Newton

\% Newton Method to Solve f(x)=x*x - 4 Sin(x)=0,
\% Date: October 16,2002
\% by Yuri V Lvov
  format long;
  tolerance=.001; \%tolerance to the result
pr = 3; next=0;
while ((abs(pr*pr -  4 * sin(pr))>tolerance)),
  next=(pr-((pr*pr -  4 * sin(pr))/(2*pr-4*cos(pr))));
\% This is x[{k+1}]=x[k]-f(x)/f'(x)
pr=next;
x=[next, next^ 2-4*sin(next)]
 end
\end{vervatun}
\subsection{Secant}

\% Secant Method to Solve f(x)=0, for user supplied f(x)
 
\% Date: October 16,2002

\% by Yuri V Lvov


 format long;

x=[];x(1)=1;x(2)=2;

  tolerance=.001; %tolerance to the result

i=3;

while (abs(x(i-1)-x(i-2))>tolerance),

    x(i)=(x(i-1)- f(x(i-1))*(x(i-1)-x(i-2))/(f(x(i-1))-f(x(i-2))));

disp(i);disp(x(i));disp(' ');

     i=i+1;

end

\subsection{Exploring Interpolation}

\begin{verbatim}
\% Interpolation examples

\% adapted by Yuri Lvov 

\% Choose Number of points to play with: 


Number=10;

t=linspace(.2,2.3,Number)

y=1./t+t.$\hat{\ }$2

a=polyfit(t,y,Number-1)

x=linspace(.2,2.3,100);

f=1./x+x.$\hat{\ }$ 2;

ftilde=polyval(a,x);

plot(x,f,'r',x,ftilde,'g',t,y,'mo')

\% Shows Function, interpolation and dots on the same plot

\% Try unequal spaced points, result is strange

t=[.2 .3 .5 .6 1.6 2.3]

y=1./t+t.$\hat{\ }$ 2

a=polyfit(t,y,Number-1)

x=linspace(.2,2.3,100);

f=1./x+x.$\hat{\ }$ 2;

ftilde=polyval(a,x);

plot(x,f,'r',x,ftilde,'g',t,y,'mo')

     
Number=20;

\% This is the classical example producing wiggles

 t=linspace(-1,1,Number);


\% t=-cos((2*(1:Number)-1)*pi./(2*Number)); 

\% Uncomment the preveous  line for C. points

y=1./ (1+25* t.$\hat{\ }$ 2);

a=polyfit(t,y,Number-1);

x=linspace(-1,1,200);

f=1./(1+25*x.$\hat{\ }$ 2);

ftilde=polyval(a,x);

plot(x,f,'r',x,ftilde,'g',t,y,'mo')

\% Do the same with SPLINE interpolation

\% Type 'help spline' to get more info

\% type  'help interp1' to get even more information
  

 x = -1:.1:1;  

y = 1./(1+25*x.$\hat{\ }$ 2);

        xx = -1:.01:1;

        yy = spline(x,y,xx);

        plot(x,y,'o',xx,yy)