QUIZ 2

1. Describe Secant method as a linear interpolation method, briefly.

   Figure 1: Graphical illustration of the Secant Method.

   
   • As Figure 1 illustrates, we draw the line through these two points and find where it intersects the x-axis.
   • The intersection of the line with the x-axis is not at $x=r$ but it should be close to it. From the obvious similar triangles we can write
     
     $$\frac{(x_1 - x_2)}{f(x_1)}=\frac{ (x_0 - x_1)}{f(x_0) - f(x_1)}\Longrightarrow x_2=x_1-f(x_1)\frac{(x_0-x_1)}{f(x_0)-f(x_1)}$$
     
     
     and the general term:
     
     $$x_{n+1}=x_n-f(x_n)\frac{(x_{n-1}-x_n)}{f(x_{n-1})-f(x_n)}$$
     
     

2. Describe Newton's method for solving nonlinear eqns, briefly.

   Figure 2: Graphical illustration of the Newton's Method.

   
   • Like the previous ones, this method is also based on a linear approximation of the function, but does so using a tangent to the curve. Figure 2 gives a graphical description
   • The calculation scheme follows immediately from the right triangle shown in Fig. 2.
     
     $$tan\theta=f^{'}(x_0)=\frac{f(x_0)}{x_0-x_1}\Rightarrow x_1=x_0-\frac{f(x_0)}{f^{'}(x_0)}$$
     
     
     and the general term is:
     
     $$x_{n+1}=x_{n}-\frac{f(x_n)}{f^{'}(x_n)},n ~0,1,2,\ldots$$
     
     

   • Newton's algorithm is widely used because, it is more rapidly convergent than any of the methods discussed so far.