Re: Bad octave norms

[Vic Norton]

Sorry to disagree with you, Glen.

Octave's definition may be in agreement with MATLAB, but it is not in 
agreement with the accepted mathematical definition. Reread Chapter 
2.3, Matrix Norms, of Golub & Van Loan's Matrix Computations.

The p-norm of an (m x n)-matrix matrix A is, by definition,
    norm(A, p) = sup { norm(A * x, p) : norm(x, p) = 1 },
where x is an n x 1 vector. It follows that, if the 1-norm of an n x 
1 vector x is
    sum { abs(x(i)) : i = 1,..,n },
then the 1-norm of an m x n matrix A must be
    max { sum { abs(A(i, j)) : i = 1,..,m } : j = 1,..,n }.
Likewise, if the infinity-norm of an n x 1 vector x is
    max { abs(x(i)) : i = 1,..,n },
then the infinity-norm of an m x n matrix A must be
    max { sum { abs(A(i, j)) : j = 1,..,m } : i = 1,..,n }.

Octave and MATLAB have got their definitions right for every m other 
than 1. Why in the world would they think they should change things 
when m = 1? Is this just the computer way of doing things? Anything 
for expediency? But we are talking mathematics here. Mathematics 
doesn't work this way. Mathematics insists on elegance, on logical 
consistency. Mathematics is not just a bag of tricks -- no matter 
what MATLAB thinks!

I suggest suggest that Octave rename its function morn(A, p). That 
way everyone will know that it just an ad hoc creation, not what 
mathematicians call the p-norm.

And then there is duality and the relation (1/p) + (1/q) = 1. Start 
with R^n. The dual space of R^n, (R^n)*, looks just like R^n. So what 
has this got to do with p's and q's? Just this: the p-norm on (R^n)* 
is the same as the q-norm on R^n. Column vectors are in R^n; row 
vectors are in (R^n)*. The only time that R^n and (R^n)* are the same 
normed spaces is when p = q = 2 -- no matter what MATLAB and Octave 
say!

BTW, if I recall correctly, complete normed algebras that obey the rule
    norm(A * B) <= norm(A) * norm(B)
are called Banach algebra's, and Banach was doing this stuff way before MATLAB.


Regards,

Vic

At 12:57 PM -0700 1/1/04, Glenn Golden wrote:
> 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

--
*---* mailto:address@hidden
|     Victor Thane Norton, Jr.
|     Mathematician and Motorcyclist
|     phone: 419-353-3399
*---* http://vic.norton.name



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