John Cremona wrote: > 2008/9/23 John Cremona <[EMAIL PROTECTED]>: >> 2008/9/23 pong <[EMAIL PROTECTED]>: >>> >>> I tried >>> >>> sage: A=matrix([[1,1],[2,3]]) >>> sage: A.eigenvalues() >>> [0.2679491924311228?, 3.732050807568878?] >>> >>> My question is why the last digit before ? in 0.2679491924311228? is >>> '8'? Shouldn't it be "7" according to >>> >>> sage: ch=characteristic_polynomial(A) >>> sage: solve(ch(x)==0,x) >>> [x == 2 - sqrt(3), x == sqrt(3) + 2] >>> sage: N(2-sqrt(3), digits=18) >>> 0.267949192431122706 >>> >>> >>> >>> >> Alternatively: >> >> sage: e1,e2= A.eigenvalues() >> sage: e1 >> 0.2679491924311228? >> sage: e1.interval(ComplexIntervalField(53)) # the default is 53 bits >> precision >> 0.2679491924311228? >> sage: e1.interval(ComplexIntervalField(100)) >> 0.267949192431122706472553658494? >> sage: e1.interval(ComplexIntervalField(200)) >> 0.267949192431122706472553658494127633057194746189619371944193? >> >> i.e. e1 is a number of a type which is essentially infinite precision, >> and you can ask for it to any desired precision later. >> >> However I'm still not sure why the original display ends 8? and not >> 7?. The ? notatation was introduced quite recently, so I am not sure >> whether this is a feature or a bug. > > OK, it is a feature. To quote from > http://wiki.sagemath.org/sage-3.1.2 (Release notes for 3.1.2): > > The question marks at the end of the numbers in the previous example > mean that Sage is printing out an approximation of an exact value that > it uses. In particular, the question mark means that the last digit > can vary by plus or minus 1. In other words, 32.46424919657298? means > that the exact number is really between 32.46424919657297 and > 32.46424919657299. Sage knows what the exact number is and uses the > exact number in calculations.
That last sentence is technically incorrect, I think (sorry; I wrote it). It might be better to say, "Sage carries out the calculations using interval arithmetic and essentially uses infinite precision in these cases." Carl, is there some way to word things better? Thanks, Jason --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
