From bug-octave-request at bevo dot che dot wisc dot edu Thu Jan  1 13:58:12 2004
Subject: Re: Bad octave norms 
From: Glenn Golden <gdg at zplane dot com>
To: Vic Norton <vic at norton dot name>
Cc: bug-octave at bevo dot che dot wisc dot edu
Date: Thu, 01 Jan 2004 12:57:33 -0700

Vic,

> The 1 and inf octave norms are not defined correctly for row vectors.
> The definitions should be
>     norm(A, 1) = max(abs(A))
>     norm(A, inf) = sum(abs(A))
> when rows(A) = 1.
> 

When rows(A) == 1 or columns(A) == 1, then norm(A, p) are vector p-norms
(as opposed to matrix p-norms).  Octave's implementation of them agrees
with Matlab's, and both behave in accordance with the definition of
vector p-norm,

    norm(A, p)   =def=   ( SUM ( |A(k)| ** p) ) ** (1/p)
                            k

in which case

    norm(A, 1)   = sum(abs(A))
    norm(A, inf) = max(abs(A)) .




>
> A reasonable matrix norm should satisfy
>     norm(A * B) <= norm(A) * norm(B).
>

Octave's matrix p-norms satisfy this.


>
> The 1 and inf octave norms can violate this condition when A is a row
> vector. For example, set A = [1 0] and B = [1 1; 0 0]. Then
>    2 = norm(A * B, 1) > norm(A, 1) * norm(B, 1) = 1.
> Again, set A = [1 1] and B = [1 0; 1 0]. Then
>    2 = norm(A * B, inf) > norm(A, inf) * norm(B, inf) = 1.
>


In your examples, A is a vector, B is a matrix.  In this case, the
requirement for "reasonable" norm behavior (where "reasonable" amounts
to subordinate behavior of matrix p-norms w.r.t. their corresponding
vector p-norms) is

	norm(B * A', p)   <=   norm(B, p) * norm(A', p)

and Octave's (and Matlab's) behavior satisfy this.



Ref: Golub & Van Loan, 1/e, Sec. 2.1, 2.2.

Glenn Golden



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