
From:  bewantbe 
Subject:  Re: Octave criterion for matrix singularity 
Date:  Tue, 26 Feb 2019 23:01:00 +0800 
Useragent:  Mozilla/5.0 (X11; Linux x86_64; rv:60.0) Gecko/20100101 Thunderbird/60.5.1 
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 = 1.56508e17
You can search the source code by key words $ grep IR 'matrix singular to machine precision'This direct you to file "liboctave/util/loarrayerrwarn.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 Linfinity) 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 superlow reciprocal condition number compared to cond(), which is based on SVD. e.g. rcond(vander((1:50)/50))=1.4262e31, 1/cond(vander((1:50)/50))=4.8811e20 (which is well under estimated).
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