Re: Efficient multiplication by a diagonal matrix

[Ted Harding]

( Re Message From: Mario Storti )
> 
> 
> I found  myself repeatedly with the following  problem. Given a matrix
> A(n,m)  and a vector  v(n), I  have to  multiply   each row  A(j,:) by
> v(j). This is equivalent to compute:
> 
> B = diag(v) * A                     (1)
> 
> Now, for large    n, (1) is  very  inefficient,   because it  requires
> constructing the square matrix diag(v) which requires storage and many
> redundant operations since most elements  of diag(v) are null.

A good while ago I wrote to John Eaton suggesting an extension of the
octave ".*" multiplication operator so that, if A (m x n) is a matrix,
u (m x 1) and v (1 x n) are vectors, then the following could be written:

        u .* A  =  A .* u
        =  rows of A times corresponding elements of u (the above case)

        v .* A  =  A .* v
        =  columns of A times corresponding elements of v

If this were implemented by built-in code it could be done very
efficiently and fast. The decision between row-wise or column-wise
multiplication would be taken according to the dimensions of u (or v).
Any other dimensional relationships (except u or v scalar) would be an
error.

John reacted favourably at the time, but I have heard no more since.

I certainly have frequent call for such operations: in Statistics,
applying the same weighting factor along a whole row (or column) is a
daily requirement; I have tended to adopt the diag(u)*A method, but as
Mario points out this can be inefficient, especially in computations
involving iterative re-weighting and large matrices (a matrix with
several thousand rows -- "cases" -- is not uncommon).

Best wishes to all,
Ted.                                    (address@hidden)