College Math Teaching

May 23, 2014

Math before symbolic notation had been invented…

Filed under: editorial, elementary mathematics, media — Tags: , , — blueollie @ 7:02 pm

There is a tendency to take modern symbolic notation for granted. Here is an excellent “popular” article about the invention of algebraic notation.

This might be something nice to share with calculus students.

Blogging will be very light as I work on revisions for a paper.

May 4, 2014

How to create tridiagonal matrices in Matlab (any size)

Filed under: linear albegra, matrix algebra, numerical methods, pedagogy — Tags: , , — collegemathteaching @ 1:38 am

Suppose we wanted to create a tridiagonal matrix in Matlab and print it to a file so it would be used in a routine. For demonstration purposes: let’s create this 10 x 10 matrix:

\left( \begin{array}{cccccccccc}  1 & e^{-1} &0 & 0 & 0 &0 &0 &0 &0 &0\\  \frac{1}{4} & \frac{1}{2} & e^{-2} &0 &0 &0 &0 &0 &0 &0\\  0 &  \frac{1}{9} & \frac{1}{3} & e^{-3} & 0 &0 &0 &0 &0 & 0 \\  0 & 0 &  \frac{1}{16}  & \frac{1}{4} & e^{-4} & 0 &0 &0 &0 &0 \\  0 & 0 & 0 & \frac{1}{25} & \frac{1}{5} & e^{-5} & 0 &0 &0 &0 \\  0 & 0 & 0 & 0 & \frac{1}{36} & \frac{1}{6} & e^{-6} & 0 & 0 &0 \\  0 & 0 & 0 & 0 & 0 & \frac{1}{49} & \frac{1}{7} & e^{-7} & 0 & 0 \\  0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{64} & \frac{1}{8} & e^{-8} & 0 \\  0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{81} & \frac{1}{9} & e^{-9} \\  0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{100} & \frac{1}{10}   \end{array} \right)

To take advantage of Matlab’s “sparse” command we should notice the pattern of the entries.
The diagonals: m_{(i,i)} = \frac{1}{i} for i \in \{1, 2, ...10 \} .
The “right of diagonal” entries: m_{(i, i+1)} = e^{-i} for i \in \{1, 2, ...9 \}
The “left of diagonal” entries: m_{(i-1, i)} = \frac{1}{i^2} for i \in \{2, 3, ...10 \}

Now we need to set up Matlab vectors to correspond to these indices.

First, we need to set up a vector for the “row index” entries for: the diagonal entries, the right of diagonal entries, then the left of diagonal entries.
One way to do this is with the command “i = [1:10 1:9 2:10]; ” (without the quotes, of course).
What this does: it creates a list of indices for the row value of the entries: 1, 2, 3,…10 for the diagonals, 1, 2, 3, …9 for the right of diagonals, and 2, 3, …10 for the left of diagonals.

Now, we set up a vector for the “column index” entries for: the diagonal entries, the right of diagonal entries and the left of diagonal entries.
We try: “j = [1:10 2:10 1:9]; ”
What this does: it creates a list of indices for the column value of the entries: 1, 2, 3, ..10 for the diagonals, 2, 3, …10 for the right of diagonals and 1, 2, …9 for the left of diagonals.

As a pair, (i,j) goes (1,1), (2,2), …(10,10), (1,2), (2,3), ….(9, 10), (2,1), (3,2)…..(10, 9).

Now, of course, we need to enter the desired entries in the matrix: we want to assign \frac{1}{i} to entry (i,i), \frac{1}{i^2} to entry (i, i-1) and e^{-i} to entry (i, i+1).

So we create the following vectors: I’ll start with “MD = 1:10 ” to get a list 1, 2, 3, ..10 and then “D = 1./M” to get a vector with the reciprocal values. To get the left of diagonal values, use “ML = 2:10” and then “L = 1./ML.^2 “. Now get the list of values for the right of diagonal entries: “MU = 1:9” then “U = exp(-MU)”.

Now we organize the values as follows: “s = [D L U]”. This provides a vector whose first 10 entries are D, next 9 are L and next 9 are U (a list concatenation) which is in one to one correspondence with (i,j).

We can then generate the matrix with the command (upper/lower cases are distinguished in Matlab): “S =sparse(i, j, s)”.

What this does: this creates a matrix S and assigns each (i,j) entry the value stored in s that corresponds to it. The remaining entries are set to 0 (hence the name “sparse”).

Let’s see how this works:

tridiag1

(click to see a larger size)

Now what does this put into the matrix S?

tridag2

Note how the non-zero entries of S are specified; nothing else is.

Now suppose you want to store this matrix in a file that can be called by a Matlab program. One way is to write to the file with the following command:

“dlmwrite(‘c:\matlabr12\work\tridag.dat’, S , ‘ ‘)”

The first entry tells what file to send the entries to (this has to be created ahead of time). The next is the matrix (in our case, S) and the last entry tells how to delineate the entries; the default is a “comma” and the Matlab programs I am using requires a “space” instead, hence the specification of ‘ ‘ .

Now this is what was produced by this process:

tridiag3

Suppose now, you wish to produce an augmented matrix. We have to do the following:
add row indices (which, in our case, range from 1 to 10), column indices (column 11 for each row) and the augmented entries themselves, which I’ll call 1, 2, 3, …10.

Here is the augmenting vector:

>> B = [1 2 3 4 5 6 7 8 9 10];

Here is how we modify i, j, and s:

>> i =[1:10 2:10 1:9 1:10];
>> j = [1:10 1:9 2:10 11*ones(1,10)];
>> s = [D L U B];
>> S = sparse(i, j, s);

For “i”: we append 1:10 which gives rows 1 through 10.
For “j”: we created a vector of all 11’s as each augmented entry will appear in the 11’th column of each row.
That is, to our (i,j) list we added (1,11), (2,11), (3, 11), ….(10, 11) to the end.
Now we add B to our S: the list of non-zero matrix entries.

Then

“dlmwrite(‘c:\matlabr12\work\tridage.dat’, S , ‘ ‘)”

produces:

tridag4

So now you are ready to try out matrix manipulation and algebra with (relatively) large size matrices; at least sparse ones.

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