A3, A4, …, A10 generated with test_arrays.m Solving for b=ones(N,1); where N is the size of A. In your README.md file, compare the norm of the error between your result and the result of AN.
+| size of A | norm(error) |
+|-----------|-------------|
+| 3 | ... |
+| 4 | ... |
+In the system shown above, determine the three differential equations for the position of masses 1, 2, and 3. Solve for the vibrational modes of the spring-mass system if k1=10 N/m, k2=k3=20 N/m, and k4=10 N/m. The masses are m1=1 kg, m2=2 kg and m3=4 kg. Determine the eigenvalues and natural frequencies.
The curvature of a slender column subject to an axial load P (Fig. P13.10) can be modeled by
+\(\frac{d^{2}y}{dx^{2}} + p^{2} y = 0\)
+where \(p^{2} = \frac{P}{EI}\)
+where E = the modulus of elasticity, and I = the moment of inertia of the cross section about its neutral axis.
+This model can be converted into an eigenvalue problem by substituting a centered finite-difference approximation for the second derivative to give \(\frac{y_{i+1} − 2y_{i} + y_{i−1} }{\Delta x^{2}}+ p^{2} y_{i}\)
+where i = a node located at a position along the rod’s interior, and \(\Delta x\) = the spacing between nodes. This equation can be expressed as \(y_{i−1} − (2 − \Delta x^{2} p^{2} )y_{i} + y_{i+1} = 0\) Writing this equation for a series of interior nodes along the axis of the column yields a homogeneous system of equations. (See 13.10 for 4 interior-node example)
+Determine the eigenvalues for a 5-segment (4-interior nodes), 6-segment (5-interior nodes), and 10-segment (9-interior nodes). Using the modulus and moment of inertia of a pole for pole-vaulting ( http://people.bath.ac.uk/taf21/sports_whole.htm ) E=76E9 Pa, I=4E-8 m^4, and L= 5m.
+Include a table in the README.md that shows the following results: What are the largest and smallest eigenvalues for the beam? How many eigenvalues are there?
+| # of segments | largest | smallest | # of eigenvalues |
+| --- | --- | --- | --- |
+| 5 | ... | ... | ... |
+| 6 | ... | ... | ... |
+| 10 | ... | ... | ... |If the segment length approaches 0, how many eigenvalues would there be?
+
+
diff --git a/HW5/README.md b/HW5/README.md
new file mode 100644
index 0000000..a8495ea
--- /dev/null
+++ b/HW5/README.md
@@ -0,0 +1,83 @@
+# Homework #5
+## due 3/28/17 by 11:59pm
+
+*Include all work as either an m-file script, m-file function, or example code included
+with \`\`\` and document your code in the README.md file*
+
+1. Create a new github repository called 'linear_algebra'.
+
+ a. Add rcc02007 and pez16103 as collaborators.
+
+ b. Clone the repository to your computer.
+
+2. Create an LU-decomposition function called `lu_tridiag.m` that takes 3 vectors as inputs
+and calculates the LU-decomposition of a tridiagonal matrix. The output should be 3
+vectors, the diagonal of the Upper matrix, and the two off-diagonal vectors of the Lower
+and Upper matrices.
+
+ ```[ud,uo,lo]=lu_tridiag(e,f,g);```
+
+3. Use the output from `lu_tridiag.m` to create a forward substitution and
+back-substitution function called `solve_tridiag.m` that provides the solution of
+Ax=b given the vectors from the output of [ud,uo,lo]=lu_tridiag(e,f,g). *Note: do not use
+the backslash solver `\`, create an algebraic solution*
+
+ ```x=solve_tridiag(ud,uo,lo,b);```
+
+4. Test your function on the matrices A3, A4, ..., A10 generated with `test_arrays.m`
+Solving for `b=ones(N,1);` where N is the size of A. In your `README.md` file, compare
+the norm of the error between your result and the result of AN\b.
+
+```
+| size of A | norm(error) |
+|-----------|-------------|
+| 3 | ... |
+| 4 | ... |
+```
+
+
+
+5. In the system shown above, determine the three differential equations for the position
+of masses 1, 2, and 3. Solve for the vibrational modes of the spring-mass system if k1=10
+N/m, k2=k3=20 N/m, and k4=10 N/m. The masses are m1=1 kg, m2=2 kg and m3=4 kg. Determine
+the eigenvalues and natural frequencies.
+
+6. The curvature of a slender column subject to an axial load P (Fig. P13.10) can be
+modeled by
+
+$\frac{d^{2}y}{dx^{2}} + p^{2} y = 0$
+
+where $p^{2} = \frac{P}{EI}$
+
+where E = the modulus of elasticity, and I = the moment of inertia of the cross section
+about its neutral axis.
+
+This model can be converted into an eigenvalue problem by
+substituting a centered finite-difference approximation for the second derivative to give
+$\frac{y_{i+1} -2y_{i} + y_{i-1} }{\Delta x^{2}}+ p^{2} y_{i}$
+
+where i = a node located at a position along the rod’s interior, and $\Delta x$ = the
+spacing between nodes. This equation can be expressed as $y_{i-1} - (2 - \Delta x^{2}
+p^{2} )y_{i} y_{i+1} = 0$ Writing this equation for a series of interior nodes along the
+axis of the column yields a homogeneous system of equations. (See 13.10 for 4
+interior-node example)
+
+Determine the eigenvalues for a 5-segment (4-interior nodes), 6-segment (5-interior
+nodes), and 10-segment (9-interior nodes). Using the modulus and moment of inertia of a
+pole for pole-vaulting (
+[http://people.bath.ac.uk/taf21/sports_whole.htm](http://people.bath.ac.uk/taf21/sports_whole.htm))
+E=76E9 Pa, I=4E-8 m^4, and L= 5m.
+
+Include a table in the `README.md` that shows the following results:
+What are the largest and smallest eigenvalues for the beam? How many eigenvalues are
+there?
+
+```
+| # of segments | largest | smallest | # of eigenvalues |
+| --- | --- | --- | --- |
+| 5 | ... | ... | ... |
+| 6 | ... | ... | ... |
+| 10 | ... | ... | ... |
+```
+
+If the segment length ($\Delta x$) approaches 0, how many eigenvalues would there be?
diff --git a/HW5/README.pdf b/HW5/README.pdf
new file mode 100644
index 0000000..fdb31a3
Binary files /dev/null and b/HW5/README.pdf differ
diff --git a/HW5/lu_tridiag.m b/HW5/lu_tridiag.m
new file mode 100644
index 0000000..1339973
--- /dev/null
+++ b/HW5/lu_tridiag.m
@@ -0,0 +1,12 @@
+function [ud,uo,lo]=lu_tridiag(e,f,g);```
+ % lu_tridiag calculates the components for LU-decomposition of a tridiagonal matrix
+ % given its off-diagonal vectors, e and g
+ % and diagonal vector f
+ % the output is
+ % the diagonal of the Upper matrix, ud
+ % the off-diagonal of the Upper matrix, uo
+ % and the off-diagonal of the Lower matrix, lo
+ % note: the diagonal of the Lower matrix is all ones
+
+end
+
diff --git a/HW5/solve_tridiag.m b/HW5/solve_tridiag.m
new file mode 100644
index 0000000..0c9e510
--- /dev/null
+++ b/HW5/solve_tridiag.m
@@ -0,0 +1,11 @@
+function x=solve_tridiag(ud,uo,lo,b);```
+ % solve_tridiag solves Ax=b for x
+ % given
+ % the diagonal of the Upper matrix, ud
+ % the off-diagonal of the Upper matrix, uo
+ % the off-diagonal of the Lower matrix, lo
+ % the vector b
+ % note: the diagonal of the Lower matrix is all ones
+
+end
+
diff --git a/HW5/spring_mass.png b/HW5/spring_mass.png
new file mode 100644
index 0000000..274c049
Binary files /dev/null and b/HW5/spring_mass.png differ
diff --git a/HW5/spring_mass.svg b/HW5/spring_mass.svg
new file mode 100644
index 0000000..b854162
--- /dev/null
+++ b/HW5/spring_mass.svg
@@ -0,0 +1,299 @@
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ image/svg+xml
+
+
+
+
+ x1 x2 x3 m1 m2 m3 k1 k2 k3 k4
+
diff --git a/HW5/test_arrays.m b/HW5/test_arrays.m
new file mode 100644
index 0000000..002d79c
--- /dev/null
+++ b/HW5/test_arrays.m
@@ -0,0 +1,8 @@
+rand("seed",1);
+
+for i = 3:10
+ e=sort(randi(6,[i-1,1]));
+ f=sort(randi(10,[i,1]));
+ g=sort(randi(9,[i-1,1]));
+ eval(['A',int2str(i),'=diag(f)+diag(e,-1)+diag(g,1);']);
+end
diff --git a/lecture_10/octave-workspace b/lecture_10/octave-workspace
index 8a9aba2..8c437bb 100644
Binary files a/lecture_10/octave-workspace and b/lecture_10/octave-workspace differ
diff --git a/lecture_12/octave-workspace b/lecture_12/octave-workspace
index 41ef164..8c437bb 100644
Binary files a/lecture_12/octave-workspace and b/lecture_12/octave-workspace differ
diff --git a/lecture_15/eigenvalues.gif b/lecture_15/eigenvalues.gif
new file mode 100644
index 0000000..1286762
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diff --git a/lecture_15/lecture_15.ipynb b/lecture_15/lecture_15.ipynb
index 2541de3..49e4b52 100644
--- a/lecture_15/lecture_15.ipynb
+++ b/lecture_15/lecture_15.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 40,
+ "execution_count": 1,
"metadata": {
"collapsed": true
},
@@ -13,7 +13,7 @@
},
{
"cell_type": "code",
- "execution_count": 41,
+ "execution_count": 2,
"metadata": {
"collapsed": true
},
@@ -22,6 +22,15 @@
"setdefaults"
]
},
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## UConn Hackathon Mar 24-25\n",
+ "\n",
+ "[https://www.hackuconn.org/](https://www.hackuconn.org/)"
+ ]
+ },
{
"cell_type": "markdown",
"metadata": {},
@@ -115,7 +124,7 @@
},
{
"cell_type": "code",
- "execution_count": 2,
+ "execution_count": 3,
"metadata": {
"collapsed": false
},
@@ -182,8 +191,8 @@
"or\n",
"\n",
"$|K-\\lambda I| = \\left|\\begin{array}{cc}\n",
- "\\left(\\frac{2k}{m_{1}}-\\omega^{2}\\right) & -\\frac{k}{m_{1}}X_{2} \\\\\n",
- "-\\frac{k}{m_{2}}X_{1} & \\left(\\frac{2k}{m_{2}}-\\omega^{2}\\right)\n",
+ "\\left(\\frac{2k}{m_{1}}-\\omega^{2}\\right) & -\\frac{k}{m_{1}} \\\\\n",
+ "-\\frac{k}{m_{2}} & \\left(\\frac{2k}{m_{2}}-\\omega^{2}\\right)\n",
"\\end{array}\\right|=0$\n",
"\n",
"where $\\lambda = \\omega^{2}$\n"
@@ -191,7 +200,7 @@
},
{
"cell_type": "code",
- "execution_count": 34,
+ "execution_count": 4,
"metadata": {
"collapsed": false
},
@@ -200,38 +209,40 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "K =\n",
+ "'eig' is a built-in function from the file libinterp/corefcn/eig.cc\n",
"\n",
- " 10 -5\n",
- " -5 10\n",
+ " -- Built-in Function: LAMBDA = eig (A)\n",
+ " -- Built-in Function: LAMBDA = eig (A, B)\n",
+ " -- Built-in Function: [V, LAMBDA] = eig (A)\n",
+ " -- Built-in Function: [V, LAMBDA] = eig (A, B)\n",
+ " Compute the eigenvalues (and optionally the eigenvectors) of a\n",
+ " matrix or a pair of matrices\n",
"\n",
- "v =\n",
+ " The algorithm used depends on whether there are one or two input\n",
+ " matrices, if they are real or complex, and if they are symmetric\n",
+ " (Hermitian if complex) or non-symmetric.\n",
"\n",
- " -0.70711 -0.70711\n",
- " -0.70711 0.70711\n",
+ " The eigenvalues returned by 'eig' are not ordered.\n",
"\n",
- "e =\n",
+ " See also: eigs, svd.\n",
"\n",
- "Diagonal Matrix\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",
- " 5 0\n",
- " 0 15\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": [
- "m=40; % mass in kg\n",
- "k=200; % spring constant in N/m\n",
- "\n",
- "K=[2*k/m,-k/m;-k/m,2*k/m]\n",
- "\n",
- "[v,e]=eig(K)"
+ "help eig"
]
},
{
"cell_type": "code",
- "execution_count": 35,
+ "execution_count": 5,
"metadata": {
"collapsed": false
},
@@ -240,59 +251,38 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "ans =\n",
- "\n",
- " -10.607\n",
- " 10.607\n",
+ "K =\n",
"\n",
- "ans =\n",
+ " 10 -5\n",
+ " -5 10\n",
"\n",
- " -10.607\n",
- " 10.607\n",
- "\n"
- ]
- }
- ],
- "source": [
- "K*v(:,2)\n",
- "e(2,2)*v(:,2)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 26,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
"v =\n",
"\n",
- " 0.78868 -0.57735 0.21132\n",
- " -0.57735 -0.57735 0.57735\n",
- " 0.21132 0.57735 0.78868\n",
+ " -0.70711 -0.70711\n",
+ " -0.70711 0.70711\n",
"\n",
"e =\n",
"\n",
"Diagonal Matrix\n",
"\n",
- " 1.2679 0 0\n",
- " 0 3.0000 0\n",
- " 0 0 4.7321\n",
+ " 5 0\n",
+ " 0 15\n",
"\n"
]
}
],
"source": [
- "[v,e]=eig(A)"
+ "m=40; % mass in kg\n",
+ "k=200; % spring constant in N/m\n",
+ "\n",
+ "K=[2*k/m,-k/m;-k/m,2*k/m]\n",
+ "\n",
+ "[v,e]=eig(K)"
]
},
{
"cell_type": "code",
- "execution_count": 32,
+ "execution_count": 6,
"metadata": {
"collapsed": false
},
@@ -303,22 +293,20 @@
"text": [
"ans =\n",
"\n",
- " 1.00000\n",
- " -0.73205\n",
- " 0.26795\n",
+ " -10.607\n",
+ " 10.607\n",
"\n",
"ans =\n",
"\n",
- " 1.00000\n",
- " -0.73205\n",
- " 0.26795\n",
+ " -10.607\n",
+ " 10.607\n",
"\n"
]
}
],
"source": [
- "A*v(:,1)\n",
- "e(1,1)*v(:,1)"
+ "K*v(:,2)\n",
+ "e(2,2)*v(:,2)"
]
},
{
@@ -364,7 +352,7 @@
},
{
"cell_type": "code",
- "execution_count": 55,
+ "execution_count": 8,
"metadata": {
"collapsed": false
},
@@ -411,7 +399,7 @@
},
{
"cell_type": "code",
- "execution_count": 57,
+ "execution_count": 9,
"metadata": {
"collapsed": false
},
@@ -563,6 +551,39 @@
""
]
},
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "e =\n",
+ "\n",
+ " -10\n",
+ " 0\n",
+ " 10\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "e=eig(s)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": []
+ },
{
"cell_type": "code",
"execution_count": null,
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diff --git a/lecture_16/lecture_16.ipynb b/lecture_16/lecture_16.ipynb
new file mode 100644
index 0000000..74bdda2
--- /dev/null
+++ b/lecture_16/lecture_16.ipynb
@@ -0,0 +1,54 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Curve-Fitting\n",
+ "## Linear Regression\n",
+ "\n",
+ "Often, we have a model with empirical parameters. (e.g. Young's modulus, Poisson's ratio, drag coefficient, coefficient of restitution, spring constant)\n",
+ "\n",
+ "Experimental measurements are prone to a number of stochastic (random) factors:\n",
+ "\n",
+ "- Environmental noise\n",
+ " \n",
+ "- Measurement Uncertainty\n",
+ " \n",
+ "- Factors not accounted in model (e.g. 2D effects of 1D approximation)\n",
+ "\n",
+ "These can lead to **noise** (lack of precision) and **bias** (lack of accuracy)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "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_16/octave-workspace b/lecture_16/octave-workspace
new file mode 100644
index 0000000..8c437bb
Binary files /dev/null and b/lecture_16/octave-workspace differ
+