updates

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_07/lecture_07.md

@@ -18,20 +18,22 @@ $f'(x_{i})=\frac{f(x_{i})-0}{x_{i}-x_{i+1}}$
 
 $x_{i+1}=x_{i}-\frac{f(x_{i})}{f'(x_{i})}$
 
-Use Newton-Raphson to find solution when $e^{x}=x$
+Use Newton-Raphson to find solution when $e^{-x}=x$
 
 
 ```octave
 f= @(x) exp(-x)-x;
 df= @(x) -exp(-x)-1;
 
 x_i= 0;
+x_r = x_i-f(x_i)/df(x_i)
 error_approx = abs((x_r-x_i)/x_r)
-x_r=x_i;
+x_i=x_r;
 
 ```
 
-    error_approx =  1
+    x_r =  0.50000
+    error_approx =  1
 
 
 
@@ -41,8 +43,8 @@ error_approx = abs((x_r-x_i)/x_r)
 x_i=x_r;
 ```
 
-    x_r =  0.50000
-    error_approx =  1
+    x_r =  0.56631
+    error_approx =  0.11709
 
 
 
@@ -52,8 +54,8 @@ error_approx = abs((x_r-x_i)/x_r)
 x_i=x_r;
 ```
 
-    x_r =  0.56631
-    error_approx =  0.11709
+    x_r =  0.56714
+    error_approx =  0.0014673
 
 
 
@@ -64,7 +66,7 @@ x_i=x_r;
 ```
 
     x_r =  0.56714
-    error_approx =  0.0014673
+    error_approx =    2.2106e-07
 
 
 In the bungee jumper example, we created a function f(m) that when f(m)=0, then the mass had been chosen such that at t=4 s, the velocity is 36 m/s. 
@@ -90,10 +92,12 @@ df_m = @(m) 1/2*sqrt(g./m/c_d).*tanh(sqrt(g*c_d./m)*t)-g/2./m*sech(sqrt(g*c_d./m
 
 
 ```octave
-newtraph(f_m,df_m,140,0.00001)
+[root,ea,iter]=newtraph(f_m,df_m,140,0.00001)
 ```
 
-    ans =  142.74
+    root =  142.74
+    ea =    8.0930e-06
+    iter =  48
 
 
 ## Secant Methods
@@ -119,32 +123,46 @@ $x_{i+1}=x_{i}-\frac{f(x_{i})(\delta x_{i})}{f(x_{i}+\delta x_{i})-f(x_{i})}$
 
 
 ```octave
-mod_secant(f_m,1e-6,50,0.00001)
+[root,ea,iter]=mod_secant(f_m,1,50,0.00001)
 ```
 
-    ans =  142.74
+    root =  142.74
+    ea =    3.0615e-07
+    iter =  7
 
 
 
 ```octave
-Amt_numerical=mod_secant(@(A) car_payments(A,30000,0.05,5),1e-6,50,0.001)
-car_payments(Amt,30000,0.05,5)
+car_payments(400,30000,0.05,5,1)
 ```
 
-    error: 'plot_bool' undefined near line 12 column 6
-    error: called from
-        car_payments at line 12 column 3
-        mod_secant at line 22 column 8
-    error: 'Amt' undefined near line 1 column 14
-    error: evaluating argument list element number 1
+    ans =    1.1185e+04
+
+
+
+![svg](lecture_07_files/lecture_07_15_1.svg)
 
 
 
 ```octave
-Amt*12*5
+Amt_numerical=mod_secant(@(A) car_payments(A,700000,0.0875,30,0),1e-6,50,0.001)
+car_payments(Amt_numerical,700000,0.0875,30,1)
 ```
 
-    error: 'Amt' undefined near line 1 column 1
+    Amt_numerical =  5467.0
+    ans =    3.9755e-04
+
+
+
+![svg](lecture_07_files/lecture_07_16_1.svg)
+
+
+
+```octave
+Amt_numerical*12*30
+```
+
+    ans =    1.9681e+06
 
 
 Amortization calculation makes the same calculation for the monthly payment amount, A, paying off the principle amount, P, over n pay periods with monthly interest rate, r. 
@@ -257,12 +275,13 @@ ea_ms=zeros(1,N); % appr error Modified Secant
 ea_fp=zeros(1,N); % appr error false point method
 ea_bs=zeros(1,N); % appr error bisect method
 for i=1:length(iterations)
-    [root_nr,ea_nr(i),iter_nr]=newtraph(f_m,df_m,200,0,iterations(i));
+    [root_nr,ea_nr(i),iter_nr]=newtraph(f_m,df_m,300,0,iterations(i));
     [root_ms,ea_ms(i),iter_ms]=mod_secant(f_m,1e-6,300,0,iterations(i));
     [root_fp,ea_fp(i),iter_fp]=falsepos(f_m,1,300,0,iterations(i));
     [root_bs,ea_bs(i),iter_bs]=bisect(f_m,1,300,0,iterations(i));
 end
-        
+
+setdefaults
 semilogy(iterations,abs(ea_nr),iterations,abs(ea_ms),iterations,abs(ea_fp),iterations,abs(ea_bs))
 legend('newton-raphson','mod-secant','false point','bisection')
 ```
@@ -281,31 +300,27 @@ legend('newton-raphson','mod-secant','false point','bisection')
 
 
 
-![svg](lecture_07_files/lecture_07_22_1.svg)
+![svg](lecture_07_files/lecture_07_24_1.svg)
 
 
 
 ```octave
-ea_ms
+ea_nr
 ```
 
-    ea_ms =
+    ea_nr =
 
-     Columns 1 through 6:
+     Columns 1 through 8:
 
-       2.3382e+03   1.9171e-14   1.9171e-14   1.9171e-14   1.9171e-14   1.9171e-14
+       6.36591   0.06436   0.00052   0.00000   0.00000   0.00000   0.00000   0.00000
 
-     Columns 7 through 12:
+     Columns 9 through 16:
 
-       1.9171e-14   1.9171e-14   1.9171e-14   1.9171e-14   1.9171e-14   1.9171e-14
+       0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000
 
-     Columns 13 through 18:
+     Columns 17 through 20:
 
-       1.9171e-14   1.9171e-14   1.9171e-14   1.9171e-14   1.9171e-14   1.9171e-14
-
-     Columns 19 and 20:
-
-       1.9171e-14   1.9171e-14
+       0.00000   0.00000   0.00000   0.00000
 
 
 
@@ -344,32 +359,28 @@ legend('newton-raphson','mod-secant','false point','bisection')
 
 
 
-![svg](lecture_07_files/lecture_07_24_1.svg)
+![svg](lecture_07_files/lecture_07_26_1.svg)
 
 
 
 ```octave
-ea_bs
+ea_nr
 newtraph(f,df,0.5,0,12)
 ```
 
-    ea_bs =
-
-     Columns 1 through 6:
+    ea_nr =
 
-       9.5357e+03  -4.7554e-01  -2.1114e-01   6.0163e-02  -2.4387e-03   6.1052e-04
+     Columns 1 through 7:
 
-     Columns 7 through 12:
+       99.03195   11.11111   11.11111   11.11111   11.11111   11.11111   11.11111
 
-       2.2891e-04  -9.5367e-06   2.3842e-06   8.9407e-07  -2.2352e-07   9.3132e-09
+     Columns 8 through 14:
 
-     Columns 13 through 18:
+       11.11111   11.11111   11.11111   11.11109   11.11052   11.10624   10.99684
 
-      -2.3283e-09  -8.7311e-10   3.6380e-11  -9.0949e-12  -3.4106e-12   8.5265e-13
+     Columns 15 through 20:
 
-     Columns 19 and 20:
-
-      -3.5527e-14   8.8818e-15
+        8.76956    2.12993    0.00000    0.00000    0.00000    0.00000
 
     ans =  16.208
 
@@ -383,6 +394,55 @@ df(300)
 
 
 
+```octave
+% our class function
+f= @(x) tan(x)-(x-1).^2
+mod_secant(f,1e-3,1)
+```
+
+    f =
+
+    @(x) tan (x) - (x - 1) .^ 2
+
+    ans =  0.37375
+
+
+
+```octave
+f(ans)
+```
+
+    ans =   -3.5577e-13
+
+
+
+```octave
+tan(0.37375)
+(0.37375-1)^2
+```
+
+    ans =  0.39218
+    ans =  0.39219
+
+
+
+```octave
+f([0:10])
+```
+
+    ans =
+
+     Columns 1 through 8:
+
+       -1.0000    1.5574   -3.1850   -4.1425   -7.8422  -19.3805  -25.2910  -35.1286
+
+     Columns 9 through 11:
+
+      -55.7997  -64.4523  -80.3516
+
+
+
+
 ```octave
 
 ```