% what does matlab do when you want it to treat a matrix as a  vector?

>> a = sin(rand(3,4))
a =
    0.7967    0.3947    0.3989    0.3456
    0.6730    0.8049    0.7794    0.7265
    0.1754    0.7937    0.0579    0.0099
>> a(4)
ans = 0.3947

% Apparently it treats is as the column vector made up from its columns, from
% first to last.

% In fact,  a(:) is explicitly that equivalent column vector, as you can see:

>> a(:) = 1:12
a =
     1     4     7    10
     2     5     8    11
     3     6     9    12

% question from the audience: suppose you would rather have the entries of
% 1:12 put in row by row, how would you do it?
% Several possibilities:
>> b = zeros(4,3); b(:) =1:12; 
>> b'
ans =
     1     2     3     4
     5     6     7     8
     9    10    11    12

%  perhaps better:

>> reshape(a,4,3)'
ans =
     1     2     3     4
     5     6     7     8
     9    10    11    12

%  matlab commands come in (at least) three flavors: those implementing 
% functions of scalars,  like  sin, exp
% functions of vectors,  like  sum, prod, max, min, cumsum, cumprod, sort, ...
% functions of matrices, like  size, ...

% What happens if their argument is not of the `right' type?

% matlab etiquette: a function of a scalar should work on a matrix by applying
% the function entrywise.
>> sin(a)
ans =
    0.8415   -0.7568    0.6570   -0.5440
    0.9093   -0.9589    0.9894   -1.0000
    0.1411   -0.2794    0.4121   -0.5366

% matlab etiquette: a function of a vector works on a matrix by applying the
% function to column-wise:

>> sum(a)
ans =
     6    15    24    33
%  What if you would rather have it row-wise?
% Then you can say so by specifying an additional input argument as 2:
>> a
a =
     1     4     7    10
     2     5     8    11
     3     6     9    12
>> sum(a,2)
ans =
    22
    26
    30
>> max(a)
ans =
     3     6     9    12
>> max(a,2)
ans =
     2     4     7    10
     2     5     8    11
     3     6     9    12
%  Well, that's unexpected! There is an additional wrinkle with max and min:
%  max(A,B), if legal, will return the matrix whose (i,j)-entry is the 
%  max of A(i,j) and B(i,j).
% Also, remember matlab's wrinkle concerning scalars: If matlab expects a
% matrix of a certain size and is, instead, given a scalar, it will replace
% that scalar by the matrix of the expected size having all its entries 
% equal to the scalar. E.g.,
>> b = a+1
b =
     2     5     8    11
     3     6     9    12
     4     7    10    13

% So, the statement max(a,2) compared each entry of a with 2 and chose the
% bigger of the two.
%
%  To get the max for each row, you have to specify that optional second
% argument as empty, [], and put in the 2 as the optional third argument:

>> max(a,[],2)
ans =
    10
    11
    12
% be aware of cumsum, and cumprod:

>> cumsum(1:4)
ans = 1     3     6    10

>> cumprod(1:4)
ans = 1     2     6    24

% the rest of the class was devoted to polynomial interpolation, starting off
% with matlab's polyfit command:
% To get the polynomial interpolant to given data x,y, use 
%  d = length(x)-; p = polyfit(x,y,d)
% What if you use larger d???

>> x = 1:3; y = x.^4;
>> p = polyfit(x,y,4)
p = 1.0000    0.0000         0         0   -0.0000

>> p = polyfit(x,y,5)
p = 0.0000    1.0000         0         0         0    0.0000

>> p = polyfit(x,y,7)
p =
  Columns 1 through 7 
   -0.1299    0.5000         0         0         0         0         0
  Column 8 
    0.6299
>> p = polyfit(x,y,10)
p =
  Columns 1 through 7 
   -0.0253    0.0801         0         0         0         0         0
  Columns 8 through 11 
         0         0         0    0.9453
>> xx = linspace(1,3);
>> plot(xx,polyval(p,xx),x,y,'o')

% ... so, this polynomial does interpolate to the data. Let's compare it
% with the standard one:

>> ps = polyfit(x,y,length(x)-1)
ps = 25.0000  -60.0000   36.0000
>> hold on
>> plot(xx,polyval(ps,xx),'r','linew',2)
>> hold off

%  It seems that polyfit constructs the unique interpolating polynomial 
% of the form 
%  a(1)x^(d-1) + a(2) x^(d-2) + ... + a(n-1) x^(d+1-n) + a(n)

% I would have preferred the polynomial of lowest possible degree, i.e.,
% of the form
%  a(1)x^(n-1) + a(2) x^(n-2) + ... + a(n-1) x^(1) + a(n)

% The rest of the class time was spent on discussing the Newton form of the
% interpolating polynomial and the construction of divided differences, etc.

diary off