Peter Langfelder wrote: > > On Wed, Mar 16, 2011 at 8:28 AM, Feng Li <m...@feng.li> wrote: >> Dear R, >> >> If I have remembered correctly, a square matrix is singular if and only >> if >> its determinant is zero. I am a bit confused by the following code error. >> Can someone give me a hint? >> >>> a <- matrix(c(1e20,1e2,1e3,1e3),2) >>> det(a) >> [1] 1e+23 >>> solve(a) >> Error in solve.default(a) : >> system is computationally singular: reciprocal condition number = 1e-17 >> > > You are right, a matrix is mathematically singular iff its determinant > is zero. However, this condition is useless in practice since in > practice one cares about the matrix being "computationally" singular, > i.e. so close to singular that it cannot be inverted using the > standard precision of real numbers. And that's what your matrix is > (and the error message you got says so). > > You can write your matrix as > > a = 1e20 * matrix (c(1, 1e-18, 1e-17, 1e-17), 2, 2) > > Compared to the first element, all of the other elements are nearly > zero, so the matrix is numerically nearly singular even though the > determinant is 1e23. A better measure of how numerically unstable the > inversion of a matrix is is the condition number which IIRC is > something like the largest eigenvalue divided by the smallest > eigenvalue. >
svd(a) indicates the problem. largest singular value / smallest singular value=1e17 (condition number) --> reciprocal condition number=1e-17 and the standard solve can't handle that. (pivoted) QR decomposition does help. And so does SVD. Berend -- View this message in context: http://r.789695.n4.nabble.com/Singularity-problem-tp3382093p3382465.html Sent from the R help mailing list archive at Nabble.com. ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
