README.md

|solver | initial guess(es) | ea | number of iterations|
| --- | --- | --- | --- |
|falsepos   | 1, 300 | 3.6148e-05 | 18 |
|bisect     | 1, 300 | 5.3371e-05 | 21 |
|newtraph   |  0 | 0.0015 | 400 |
|mod_secant | 0 | 1.7974e-07 | 5 |

The above iteration test, makes it so that the mod_secant and newtraph methods are the beth methods for error approximation since their is less error and more iterations. Bisect, falsepos, and mod_secant have the lowest number of iterations.  

![Plot of convergence for four numerical solvers.](convergence.png)

This is the plot for the second part of the question which illustrated the general trend of each method.

In the data above, the bisect, the mod_secant, and the falsepos required signicantly less iterations than the newtraph. 

### Divergance of Newton-Raphson method

| iteration | x_i | approx error |
| --- | --- | --- |
| 0 | 2 | n/a |
| 1 | 2.2857 | 12.5 |
| 2 | 2.5276 | 9.5703 |
| 3 | 2.7422 | 7.8262 |
| 4 | 2.9375 | 6.6491 |
| 5 | 3.1182 | 5.7943 |

For this method, I debugged the nethraphsons method and ran the iteration six times to get the error approximation. For the divergence problem, I used x_1 = 2 as the initial guess and 'ea' as the error approximation and the 'e_r old' was the x_i reading for every iteration.  

### Convergance of Newton-Raphson method

| iteration | x_i | approx error |
| --- | --- | --- |
| 0 | .2 | n/a |
| 1 | .2  | 1.25e03v|
| 2 | -0.0174  | 1.6531e05 |
| 3 | 1.0527e-5 | 4.5122e11 |
| 4 | -2.3329e-15 | 4.5122e11 |
| 5 | 0 | 4.5122e11 |

#Homework #4 

Part B) Minimum: 0.5

Part C) The minimum potential energy position is X_c = 0.9706 at theta = 0.9 degrees. This value was calculated using the for loop function in For_Loop.m.

Part D) 
Graph is included with the other files in the repository.