It looks like those who think this is an efficient tool (Hongjun, etc.) and therefore give it a good rating, don't understand how to use sparse matrices in MATLAB. In fact, a couple of those people talk about using inv. SIGH.
In fact, this tool REALLY is considerably slower than backslash, as long as you understand what you are doing. If you have no idea what you are doing, then quite likely you are using the wrong tools in MATLAB in any event.
n = 1000000;
B = rand(n,3);
B(:,2) = B(:,2) + 1;
A = spdiags(B,-1:1,n,n);
d = rand(n,1);
a = B(:,2);
b = B(2:n,3);
c = B(1:n-1,1);
tic
xt = thomas(a,b,c,d);
toc
Elapsed time is 1.727341 seconds.
tic
xs = A\d;
toc
Elapsed time is 0.125294 seconds.
See that thomas was roughly 14 times slower than backslash on the same problem. This is a major difference. backslash was more efficient for ANY size matrix, ranging from 5 to 20 times as fast in my tests on different size matrices.
Someone will be sure to think that I've not included the time to generate the sparse matrix itself, but the call to spdiags itself took only about 0.5 seconds. So even if I combine the two calls, it still is twice as fast to use backslash. If you will do other things with the sparse matrix A, then the call to spdiags is arguably spread across all the work you do, so it is more efficient yet.
The point is, there really is no need for thomas to exist at all, as far batter tools are available in MATLAB. Use backslash. And for gods sake, do NOT use inv.

I rewrote this with all of the matrix operations in the form of operators on A(:,i) rather than on A(i,:). As a result it does large cases such as 5000 instances of a 5000x5000 array about 13x faster. Presumably this is because MATLAB stores data by columns.

My test result is the same as Tim's and Alex's. My A matrix is only 20*20 and is strictly diagonally dominant tridiagonal.
It took "thomas(A,B)" 62.7 seconds to complete 4000 time-steps, while "A\B" used only 4.5 seconds.

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