[sage-devel] polynomials are power series?
While the ring type hierarchy does not reflect that polynomials are power series, you can have a power series without bigoh which is pratically a polynomial but, being a power series, has much less member functions available. I think Sage shouldn't allow a zero bigoh term in power series. It should avoid unexpected behaviour, eg. users complaining that a polynomial isn't what it seems. But I'm writing here to ask for your opinion before I think about patching, because I'm only beginning to understand Sage, and I'm not even a mathematician! Regards, Ralf Stephan -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/groups/opt_out.
Re: [sage-devel] polynomials are power series?
Surely all Ralf meant was that R[X] is a subring of R[[X]], i.e. some elements of R[[X]] are exact, just as some decimal numbers like 0.25 are exact (in binary), and just as we might want to define a real number as having *exactly* the value 0.25 and not just 0.25 + O(10^-1000) one might want to consider 1+X as an exact power series and not just 1+X+O(X^1000). Of course I amy have misunderstood Ralf (or you)! John On 22 January 2014 11:49, Ralf Stephan gtrw...@gmail.com wrote: While the ring type hierarchy does not reflect that polynomials are power series, you can have a power series without bigoh which is pratically a polynomial but, being a power series, has much less member functions available. I think Sage shouldn't allow a zero bigoh term in power series. It should avoid unexpected behaviour, eg. users complaining that a polynomial isn't what it seems. But I'm writing here to ask for your opinion before I think about patching, because I'm only beginning to understand Sage, and I'm not even a mathematician! Regards, Ralf Stephan -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/groups/opt_out. -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/groups/opt_out.
Re: [sage-devel] polynomials are power series?
I understand precision as being independent from element properties (as it is in Pari). Note also that R.random_element() always has O(x^20) so a fixed precision is already implemented. John is right that I see polynomials as a subring to power series. I would not be able to give references to that however. -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/groups/opt_out.
Re: [sage-devel] polynomials are power series?
Hi Ralf, I understand precision as being independent from element properties (as it is in Pari). In Sage, there are two kinds of precision: the precision of an individual element and the default precision of the power series ring. The same power series ring can contain elements that are represented using different precisions; for example, you can have a power series ring R with default precision 20, an element f in R with precision 10, and another element g in R with infinite precision. An operation on power series (addition, inversion etc.) return the result in the highest precision to which it is defined; this depends on the precision of the elements, not on the default precision. The exception is when the input has infinite precision and the output cannot be represented with infinite precision. This is where the default precision comes in. For example, 1 - x has infinite precision, but 1/(1 - x) = 1 + x + x^2 + x^3 + ... cannot be represented exactly as a power series, so it is truncated to the default precision. In PARI the situation is similar, except for two things: (1) there is no distinction between polynomials and power series of infinite precision that happen to be polynomials, and (2) the default precision is a global setting, not tied to any specific ring. Both of these are simply because PARI has no (explicit) concept of polynomial rings and power series rings. Peter -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/groups/opt_out.
Re: [sage-devel] polynomials are power series?
Thanks Travis, so there is coercion already. Now I think it natural to also have coercion from the polynomial fractions to power series, or at least have an expand() member function with a precision parameter and coercion in case of addition with some bigoh, see http://trac.sagemath.org/ticket/15698 And thanks to Peter for completely clarifying power series precision. On Wed, Jan 22, 2014 at 5:36 PM, Peter Bruin pjbr...@gmail.com wrote: Hi Ralf, I understand precision as being independent from element properties (as it is in Pari). In Sage, there are two kinds of precision: the precision of an individual element and the default precision of the power series ring. The same power series ring can contain elements that are represented using different precisions; for example, you can have a power series ring R with default precision 20, an element f in R with precision 10, and another element g in R with infinite precision. An operation on power series (addition, inversion etc.) return the result in the highest precision to which it is defined; this depends on the precision of the elements, not on the default precision. The exception is when the input has infinite precision and the output cannot be represented with infinite precision. This is where the default precision comes in. For example, 1 - x has infinite precision, but 1/(1 - x) = 1 + x + x^2 + x^3 + ... cannot be represented exactly as a power series, so it is truncated to the default precision. In PARI the situation is similar, except for two things: (1) there is no distinction between polynomials and power series of infinite precision that happen to be polynomials, and (2) the default precision is a global setting, not tied to any specific ring. Both of these are simply because PARI has no (explicit) concept of polynomial rings and power series rings. Peter -- You received this message because you are subscribed to a topic in the Google Groups sage-devel group. To unsubscribe from this topic, visit https://groups.google.com/d/topic/sage-devel/APnWIGYcq3M/unsubscribe. To unsubscribe from this group and all its topics, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/groups/opt_out. -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/groups/opt_out.
