Re: [eigen] Machine precision<> too coarse |

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*To*: eigen@xxxxxxxxxxxxxxxxxxx*Subject*: Re: [eigen] Machine precision<> too coarse*From*: Benoit Jacob <jacob.benoit.1@xxxxxxxxx>*Date*: Tue, 3 Feb 2009 19:32:07 +0100*Dkim-signature*: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey-signature:mime-version:received:in-reply-to:references :date:message-id:subject:from:to:content-type :content-transfer-encoding; bh=cticBY7JjmQ4gyxFEXvujVZe1aI1vKxaneggbdxbgD4=; b=Sc51gFIT7PRiQdBzZfOIcsn0OQUgqDirNHDlCOjFC5BWR0ZDYDoOfCwAgJFV6yOrOO UwYnbXOOMY7/EFS+Rd5elollWjK25I77V5I9A4Cq/Y/q2c7kLaGnyJGuS1+EWAJwegyC SHuqOfT2nma1GYr2oRec+VH75hHZsCGT7wrJA=*Domainkey-signature*: a=rsa-sha1; c=nofws; d=gmail.com; s=gamma; h=mime-version:in-reply-to:references:date:message-id:subject:from:to :content-type:content-transfer-encoding; b=Vu2/syCPttyr6q6v3eZ45iAqNJgjIJPkSHWWlD72UlrxrEvMmwcbeUjLkPQEc2v73x CbjvD29vgvEbMfiR4t7v0zHVruObZxyoPUhrkS0SQ2tUqfjfAZcq30/GBWeHRCuV1uh0 Aie8ITVd7KHOS/XXp5YfD6pVmviKz8J6fSVU4=

2009/2/3 Keir Mierle <mierle@xxxxxxxxx>: > On Tue, Feb 3, 2009 at 8:59 AM, Benoit Jacob <jacob.benoit.1@xxxxxxxxx> > wrote: >> >> Also, look at things this way: >> >> Suppose that you have a NxN matrix of doubles, with entries between -1 >> and 1, except in one row where entries are smaller than 1e-12. >> >> For example (N=2): >> 1 0 >> 0 1e-12 >> >> Do you want to consider this matrix invertible or not? > > The condition number is bad, but the matrix is certainly invertible to > machine precision. Indeed. Yet the condition number is so bad that in practice, in most cases where this happens, the matrix is meant to be non-invertible. What this shows, is that "invertible to machine precision" is not a particularly useful notion. To take a more extreme example: 1 0 0 1e-15 is also "invertible to machine precision" and there it's 99.99% guaranteed that the 1e-15 is noise (and hence it should be considered non invertible), as this imprecision can be realized by just a few floating point operations. Cheers, Benoit

**References**:**[eigen] Machine precision<> too coarse***From:*Keir Mierle

**Re: [eigen] Machine precision<> too coarse***From:*Benoit Jacob

**Re: [eigen] Machine precision<> too coarse***From:*Keir Mierle

**Re: [eigen] Machine precision<> too coarse***From:*Benoit Jacob

**Re: [eigen] Machine precision<> too coarse***From:*Keir Mierle

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