Appendix: Matlab Examples

Here's how Fig. 6.1 was generated in Matlab:

>> x = [2 3];                  % coordinates of x
>> origin = [0 0];             % coordinates of the origin
>> xcoords = [origin(1) x(1)]; % plot() expects coordinate lists, not endpoints
>> ycoords = [origin(2) x(2)];
>> plot(xcoords,ycoords);      % Draw a line from origin to x

Mathematica can plot a list of ordered pairs:

In[1]:
    ListPlot[{{0,0},{2,3}},PlotJoined->True]; (* Draw a line from (0,0) to (2,3) *)

In Matlab, the mean of the row-vector can be computed as

or by using the built-in function mean().

In Matlab, if x = [x1 ... xN] is a row vector, we can compute thetotal energy as

Matlab has a function orth() which will compute an orthonormal basis for a space given any set of vectors which span the space.

>> help orth

ORTH   Orthogonalization.
Q = orth(A) is an orthonormal basis for the range of A.
Q’*Q = I, the columns of Q span the same space as the columns
of A and the number of columns of Q is the rank of A.

    See also QR, NULL.</pre>Below is an example of using <tt>orth()</tt> to orthonormalize a <a href="http://mathworld.wolfram.com/linearlyindependent.php">linearly


independent basis set for :

% Demonstration of the Matlab function orth() for
% taking a set of vectors and returning an orthonormal set
% which span the same space.
v1 = [1; 2; 3];  % our first basis vector (a column vector)
v2 = [1; -2; 3]; % a second, linearly independent column vector
v1’ * v2         % show that v1 is not orthogonal to v2

ans =



<< Previous page  TOC  INDEX  Next page >>


V = [v1,v2]      % Each column of V is one of our vectors

V =

 1     1
 2    -2
 3     3


W = orth(V)      % Find an orthonormal basis for the same space

W =

0.2673    0.1690
0.5345   -0.8452
0.8018    0.5071


w1 = W(:,1)      % Break out the returned vectors

w1 =

0.2673
0.5345
0.8018


w2 = W(:,2)

w2 =

0.1690


-0.8452
0.5071

w1’ * w2  % Check that w1 is orthogonal to w2 (to working precision)

ans =

2.5723e-17

w1’ * w1  % Also check that the new vectors are unit length in 3D

ans =


w2’ * w2

ans =


W’ * W    % faster way to do the above checks (matrix multiplication)

ans =

1.0000    0.0000
0.0000    1.0000


% Construct some vector x in the space spanned by v1 and v2:
x = 2 * v1 - 3 * v2

x =

-1
10
-3


% Show that x is also some linear combination of w1 and w2:
c1 = x’ * w1      % Coefficient of projection of x onto w1

c1 =

2.6726


c2 = x’ * w2      % Coefficient of projection of x onto w2

c2 =

-10.1419

xw = c1 * w1 + c2 * w2  % Can we make x using w1 and w2?

xw =

-1.0000
10.0000
-3.0000

error = x - xw

error =

1.0e-14 *

0.1332
     0
     0<a href="https://ccrma.stanford.edu/~jos//interpolation/lp_norms.php">norm</a>(error)       % typical way to summarize a vector error


ans =

1.3323e-15

% It works!

% Now, construct some vector x NOT in the space spanned by v1 and v2:
y = [1; 0; 0];     % Almost anything we guess in 3D will work

%  Try to express y as a linear combination of w1 and w2:
c1 = y’ * w1;      % Coefficient of projection of y onto w1
c2 = y’ * w2;      % Coefficient of projection of y onto w2
yw = c1 * w1 + c2 * w2  % Can we make y using w1 and w2?

yw =

0.1000
0.0000
0.3000


yerror = y - yw

yerror =

0.9000
0.0000


-0.3000

norm(yerror)

ans =

0.9487


% While the error is not zero, it is the smallest possible
% error in the least squares sense.
% That is, yw is the optimal least-squares approximation
% to y in the space spanned by v1 and v2 (w1 and w2).
% In other words, norm(yerror) <= norm(y-yw2) for any other vector yw2 made
% using a linear combination of v1 and v2.

% In yet other words, we obtain the optimal least squares approximation
% of y (which lives in 3D) in some subspace W (a 2D subspace of 3D)
% by projecting y orthogonally onto the subspace W to get yw as above.
%
% An important property of the optimal least-squares approximation
% is that the approximation error is orthogonal to the the subspace
% in which the approximation lies.  Let’s show this:

W’ * yerror % must be zero to working precision

ans =

1.0e-16 *

-0.2574
-0.0119

Appendix: Matlab Examples

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Appendix: Matlab Examples

Appendix: Matlab Examples