inconsistent condition numbers from cond, det, inv functions

[Ben Abbott]

On Feb 7, 2008, at 7:00 AM, Rolf Fabian wrote:

> [...]
>
> Unfortunately example matrix x was choosen to be symmetric.
> This wasn't a good choice  because Inf- and 1-norms are
> idential in this case.
>
> Using instead the asymmetric matrix
> octave-3.0.0.exe:> y=[7,2,3;1,3,4;6,4,5]
> y =
>   7   2   3
>   1   3   4
>   6   4   5
>
> octave-3.0.0.exe:> 1/condh(y,1)
> ans =  0.017460
> octave-3.0.0.exe:> 1/condh(y,2)
> ans =  0.019597
> octave-3.0.0.exe:> 1/condh(y,'fro')
> ans =  0.018714
> octave-3.0.0.exe:> 1/condh(y,'inf')
> ans =  0.012022
> octave-3.0.0.exe:> 1/condh(y,Inf)
> ans =  0.012022
>
> octave-3.0.0.exe:37> [dum,rc1]=det(y)
> rc1 =  0.017460
> octave-3.0.0.exe:38> [dum,rc2]=inv(y)
> rc2 =  0.017460
> octave-3.0.0.exe:39> rc3 = 1/cond(y)
> rc3 =  0.019597
>
> allows to differentiate between 1- and Inf -norm.
>
> Ben, what' s the result for 1./cond(y) ?
> Does Matlabs cond() have a similar argument 'p' too?

Yes!

 >> y=[7,2,3;1,3,4;6,4,5]

y =

      7     2     3
      1     3     4
      6     4     5

 >> 1/cond(y)

ans =

     0.0196

 >> help cond
  COND   Condition number with respect to inversion.
     COND(X) returns the 2-norm condition number (the ratio of the
     largest singular value of X to the smallest).  Large condition
     numbers indicate a nearly singular matrix.

     COND(X,P) returns the condition number of X in P-norm:

        NORM(X,P) * NORM(INV(X),P).

     where P = 1, 2, inf, or 'fro'.

     Class support for input X:
        float: double, single

     See also rcond, condest, condeig, norm, normest.

     Reference page in Help browser
        doc cond

 >> 1/cond(y,1)

ans =

     0.0175

 >> 1/cond(y,2)

ans =

     0.0196

 >> 1/cond(y,inf)

ans =

     0.0120

 >> 1/cond(y,'fro')

ans =

     0.0187

Ben