echo on
% This causes all steps and commenst to be shown/displayed.
%
%
% This is a commented version (w/ a few add-ons) of the
% file hello.m
% The "problem" is to fit (interpolate) a number of data
% points with a polynomial function. There are built-in
% MATLAB functions (see at the very end) that complete
% this task with a few key-strokes ... but here the idea
% is to learn the basic commands on a reasonable problem.
%
% The first item is to  learn about m-files, scripts, and
% setting the path so that MATLAB can find the file.....
%
% Then enter the data points -- as column vectors. Note the
% two different formats used: The x-values are equally 
% spaced and easily generated, the y-values have to be
% typed in (unless they may be read in from a file).
%
xx=[1:1:8]';
yy=[1,0,2,3,1,9,6,0]';
%
% Always "take a look", plot the data -- e.g. to enjoy the
% picture, to get an idea what is going on, to plan ahead,
% and to detect possible mistakes when entering the data.
% 
% First clear the graphics window -- clg is an alternative,
% try out "help clf" and "help clg" to find out the diffe-
% rences.
% Then plot the data -- enjoy the color and style options.
% For details look up "help plot"
clf
plot(xx,yy,'ro')
%
% The main business is to create the matrix of coefficients.
% Here we start by reserving the space -- first create a matrix
% of only ones (similar try out "help zeros" and "help eye").
%
A=ones(8,8);
%
% The second column should contain the values of the x-data.
% Assign data to a column using A(:,k)=.....
%
A(:,2)=xx;
%
% The subsequent columns contain increaisng powers of the
% values of the second column. There are many ways to create
% these data -- we use it as a typical example for "array
% multiplication" -- i.e. ".*" as opposed to "*".....
%
A(:,3)=xx.*xx;
A(:,4)=xx.*xx.*xx;
%
% A different way uses the previous column....
%
A(:,5)=A(:,4).*xx;
%
% It is easy to use CNTRL-C, CNTRL-V and editing to get the
% remaining columns -- but the last attempt suggests to simply
% use a for-loop. This is not mandatory for our class, but 
% a very basic programming construct..... it is worth learning
% the syntax for this in ANY language!
%
for i=3:8
   A(:,i)=A(:,i-1).*xx;
end
%
% Solving the system to obtain the coefficients is a real JOY,
% a minimal number of keystrokes -- dividing A (from the left) 
% into y
%
cc=A\yy
%
% These are the coefficients of the polynomial function that we
% were after! But to get a nice closure, let is overlay the
% graph of the polynomial over the data points.
% First we need to create the "vector" of x-values at which
% we want to evaluate and plot the polynomial -- we are in
% charge, MATLAB ONLY plots and connects points -- we have
% to create the table of (x,y) pairs to bve plotted. First 
% the x-values in small steps as a column vector.
%
xxx=[0.8:0.1:8.4];
%
% Then evaluate the polynomial at these x-values, i.e.
% substitute the x-values into the polynomial. 
% The following is a very crude way of doing this -- there
% are more efficient ways, but it is good for a start:
% What matters is that we can USE array-operations (either
% ".*" or ".^") to calculate all y-values from all x-values
% basically "at the same time" (both xxx and yyy are long
% column vectoirs of data).
%
yyy=cc(1)+...
   cc(2)*xxx+...
   cc(3)*xxx.^2+...
   cc(4)*xxx.^3+...
   cc(5)*xxx.^4+...
   cc(6)*xxx.^5+...
   cc(7)*xxx.^6+...
   cc(8)*xxx.^7;
%
% To overlay a plot use "hold on" -- see "help hold" for
% details, and plot the interpolating curve in your favorite
% color and style.
%
hold on
plot(xxx,yyy,'m-')
%
% Faster commands use built in functions -- strike any
% key to continue:
pause
%
% There are many matrices that occur frequently, and that
% have earned their own names. MATLAB has built-in functions
% that create most of these, it is just a matter of learning
% many names, looking around..... 
% Our matrix A is a "Vandermonde-matrix", and is easily 
% generated with a few keystrokes. However, the built-in
% function has the matrix flipped right-to-left (this
% corresponds to listing the powers c_k*x^k in the polynomial
% in decreasing rather than increasing order -- hence we "flip
% the matrix right-left"....
%
A=fliplr(vander([1:8]))
%
%Strike any key to continue:
pause
%
% But it is even easier -- the command PLOYFIT will find the
% (coefficients c_k of the) polynomial in one step.
%
ccc=polyfit(xx,yy,7)
%
% Note that these coefficients are again listed in reverse order
% (compared to our hands-on attempt). And they come as a row
% vector, rather than a column vector. Taking transposes (') and
% flipping right-to-left or up-down easily converts the format.
%
%Strike any key to continue:
pause
%
% Finally, given (the coefficients c_0, c_1,.... c_n of a 
% polynomial the function POLYVAL will evaluate the polynomial 
% at any (vector of) x-values --- i.e. there is no need to 
% laboriously type in the formula for the polynomial!
%
% The main issues are whether the coeffiencts are to be given
% in descending or ascending order of the powers, and whether
% the vectors ccc of coefficients and the vector xxx of x-values
% have to be column or row vectors -- for such issues it is best
% to quickly visit the help pages "help polyval"
%
yyy = polyval(ccc,xxx);
plot(xxx,yyy,'b-.')
%
echo off
%