I realize this is a bit naive (and not completely related to the OP) but as it currently stands the CC used in sage is essentially a subfield of QQ(i). It may be better to implement RR and CC using exact precision (though admittedly this will come at a cost in performance) and allow the real field elements to be coerced to approximations using mpfr when the user wishes. Doing so would also allow for more straightforward interaction with maxima's solver (the default solver in sage), or anything else using sage.calculus.calculus.SymbolicArithmetic.
Or is there another reason that the default RR and CC do not use exact precision, other than very slight performance issues? I suspect the current reason is that Singular is not equipped to deal with exact precision RR and CC. I'm just curious if this would be a good thing in the long run. On Jan 6, 11:57 am, "John Cremona" <[EMAIL PROTECTED]> wrote: > You are right about the correct inner product definition for complex > vectors. In your first example Sage seems not to be doing any > conjugation. > > RR is real numbers (to some precision) while QQ is rational numbers > (which are exact). If you want complex numbers with real and > imaginary parts in QQ that qould be anumber field QQ(i). But I would > not expect Sage to automatically see how to take an inner product of > vectors whose entries are in number fields (for which there is no easy > answer since number fields in general have both real and complex > embeddings). I doubt if that is what you mean, so that leaves you > working with CC which will definitely be approximate (even with exact > input). > > Now the fact that Sage does not (apparently) recognise complex vectors > and use an appropriate definition for inner_product() is surely a bug. > It should be easy to fix if the parent field is CC (or is coerciable > to CC?). > > John > > On 06/01/2008, Fabio Tonti <[EMAIL PROTECTED]> wrote: > > > > > sage: u=vector([2+3*I,5+2*I,-3+I]) > > sage: v=vector([1+2*I,-4+5*I,0+5*I]) > > sage: p1=u*v;p1.expand() > > 9*I - 39 > > sage: p2=u.inner_product(v);p2.expand() > > > 9*I - 39 > > sage: p3=u.dot_product(v);p3.expand() > > 9*I - 39 > > sage: p4=u.inner_product(vector([i.conjugate() for i in v]));p4.expand() > > 3 - 19*I > > > Am I right in the assumption that for the inner product of two complex > > vectors, the result should be the sum of the element wise multiplication of > > the element of the first vector times the complex conjugate of the element > > of the second vector? I had to do this by hand, as you can see for p4. I had > > a look at Mathematica, and it seems like they don't do it either. So I might > > be wrong. > > > Another thing: > > sage: parent(p1) > > Symbolic Ring > > sage: parent(u) > > Vector space of dimension 3 over Symbolic Ring > > is it meant to be over symbolic Ring? > > > and one more: > > sage: k=vector([complex(1,2),complex(3,4),complex(25,15)]) > > Traceback (most recent call last): > > ... > > TypeError: unable to find a common ring for all elements > > > seriously? why that? > > sage: > > u=vector(CC,[complex(1,2),complex(3,4),complex(25,15)]);u > > (1.00000000000000 + 2.00000000000000*I, 3.00000000000000 + > > 4.00000000000000*I > > , 25.0000000000000 + 15.0000000000000*I) > > And now I've got creepy precision stuff in there. And the inner product > > still doesn't do what I'd like it to. > > > Does a complex number from CC constructed by complex(<re>,<im>) have as > > resulting real and imaginary part have elements from RR? What's the > > difference between RR and QQ anyway? QQ is arbitrary precision, does RR use > > machine precision maybe? > > > Now maybe I've asked too many questions, but I've had no luck with the > > reference manual so far (maybe I just don't get the explanations in > > there...). > > I know that today there's the big AMS meeting (good luck for that), so no > > need to hurry in order to reply for anyone. And excuse my English, there > > maybe some mistakes since I'm in a rush... > > > Thanks a lot, Fabio > > -- > John Cremona --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@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-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
