constructing a difficult large matrix

Dear all,

I am interested in constructing a complicated matrix A.

The matrix A is a diagonal matrix, where each diagonal element 'a_i' is of 1X1000, having the following patter

a_1=[p 1 0 0 0 0 0 .....0 ];
a_2=[p^2 p 1 0 0 0..... 0];
a_2=[p^3 p^2 p 1 0 0......0];
a_2=[p^4 p^3 p^2 p 1 0 0......0 ];
a_2=[p^5 p^4 p^3 p^2 p 1 0 ...0];
.
.
.
a_1000=[p^1000 p^999 p^998.........1];

And I want to multiply each element "a_i" with the vector h, where h is of dimension 1000X1, which contains just numbers of no particular pattern

Is there any way of doing that fast?

Maybe by using sparse or speye?

Thank you

[per isakson]

"multiply each element "a_i" with the vector h, where h is of dimension 1000X1" the product will that be a scalar, vector or matrix?

"doing that fast" how fast is "fast" ?

[ektor]

Hi isakson,

It will be a scalar. So each element of a_i will be multiplied by the corresponding element of h to produce a scalar (like inner product). Sorry for not clarifying first.

When I say fast, I mean not time consuming.

Answer 1

[David Goodmanson]

Hi ektor,

I presume you are interested not so much in the in the matrix A as in the 1000 scalars you get after you multiply each row of A by the vector h. To make things a little easier I included an extra row a_0 at the beginning.

a_0=[1 0 0 0 0 0 0 .....0 ];
a_1=[p 1 0 0 0 0 0 .....0 ];
a_2=[p^2 p 1 0 0 0..... 0];
a_3=[p^3 p^2 p 1 0 0......0];
a_4=[p^4 p^3 p^2 p 1 0 0......0 ];
a_5=[p^5 p^4 p^3 p^2 p 1 0 ...0];
.
.
a_1000=[p^1000 p^999 p^998.........1];

then

Ah = filter(1,[1 -p],h)

is the same as A*h. It's very fast. There is one exra element at the beginning corresponding to row a_0 but you can always delete it.

[ektor]

Hi David,

I made a correction thanks to isakson. please see above

[David Goodmanson]

Hi ektor,

The filter approach is for inner products. If that's the case, suppose you take a small example, say n = 5 instead of 1000, create the matrix A and calculate A*h for some h, which gives the set of inner products. Then do the version using filter. If you have an example where the answers do not agree, I would be interested to see that.

[David Goodmanson]

Hi ektor,

I was assuming that a_1 times h is the inner product (scalar product). that is, one number for the output, is that correct? Or do you mean element-by-element multiplicaton with no sum, so that you get an output of length(a_1) numbers?

[ektor]

HI david,

Just a quick question: " There is one exra element at the beginning corresponding to row a_0 but you can always delete it. "

what does this mean?

[David Goodmanson]

Hi ektor,

I meant that your matrix has 1000 rows and 1001 columns. The output is the inner product of h with each row, so 1000 values in all. However, for the filter function to reproduce that result, it was convenient to add a row at the top that I called a0. Then there are 1001 rows and 1001 values in the result. But the first result is due to the added row, so if you delete it you are back to the 1000 values in the original problem.

Answer 2

[per isakson]

Here are three functions, two of which are based on the answer of David Goodmanson. I think all of them are fast enough. However, more important than speed is that their results are correct. More test are needed.

A small test with numbers that I'm able to check.

>> cssm(8)
ans =
     3     0     0     0     0     0     0     0
     0     7     0     0     0     0     0     0
     0     0    15     0     0     0     0     0
     0     0     0    31     0     0     0     0
     0     0     0     0    63     0     0     0
     0     0     0     0     0   127     0     0
     0     0     0     0     0     0   255     0
     0     0     0     0     0     0     0   510
>> 

>> tic,cssm(1e4);toc
Elapsed time is 1.155732 seconds.
>> tic,M=dgsp(1e4);toc
Elapsed time is 0.490225 seconds.
>> tic,M=dg(1e4);toc
Elapsed time is 0.014082 seconds.

function    M = cssm( N )
    %%
    h = ones( N, 1 );
    p = 2;
    pr = 1;
    a = zeros( N, N );
    a( N+1 : (N+1) : end ) = 1;
    for jj = 1 : N
        pr = pr * p;
        a( jj : (N+1) : end-(jj-2)*N ) = pr;
    end
    
    M = zeros( N, N );
    for jj = 1 : N
        M(jj,jj) = a(jj,:) * h; 
    end
end

function    M = dgsp( N )
    %%
    h   = ones( N, 1 );
    tmp = [ 1; h ];
    p = 2;
    %%
    Ah = filter( 1, [1,-p], tmp );
    M = sparse( zeros( N, N ) );    % better: sparse(10,10,0); 
    M( 1 : N+1 : end ) = Ah( 2 : end );
end

function    M = dg( N )
    %%
    h   = ones( N, 1 );
    tmp = [ 1; h ];
    p = 2;
    %%
    Ah = filter( 1, [1,-p], tmp );
    M = zeros( N, N );
    M( 1 : N+1 : end ) = Ah( 2 : end );
end

[David Goodmanson]

Hello per,

could you explain the idea behind putting the result onto the diagonal of a matrix which is otherwise zero?

[per isakson]

Do you ask for why or how? I answer both.

"[...] constructing a complicated matrix A. The matrix A is a diagonal matrix" but I'm still not sure exactly what OP asked for.

Internally, Matlab stores arrays as linear sequences of elements together with some metadata, including the size of the array. Matlab uses column-major (and C row-major). Accessing array elements can be done with subscripts or linear indexing. (The functions, sub2ind and ind2sub convert between the two.) 1:N+1:end is the linear indicies of the diagonal elements.

[David Goodmanson]

I'm fine with the how, I was inquiring about the why, and now I see the direction you took. In the statement "A is a diagonal matrix, where each diagonal element 'a_i' is of 1X1000", if A is interpreted to be a very large block diagonal matrix, you do end up with the large diagonal matrix you have.

[per isakson]

Yes, I need a copy of Walters Mind Reading Toolbox :(.