Re: Octave criterion for matrix singularity

[bewantbe]

> At each iteration I'm increasing the matrix order by one. At some point
> Octave issues a warning:
>
>    warning: matrix singular to machine precision, rcond = 1.56508e-17

You can search the source code by key words
  $ grep -IR 'matrix singular to machine precision'
This direct you to file "liboctave/util/lo-array-errwarn.cc" and 
function "warn_singular_matrix (double rcond)".

With this next key words, search by
$ grep -IR 'warn_singular_matrix'
get you to e.g. file "liboctave/array/dMatrix.cc" and the function 
"Matrix::fsolve".

One lesson I learn is: this "rcond" is actually (reciprocal) condition 
number of L1 (or L-infinity) norm, instead of the usual L2 norm. Thus it 
is easier and faster to compute (by LAPACK routine dgecon()). You can 
get this rcond by "rcond" function in Octave. And to my surprise, 
rcond() is much more capable to super-low reciprocal condition number 
compared to cond(), which is based on SVD. e.g. 
rcond(vander((1:50)/50))=1.4262e-31, 
1/cond(vander((1:50)/50))=4.8811e-20 (which is well under estimated).