Re: [sage-support] Re: Completely confused about simple Sage matrix operation error
I think problem is actually due to the inverse using a non-numerically stable echelon form algorithm for inexact fields. For example, if you using matrices over RDF: M=matrix(RDF,[[7,3,10,13],[1,1,2,2],[1,2,3,4],[1,3,5,7]]);det(M);invM=M^(-1);invM*M;det(invM) you don't get this problem. My guess is that if we implemented something like partial pivoting for inexact fields. --Mike --Mike On Mon, Jul 1, 2013 at 10:06 AM, Harald Schilly wrote: > > > On Monday, July 1, 2013 10:45:36 AM UTC+2, David Ingerman wrote: >> >> The following matrix operation produces wrong answer in online Sage: >> >> M=matrix(RR,[[7,3,10,13],[1,1,2,2],[1,2,3,4],[1,3,5,7]]);det(M);invM=M^(-1);invM*M;det(invM) > > > RR stands for the "real numbers" with the usual 53bits of precision, e.g. > 5.123957322…. > QQ are rational numbers where two large integers build up each number, > e.g. 4141/333000333000333000333000333000333000333 > Therefore, QQ has a much higher precision … but is much slower and uses more > memory. > > In your case, the lack of precision in RR causes you troubles and you have > to find a way to pose the problem you want to solve differently. You cannot > rely on QQ, because in bad cases, the expressions blow up and eat all your > memory. More generally, this is not a Sage related problem, but related to > all calculations your are doing "natively" with your CPU. > > To see in advance when this happens, you have to calculate the conditional > number of the matrix. I think that's only in numpy (or I haven't found it). > > sage: M=matrix(RR,[[7,3,10,13],[1,1,2,2],[1,2,3,4],[1,3,5,7]]) > sage: import numpy as np > sage: np.linalg.cond(M) > 104.85355762315329 > > http://en.wikipedia.org/wiki/Condition_number > > Here are some decomposition methods that might help: > > http://en.wikipedia.org/wiki/Matrix_decomposition > > H > > > -- > You received this message because you are subscribed to the Google Groups > "sage-support" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-support+unsubscr...@googlegroups.com. > To post to this group, send email to sage-support@googlegroups.com. > Visit this group at http://groups.google.com/group/sage-support. > For more options, visit https://groups.google.com/groups/opt_out. > > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
Re: [sage-support] Re: Completely confused about simple Sage matrix operation error
On Mon, Jul 1, 2013 at 5:27 AM, David Ingerman wrote: > On Monday, July 1, 2013 2:06:54 AM UTC-7, Harald Schilly wrote: >> On Monday, July 1, 2013 10:45:36 AM UTC+2, David Ingerman wrote: The >> following matrix operation produces wrong answer in online Sage: >> M=matrix(RR,[[7,3,10,13],[1,1,2,2],[1,2,3,4],[1,3,5,7]]);det(M);invM=M^(-1);invM*M;det(invM) >> >> RR stands for the "real numbers" with the usual 53bits of precision, e.g. >> 5.123957322…. >> QQ are rational numbers where two large integers build up each number, >> e.g. 4141/333000333000333000333000333000333000333 >> Therefore, QQ has a much higher precision … but is much slower and uses more >> memory. >> >> In your case, the lack of precision in RR causes you troubles and you have >> to find a way to pose the problem you want to solve differently. You cannot >> rely on QQ, because in bad cases, the expressions blow up and eat all your >> memory. More generally, this is not a Sage related problem, but related to >> all calculations your are doing "natively" with your CPU. >> >> To see in advance when this happens, you have to calculate the >> conditional number of the matrix. I think that's only in numpy (or I >> haven't found it). >> >> sage: M=matrix(RR,[[7,3,10,13],[1,1,2,2],[1,2,3,4],[1,3,5,7]]) >> sage: import numpy as np >> sage: np.linalg.cond(M) >> 104.85355762315329 >> >> http://en.wikipedia.org/wiki/Condition_number >> >> Here are some decomposition methods that might help: >> >> http://en.wikipedia.org/wiki/Matrix_decomposition >> >> H > > But this is a very small matrix and conditional number is not large... Using RDF is better: sage: M=matrix(RDF,[[7,3,10,13],[1,1,2,2],[1,2,3,4],[1,3,5,7]]);det(M);invM=M^(-1);invM*M;det(invM) -7.0 [ 1.00.00.0 -8.881784197e-16] [ 0.01.00.0 -8.881784197e-16] [ 4.4408920985e-160.01.08.881784197e-16] [-5.55111512313e-17 -2.77555756156e-16 -1.66533453694e-161.0] -0.142857142857 > > -- > You received this message because you are subscribed to the Google Groups > "sage-support" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-support+unsubscr...@googlegroups.com. > To post to this group, send email to sage-support@googlegroups.com. > Visit this group at http://groups.google.com/group/sage-support. > For more options, visit https://groups.google.com/groups/opt_out. > > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
