Merge pull request #6 from rcc02007/master

Update base repo

HW3/README.md

@@ -0,0 +1,73 @@
+# Homework #3
+## due 2/15/17 by 11:59pm
+
+
+1. Create a new github repository called 'roots_and_optimization'. 
+
+    a. Add rcc02007 and pez16103 as collaborators.
+
+    b. Clone the repository to your computer.
+
+    c. Copy your `projectile.m` function into the 'roots_and_optimization' folder.
+    *Disable the plotting routine for the solvers*
+
+    d. Use the four solvers `falsepos.m`, `bisect.m`, `newtraph.m` and `mod_secant.m`
+    to solve for the angle needed to reach h=1.72 m, with an initial speed of 15 m/s. 
+
+    e. The `newtraph.m` function needs a derivative, calculate the derivative of your
+    function with respect to theta, `dprojectile_dtheta.m`. This function should
+    output d(h)/d(theta). 
+
+
+    f. In your `README.md` file, document the following under the heading `#
+    Homework #3`:
+
+        i. Compare the number of iterations that each function needed to reach an
+        accuracy of 0.00001%. Include a table in your README.md with:
+
+        ```
+        | solver | initial guess(es) | ea | number of iterations|
+        | --- | --- | --- | --- |
+        |falsepos   |  |  |  |
+        |bisect     |  |  |  |
+        |newtraph   |  |  |  |
+        |mod_secant |  |  |  |
+        ```
+
+        ii. Compare the convergence of the 4 methods. Plot the approximate error vs the
+        number of iterations that the solver has calculated. Save the plot as
+        `convergence.png` and display the plot in your `README.md` with:
+
+        `![Plot of convergence for four numerical solvers.](convergence.png)`
+
+        iii. In the `README.md` provide a description of the files used to create the
+        table and the convergence plot. 
+
+2. The Newton-Raphson method and the modified secant method do not always converge to a
+solution. One simple example is the function f(x) = x*exp(-x^2). The root is at 0, but
+using the numerical solvers, `newtraph.m` and `mod_secant.m`, there are certain initial
+guesses that do not converge. 
+
+    a. Calculate the first 5 iterations for the Newton-Raphson method with an initial
+    guess of x_i=2 for f(x)=x*exp(-x^2).
+
+    b. Add the results to a table in the `README.md` with:
+
+    ```
+    ### divergence of Newton-Raphson method
+
+    | iteration | x_i | approx error |
+    | --- | --- | --- |
+    | 0 | 2 | n/a |
+    | 1 |   |     |
+    | 2 |   |     |
+    | 3 |   |     |
+    | 4 |   |     |
+    | 5 |   |     |
+    ```
+
+    c. Repeat steps a-b for an initial guess of 0.2. (But change the heading from
+    'divergence' to 'convergence')
+
+3. Commit your changes to your repository. Sync your local repository with github. Then
+copy and paste the "clone URL" into the following Google Form [Homework 3](https://goo.gl/forms/UJBGwp0fQcSxImkq2)

README.md

@@ -74,8 +74,8 @@ general, I will not post homework solutions.
 |3|1/31||Consistent Coding habits|
 |   |2/2|5|Root Finding| 
 |4|2/7|6|Root Finding con’d|      
-|   |2/9|7|Optimization|      
-|5|2/14||Intro to Linear Algebra|
+|   |2/9|7| **Snow Day**|      
+|5|2/14|| Optimization |
 |   |2/16|8|Linear Algebra|
 |6|2/21|9|Linear systems: Gauss elimination|
 |   |2/23|10|Linear Systems: LU factorization|

lecture_06/bisect.m

@@ -0,0 +1,37 @@
+function [root,fx,ea,iter]=bisect(func,xl,xu,es,maxit,varargin)
+% bisect: root location zeroes
+% [root,fx,ea,iter]=bisect(func,xl,xu,es,maxit,p1,p2,...):
+% uses bisection method to find the root of func
+% input:
+% func = name of function
+% xl, xu = lower and upper guesses
+% es = desired relative error (default = 0.0001%)
+% maxit = maximum allowable iterations (default = 50)
+% p1,p2,... = additional parameters used by func
+% output:
+% root = real root
+% fx = function value at root
+% ea = approximate relative error (%)
+% iter = number of iterations
+if nargin<3,error('at least 3 input arguments required'),end
+test = func(xl,varargin{:})*func(xu,varargin{:});
+if test>0,error('no sign change'),end
+if nargin<4|isempty(es), es=0.0001;end
+if nargin<5|isempty(maxit), maxit=50;end
+iter = 0; xr = xl; ea = 100;
+while (1)
+  xrold = xr;
+  xr = (xl + xu)/2;
+  iter = iter + 1;
+  if xr ~= 0,ea = abs((xr - xrold)/xr) * 100;end
+  test = func(xl,varargin{:})*func(xr,varargin{:});
+  if test < 0
+    xu = xr;
+  elseif test > 0
+    xl = xr;
+  else
+    ea = 0;
+  end
+  if ea <= es | iter >= maxit,break,end
+end
+root = xr; fx = func(xr, varargin{:});

lecture_06/falsepos.m

@@ -0,0 +1,39 @@
+function [root,fx,ea,iter]=bisect(func,xl,xu,es,maxit,varargin)
+% bisect: root location zeroes
+% [root,fx,ea,iter]=bisect(func,xl,xu,es,maxit,p1,p2,...):
+% uses bisection method to find the root of func
+% input:
+% func = name of function
+% xl, xu = lower and upper guesses
+% es = desired relative error (default = 0.0001%)
+% maxit = maximum allowable iterations (default = 50)
+% p1,p2,... = additional parameters used by func
+% output:
+% root = real root
+% fx = function value at root
+% ea = approximate relative error (%)
+% iter = number of iterations
+if nargin<3,error('at least 3 input arguments required'),end
+test = func(xl,varargin{:})*func(xu,varargin{:});
+if test>0,error('no sign change'),end
+if nargin<4|isempty(es), es=0.0001;end
+if nargin<5|isempty(maxit), maxit=50;end
+iter = 0; xr = xl; ea = 100;
+while (1)
+  xrold = xr;
+  xr = (xl + xu)/2;
+  % xr = (xl + xu)/2; % bisect method
+  xr=xu - (f_m(xu)*(xl-xu))/(f_m(xl)-f_m(xu)); % false position method
+  iter = iter + 1;
+  if xr ~= 0,ea = abs((xr - xrold)/xr) * 100;end
+  test = func(xl,varargin{:})*func(xr,varargin{:});
+  if test < 0
+    xu = xr;
+  elseif test > 0
+    xl = xr;
+  else
+    ea = 0;
+  end
+  if ea <= es | iter >= maxit,break,end
+end
+root = xr; fx = func(xr, varargin{:});

lecture_06/incsearch.m

@@ -0,0 +1,37 @@
+function xb = incsearch(func,xmin,xmax,ns)
+% incsearch: incremental search root locator
+%   xb = incsearch(func,xmin,xmax,ns):
+%     finds brackets of x that contain sign changes
+%     of a function on an interval
+% input:
+%   func = name of function
+%   xmin, xmax = endpoints of interval
+%   ns = number of subintervals (default = 50)
+% output:
+%   xb(k,1) is the lower bound of the kth sign change
+%   xb(k,2) is the upper bound of the kth sign change
+%   If no brackets found, xb = [].
+if nargin < 3, error('at least 3 arguments required'), end
+if nargin < 4, ns = 50; end %if ns blank set to 50
+% Incremental search
+x = linspace(xmin,xmax,ns);
+f = func(x);
+nb = 0; xb = []; %xb is null unless sign change detected
+%for k = 1:length(x)-1
+%  if sign(f(k)) ~= sign(f(k+1)) %check for sign change
+%    nb = nb + 1;
+%    xb(nb,1) = x(k);
+%    xb(nb,2) = x(k+1);
+%  end
+%end
+sign_change = diff(sign(f));
+[~,i_change] = find(sign_change~=0);
+nb=length(i_change);
+xb=[x(i_change)',x(i_change+1)'];
+
+if isempty(xb) %display that no brackets were found
+  fprintf('no brackets found\n')
+  fprintf('check interval or increase ns\n')
+else
+  fprintf('number of brackets: %i\n',nb) %display number of brackets
+end