README.md

# curve_fitting
Homework 5 

#Problem 1 
A least-squares function is created that accepts a Z-matrix and dependent variable y as input and returns the vector of best-fit constants, a, the best-fit function evaluated at each point, fx, and the coefficient of determniation, r2. 

The following results of each variable are calculated for each case function:

a) y=0.3745+0.98644x+0.84564/x


|    a    |   fx   |   r2   | 
|---------|--------|--------| 
| 0.3745  | 2.2066 | 0.9996 |
| 0.9864  | 2.7702 |        | 
| 0.8456  | 3.6157 |        |   
|         | 4.5317 |        |
|         | 5.4758 |        |

b) y=-11.4887+7.143817x-1.04121 x^2+0.046676 x^3 


|    a    |   fx   |   r2   | 
|---------|--------|--------| 
|-11.4887 | 1.8321 | 0.8290 |
| 7.1438  | 3.4145 |        | 
| -1.0412 | 4.0347 |        |   
|  0.0467 | 2.9227 |        |
|         | 2.4947 |        |
|         | 3.2330 |        |
|         | 4.9595 |        |  

c) y=4.0046e^(-1.5x)+2.9213e^(-0.3x)+1.5647e^(-0.05x)


|    a    |   fx   |   r2   | 
|---------|--------|--------| 
| 4.0046  | 5.9321 | 0.9971 |
| 2.9213  | 4.5461 |        | 
| 1.5647  | 3.2184 |        |   
|         | 2.5789 |        |
|         | 2.1709 |        |
|         | 1.8726 |        |
|         | 1.6425 |        | 
|         | 1.4605 |        | 
|         | 1.1940 |        |  
    
#Problem 2 
For this problem, we look at the case where the independent variable is temperature and the dependent variable is failure (1=fail, 0=pass). A function called cost_logistic.m takes the vector a, independent variable x and dependent variable y so that the output is [J,grad] or [cost, gradient]. Also, we solved for a0 and a1 on part b and plotted the data in part c. 

a) 
[J, grad] = cost_logistic(a, x, y) 

J = 115.5085

grad = 5.0130 

b) The result for a0 and a1 generated through cost_logistic

c) The plot is located in the repository under Bestfitgraph.PNG 
    
#Problem 3 

The function boussinesq_lookup.m is writtent so that when you enter a force , q, dimensions of rectangular area a, b, and depth, z, it uses a third-order polynomial interpolation of the four closest values of m to determine the stress in the vertical direction as shown in the file in the repository. 

b) The boussinesq_lookup.m code is copied to a file called boussinesq_spline.m using a cubic spline to interpolate in two dimensions, both m and n, that returns sigma_z.