Please help me come up with some good examples



A user can change the functions that add and multiply matrix elements.
I once tried unsuccessfully to figure out how to do this with one of
the other CAS.  At first thought, re-defining these functions might
not seem useful, but it is.  Here is one application.

Let m be the matrix of a weighted graph. Think of m as giving the
hiway distances between cities. If no road directly connects two
cities, let the matrix element be infinity. The diagonal elements of
m all vanish. We can compute the shortest paths between cities by
re-defining the functions that add and multiply matrix elements.

[barton@localhost orthopoly-0.93]$ maxima
Maxima 5.9.0rc3 http://maxima.sourceforge.net
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
This is a development version of Maxima. The function bug_report()
provides bug reporting information.
(C1) matrix_element_add : lambda([[x]], apply(min,x));

(D1)               LAMBDA([[x]], APPLY(MIN, x))
(C2) matrix_element_mult : lambda([x,y], if (x = inf) or (y = inf) then inf
else x+y);

(D2)     LAMBDA([x, y], IF x = INF OR y = INF THEN INF ELSE x + y)
(C3) m : matrix([0,3,5],[inf,0,8],[7,inf,0]);

                        [  0    3   5 ]
                        [           ]
(D3)                      [ INF   0   8 ]
                        [           ]
                        [  7   INF  0 ]


The shortest paths via two roads is given by

(C4) m.m;

                         [ 0   3   5 ]
                         [         ]
(D4)                     [ 15  0   8 ]
                         [         ]
                         [ 7   10  0 ]

The shortest paths via three roads is given by

(C5) m.m.m;

                         [ 0   3   5 ]
                         [         ]
(D5)                     [ 15  0   8 ]
                         [         ]
                         [ 7   10  0 ]
(C6) quit();

We could find the longest path by using max instead of min.  Cool.