% sample.m   
% evaluates Pn(x1), where the user is prompted to first give the 
% degree of the polynomial, then the coefficients in increasing 
% powers of x, and finally the value of x1. 

% It also plots the polynomial using the function nested.m 

% We will use this type of program later, but for now, this program 
% merely demonstrates some of the bells and whistles associated with 
% programming in matlab. 

clear  % this clears all variables from memory

%%%%%  getting user input %%%%%%%
degree = input('what is the degree of the polynomial?  ');
n = degree + 1; % the number of coefficients 

for i = 1:n
disp(['what is the order ' num2str(i-1) ' coefficient']);
a(i) = input('');
end

x1 = input('what is the value of x1? \n');


%%%%%%%%%%%%%%% evaluating Pn(x1) the long way = sol1 %%%%%
sol1 = a(1);
for i = 1:degree
sol1 = sol1 +  a(i+1) * x1^(i);
end


%%%%%%%%%%%%%%  evaluating Pn(x1) with nested multiplication = sol2 %%%%
% Make sure nested.m can be found by MATLAB
sol2 = nested(a,x1);


%%%%%%%% printing results to the screen %%%%%%%%%%%%%%
disp(['Pn(' num2str(x1) ') using the original polynomial = '...
num2str(sol1,16)]);
disp(['Pn(' num2str(x1) ') using nested multiplication = ' ...
num2str(sol2,16)]);


%%%%%%%%%%%%%%%%% Displaying the graph  %%%%%%%%%%%%%%%%%
disp(['press any key to plot the polynomial']);
pause  % this pauses the program until a key is hit

%%%% getting more user input %%%%%%
xleft = input('from x = \n');
xright = input('to x = \n'); 
pts = input('plotting how many points? \n');

%%%%%% creating the x and y vectors %%%%%%%%%
x=xleft:(xright-xleft)/pts:xright;
for i = 1:pts+1
  y(i) = nested(a,x(i));  % calls the function nested.m 
end

%%%%%%%%%%%%%%%%%  plotting and labelling %%%%%%
plot(x,y,'ro')
 xlabel('x');
 ylabel('Pn(x)');
 title(['The graph of Pn(x) from x =' num2str(xleft), ' to ' num2str(xright)]);