3 - a and b

3 a and b

README.md

@@ -24,3 +24,23 @@ cat_cable = @(T) -35+T/10.*cosh(10./T*30)+30-T/10;
 |bisect     | 900, 910 |  0.00001| 17 |
 ```
 ![Powerline Plot](./problem1c.png)
+
+Question 3
+```matlab
+f = @(x) (x-1)*exp(-(x-1)^2);
+df = @(x) -(2*(x)^2-4*x+1)*exp(-(x-1)^2);
+[root,ea,iter]=newtraph(f,df,3,.00001,5)
+```
+
+```matlab
+### divergence of Newton-Raphson method
+
+| iteration | x_i | approx error |
+| --- | --- | --- |
+| 0 | 3 |    n/a    |
+| 1 | 3 |   8.6957  |
+| 2 | 3 |   6.8573  |
+| 3 | 3 |   5.7348  |
+| 4 | 3 |   4.9605  |
+| 5 | 3 |   4.3873  |
+```

bisect_cat.m

@@ -1,2 +1,4 @@
 cat_cable = @(T) -35+T/10.*cosh(10./T*30)+30-T/10;
-[root,fx,ea,iter]=bisect(cat_cable,900,910,0.00001)
+[root,fx,ea,iter]=bisect(cat_cable,900,910,0.00001)
+
+f = @(x) (x-1)*exp(-(x-1)^2)

newtraph.m

@@ -0,0 +1,29 @@
+function [root,ea,iter]=newtraph(func,dfunc,xr,es,maxit,varargin)
+% newtraph: Newton-Raphson root location zeroes
+% [root,ea,iter]=newtraph(func,dfunc,xr,es,maxit,p1,p2,...):
+%   uses Newton-Raphson method to find the root of func
+% input:
+%   func = name of function
+%   dfunc = name of derivative of function
+%   xr = initial guess
+%   es = desired relative error (default = 0.0001%)
+%   maxit = maximum allowable iterations (default = 50)
+%   p1,p2,... = additional parameters used by function
+% output:
+%   root = real root
+%   ea = approximate relative error (%)
+%   iter = number of iterations
+if nargin<3,error('at least 3 input arguments required'),end
+if nargin<4 || isempty(es),es=0.0001;end
+if nargin<5 || isempty(maxit),maxit=50;end
+iter = 0;
+while (1)
+  xrold = xr;
+  xr = xr - func(xr)/dfunc(xr);
+  iter = iter + 1;
+  if xr ~= 0 
+    ea = abs((xr - xrold)/xr) * 100;
+  end
+  if ea <= es || iter >= maxit, break, end
+end
+root = xr;

newtraph_f.m

@@ -0,0 +1,3 @@
+f = @(x) (x-1)*exp(-(x-1)^2);
+df = @(x) -(2*(x)^2-4*x+1)*exp(-(x-1)^2);
+[root,ea,iter]=newtraph(f,df,3,.00001,5)