# Octave criterion for matrix singularity

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## Octave criterion for matrix singularity

 Dear all, I'm repeatedly solving a linear system with a Vandermonde matrix. I'm using a "backslash" operator:   y = -W'/b' 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 If possible, would you please direct me to the Octave source code, where this decision is made? I would like to understand the criterion, that triggers the warning. I found a post on StackOverflow that says MATLAB would issue a similar warning, once rcond < 1e-12. I would like to know, what threshold does Octave use and how is it defined. Thank you! -- Best wishes, Maxim -- Sent from: http://octave.1599824.n4.nabble.com/Octave-General-f1599825.html
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## Re: Octave criterion for matrix singularity

 -- maxim.abalenkov > Dear all, > > I'm repeatedly solving a linear system with a Vandermonde matrix. I'm using > a "backslash" operator: > >   y = -W'/b' > > 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 > > If possible, would you please direct me to the Octave source code, where > this decision is made? I would like to understand the criterion, that > triggers the warning. I found a post on StackOverflow that says MATLAB would > issue a similar warning, once rcond < 1e-12. I would like to know, what > threshold does Octave use and how is it defined. Thank you! > > -- > Best wishes, > Maxim > You wrote y=W’/b’ / is slash. Is the above a type miss? Tatsuro
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## Re: Octave criterion for matrix singularity

 Yes, you are right. It is a typo. I meant "backslash":   y = -W'\b'; -- Sent from: http://octave.1599824.n4.nabble.com/Octave-General-f1599825.html
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## Re: Octave criterion for matrix singularity

 In reply to this post by mabalenk > On 25 Feb 2019, at 19:34, mabalenk <[hidden email]> wrote: > > Dear all, > > I'm repeatedly solving a linear system with a Vandermonde matrix. I'm using > a "backslash" operator: > >  y = -W'/b' > > 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 > > If possible, would you please direct me to the Octave source code, where > this decision is made? I would like to understand the criterion, that > triggers the warning. I found a post on StackOverflow that says MATLAB would > issue a similar warning, once rcond < 1e-12. I would like to know, what > threshold does Octave use and how is it defined. Thank you! > > -- > Best wishes, > Maxim The code for solving A\b with A real double precision, full matrix is here :  http://hg.savannah.gnu.org/hgweb/octave/file/14815cb62fbe/liboctave/array/dMatrix.cc#l1445the condition being checked is  rcond + 1.0 == 1.0 which means rcond is less or equal than eps(1.0)/2 HTH, c.
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## Re: Octave criterion for matrix singularity

 In reply to this post by mabalenk mabalenk wrote > 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 > > If possible, would you please direct me to the Octave source code, where > this decision is made? I would like to understand the criterion, that > ... 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". I would guess it is the function "Matrix::fsolve" that accomplishes the work of "back slash". And the key lines are     volatile double rcond_plus_one = rcon + 1.0;     if (rcond_plus_one == 1.0 || octave::math::isnan (rcon)) Alternatively, if you know that the operator "back slash" is called "mldivide", then you can do "help mldivide" in Octave, it tells you:     'mldivide' is a built-in function from the file libinterp/corefcn/data.cc From there you may trace down what's going on. Though probably not that easy due to layers of abstraction. One lesson I learn is: this "rcond" is actually (reciprocal) condition number of L1 (or L-infinity) norm, instead of 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, it 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). -- Sent from: http://octave.1599824.n4.nabble.com/Octave-General-f1599825.html
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## Re: Octave criterion for matrix singularity

 In reply to this post by Carlo de Falco-2 Yes, it does help. Thank you. But I don't understand, where is the division by 2 hidden. It seems to me, that rcond is compared to eps, the machine precision. -- Sent from: http://octave.1599824.n4.nabble.com/Octave-General-f1599825.html
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## Re: Octave criterion for matrix singularity

 > On 26 Feb 2019, at 14:21, mabalenk <[hidden email]> wrote: > > Yes, it does help. Thank you. But I don't understand, where is the division > by 2 hidden. It seems to me, that rcond is compared to eps, the machine > precision. Try by yourself : >> eps ans =    2.2204e-16 >> 1 + eps(1) == 1 ans = 0 >> 1 + eps(1)/2 == 1 ans = 1 or read the manual : >> help eps ....      More precisely, 'eps' is the relative spacing between any two      adjacent numbers in the machine's floating point system.  This      number is obviously system dependent.  On machines that support      IEEE floating point arithmetic, 'eps' is approximately 2.2204e-16      for double precision and 1.1921e-07 for single precision. .... HTH, c.
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## Re: Octave criterion for matrix singularity

 In reply to this post by tmacchant Thanks a lot. It helped a lot ----- app development companies in Canada mobile app development company in Russia -- Sent from: http://octave.1599824.n4.nabble.com/Octave-General-f1599825.html