Indexing

Computational implementation of problems in $d=3$ dimensions requires careful mapping of three dimensional objects into a linear memory space. In lecture, we developed an approach to this indexing through the formation of two index matrices M and N, where

$\parbox{3in}{ \begin{equation} {\bf M} \equiv \left[ \begin{array}{c... ...s & \vdots & S_3-1 \\ \vdots & \vdots & S_3 \end{array}\right] \end{equation}}$$\parbox{3in}{ \begin{equation} {\bf N} \equiv \left[ \begin{array}{c... ...vdots & \vdots & -2 \\ \vdots & \vdots & -1 \end{array}\right] \end{equation}}$

As a specific example, if S=[3; 3; 2], then

$${\bf M}=\left[ \begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\... ...1 \\ 0 & 2 & 1 \\ 1 & 2 & 1 \\ 2 & 2 & 1 \end{array}\right]$$ (1)

This problem guides you step by step through the setup of these matrices while exploiting the interactive aspects of the octave language.

Begin by creating a file ``setup.m'' containing a single line of octave code:

S=[3; 4; 5]

Verify that you can run octave by starting octave (in the same directory with ``setup.m'') and entering ``setup'' at the octave prompt. (Don't forget to hit the enter key!) This will run the octave commands in the file ``setup.m'' Octave should respond:

S =

  3
  4
  5

Next, create the first two columns of M by adding the following lines to ``setup.m'' immediately after the statement declaring S

%# Code fragment to create columns of m1, m2, m3 indices and the matrix M
ms=[0:prod(S)-1]'; %# Count from zero to S1*S2*s3-1 in a column vector
m1=rem(ms,S(1));
m2=rem(floor(ms/S(1)),S(2));

Rerun ``setup.m'' and inspect the result by entering first just ``m1'' and then just ``m2'' at the octave command prompt. Note that you can inspect any part of the above expressions, such as ``floor(ms/S(1))'', by entering it at the command prompt. Also, by entering ``whos'', you can get a report of all of the variables currently defined and, very usefully, their dimensions.

Following the logic which led to m1 and m2, add a corresponding line to compute m3. Verify by inspection that your formula for m3 functions properly. Finally, combine each of your columns into a matrix using the octave block matrix formation

M=[m1, m2, m3];

Inspect and verify your final result by rerunning ``setup.m'' and entering ``M'' at the command prompt.

Finally, write a block of code to compute N, verifying your results in a similar way.

Hint: Compute the columns of N from the corresponding columns of M using forms such as

n1=m1-(m1>S(1)/2)*S(1);

and then combine your result into the the final matrix.