Form a huge matrix by mapping pre-existing vectors to matrix for memory saving?

I need to form a huge matrix (say million by million), but the matrix has relatively simple structure, like (not my case, just for example)

 1  2  3  4  5 10 20 30 40 50  1  2  3  4  5 10 20 30 40 50 ...
10  9  8  7  6  5  4  3  2  1 10  9  8  7  6  5  4  3  2  1 ...

Basically this matrix is composed by some pre-existing vectors with certain patterns. I do not want to physically store this matrix, so I am wondering if there is any technique that can map those vectors to the matrix? In other words, I want a matrix holding no real data but pointers to other variables. When matlab works with the matrix, it actually goes to the vectors. Possible?

Thank you.

===========================================

More specific info regarding this question:

Here is an example: There are three row vectors in workspace

v1 = [1 2 3 4 5];

v2 = [10 20 30 40 50];

v3 = [10 9 8 7 6 5 4 3 2 1];

I want a matrix A, which is composed by v1, v2 and v3. A is a 10^6 by 10^6 matrix, whose

odd row: repeat [v1 v2] for 10^5 times

even row: repeat v3 for 10^5 times

A matrix will be multiplied by another column vector. I do not want to store A in my memory so I am looking for a method that maps those vectors v1, v2 and v3 to the (virtual?) matrix.

Answer 1

[Sean de Wolski]

Yes. It's possible.

[Dikun]

Thanks, how?

[Sean de Wolski]

How are we possible supposed to know? The infinite monkey might stand a chance but you haven't given us any example of what you need. So you have a 2xn matrix, what does it do, what does it point to, what functions are you using it for etc.

Answer 2

[Matt Fig]

For a better answer, show a small example, complete with a small matrix and what you are going to do with it.

Answer 3

[Jan]

I'm very happy, that you do not want to store a "10^6 by 10^6" DOUBLE (?) matrix in the memory. Especially if it has such a low density of information.

But trying to map the vactors in a "virtual matrix" is either too vaguely explained or nonsense. What is a "virtual matrix"?!

Obviously there is no need to store the rows repeatedly. Create a 10^6 x 2 matrix and solve the rest by intelligent indexing:

v1 = [1 2 3 4 5];
v2 = [10 20 30 40 50];
v3 = [10 9 8 7 6 5 4 3 2 1];
M = [repmat([v1 v2], 1, 1E5); repmat(v3, 1, 1E5)];

Answer 4

[Walter Roberson]

define

Vsmall = [v1 v2;v3]

Then

V = @(R,C) Vsmall(1-mod(R-1,size(Vsmall,1)), 1 + mod(C-1,size(Vsmall,2)));

V is now your virtual array as far as indexing goes (i.e., suitable for your own matrix multiply algorithm)

When you are multiplying by a column vector, each row of A that is a repeat of an earlier row will have exactly the same result as the previous row.

The column-vector multiplication could potentially be sped up by knowing the structure of the columns of A. Or you could do something like

[B,m,n] = unique(A(P,:),'first');
sum(B .* accumarray(n, ColumnVector))

this would give you the result for the row P (and thus all rows that are duplicates of row P).