Hi Simon,
thanks a lot for your long answer. Unfortunately, it doesn't apply to
my question.
So let me answer your questions. I hope it will help to achieve a
better understanding of my problem.
I defined s as the variable of the polynomial ring over the complex
numbers. In particular, I already know how to define the polynomial
ring over a given ring. All my calculations are done in the quotient
field of the polynomial ring over the complex numbers.
I guess that the square roots in my expressions aren't the principal
point in my problems. The roots aren't roots of polynomials. They are
roots of the form sqrt(1-a/b) where a and b are some variables that
take real values. We could evaluate it to a real numerical value, but
I would prefer to keep them as (symbolic) variables.
To my understanding, the main problem are the very extensive terms in
our expressions and the fact that the occurring mathematical
operations are nested.
My coefficients aren't naked complex numbers. I did not define the
coefficients directly. I defined my function just as some expression
with the (main) variable s and some other technological variables
(which can be computed to complex numbers) in the quotient field of
the ring over polynomials over the complex numbers. My main problem is
that the function is not given in the form a/b with a, b polynomials
in s over the complex numbers. I wonder whether sage can transform the
function from the given form into the form as a/b with a and b
polynomials over the complex numbers.
Thanks a lot in advance,
Urs
On Oct 13, 7:56 am, Simon King <simon.k...@uni-jena.de> wrote:
> Hi Urs,
>
> On 12 Okt., 17:00, Urs Hackstein <urs.hackst...@googlemail.com> wrote:
>
> > I have the following problem: Let f be a complex function in the
> > complex variable s that is given by a long (approximately 3 pages)
> > term consisting of a lot of fractions nested into each other, basic
> > arithmetic operations and square roots.
>
> How did you define s? By var('s'), hence, as a symbolic variable? Or
> is s actually a variable in a polynomial ring? Since your expression
> contain square roots of polynomials (or only of coefficients?), power
> series might be an option as well.
>
> That may make a huge difference. Depending on the problem, each can be
> better than the other.
>
> And how did you define the coefficients? There are different
> structures in Sage that behave like complex numbers.
>
> I'm starting with arithmetic operations, postponing square roots. In
> that case, the polynomial ring approach will readily solve your
> problem. Symbolic expressions fail to the extent that I would call it
> a bug - I'll report it to sage-devel.
> I am also showing you a pitfall: It is quite easy to *accidentally*
> define a symbolic expression when you actually want to create a
> polynomial over a complex field or when taking the square root of a
> polynomial.
> Finally, I briefly indicate how power series may help you with square
> roots.
>
> How to create polynomials
> --------------------------------------
>
> To create s as an element of a polynomial ring with coefficients in K,
> you can do
> P = K['s']
> s = P.gen()
> in a script, or you can more pleasantly do
> sage: P.<s> = K[]
> in an interactive session.
>
> You say that the coefficients are complex - but are they really
> "naked" complex numbers? Again, depending on the problem, you might
> work with complex floats, with the complex interval field, or with the
> field of algebraic numbers (when your coefficients are solutions to
> algebraic equations).
>
> Let us assume that you work over the complex interval field. If you
> decide to consider "s" as element of a polynomial ring, then your
> definitions may look as follows:
> sage: K = CIF
> sage: K
> Complex Interval Field with 53 bits of precision
> sage: P.<s> = K[]
> sage: P
> Univariate Polynomial Ring in s over Complex Interval Field with 53
> bits of precision
>
> And then, simplification should just work.
>
> Let us create some more complicated arithmetic expression:
> sage: p1 = P.random_element()
> sage: p2 = P.random_element()
> sage: p3 = P.random_element()
> sage: p4 = P.random_element()
> sage: p5 = P.random_element()
> sage: p = ((p1/p2+p3)^2+(p3/(p4+p5)))/((p3/(p4+p5/(p1+p2)))+p2)
> sage: p
> ((-0.014910171719530? - 0.0819029681647261?*I)*s^14 +
> (0.334831133184198? + 0.226920835016301?*I)*s^13 + (-1.31689500460014?
> + 0.33100925619084?*I)*s^12 + (1.12271025690267? - 2.9711886585730?
> *I)*s^11 + (4.1636372058951? + 5.3375810146048?*I)*s^10 +
> (-11.7982240203551? - 0.5358729413119?*I)*s^9 + (9.1839609650226? -
> 13.3936564432300?*I)*s^8 + (6.2198719561828? + 20.818526779122?*I)*s^7
> + (-22.0256922983008? - 5.4198967910775?*I)*s^6 + (14.576686446425? -
> 11.706831897246?*I)*s^5 + (5.6758660676201? + 14.853679842037?*I)*s^4
> + (-6.8405250783868? - 2.8000312050867?*I)*s^3 + (1.7390862645398? -
> 4.3186765269158?*I)*s^2 + (-0.85548154093647? + 1.01546197758495?*I)*s
> - 0.895136615170288? + 0.554040475082552?*I)/((-0.169080624065409? +
> 0.012578831775274?*I)*s^12 + (-0.05414287388546? - 0.737750690892271?
> *I)*s^11 + (1.52184676823240? + 1.16470671744095?*I)*s^10 +
> (-5.0190991302301? + 2.36713627116690?*I)*s^9 + (1.9657655611365? -
> 9.4018627113920?*I)*s^8 + (12.6310011693864? + 13.2679738013862?
> *I)*s^7 + (-23.8796945790930? + 2.7835626926490?*I)*s^6 +
> (15.6411210970610? - 24.7657676923677?*I)*s^5 + (7.6541549026413? +
> 27.8461228023130?*I)*s^4 + (-17.7687638664456? - 12.1686300681916?
> *I)*s^3 + (14.3341637239600? - 2.3920514116429?*I)*s^2 +
> (-7.0332616091283? + 2.6003363350971?*I)*s + 1.28808529738263? -
> 0.24159917788167?*I)
>
> So, the result is just a fraction of two polynomials.
>
> How to create a symbolic expression
> -----------------------------------------------------
>
> If you insist on using symbolic variables (I *never* consciously used
> symbolic variables for my own work, but there certainly are
> applications), simplification will not automatically happen. Let us
> compare with the "same" example in the symbolic ring:
>
> sage: SR
> Symbolic Ring
> sage: q1 = SR(p1)
> sage: q2 = SR(p2)
> sage: q3 = SR(p3)
> sage: q4 = SR(p4)
> sage: q5 = SR(p5)
> sage: q = ((q1/q2+q3)^2+(q3/(q4+q5)))/((q3/(q4+q5/(q1+q2)))+q2)
> sage: q
> ((((0.61891108886609514? - 0.092586296245679601?*I)*s -
> 0.98781137976673450? + 0.11138924672910667?*I)*s +
> (((0.29144707263062198? - 0.42564378360017075?*I)*s -
> 0.61884952759953405? - 0.91492158538646940?*I)*s -
> 0.24947131951698487? - 0.096607726700608688?*I)/((-
> (0.15155075241697081? + 0.78306247143200070?*I)*s +
> 0.57637682747181885? - 0.29257682595384749?*I)*s -
> 0.86140746403038726? + 0.51624793761351229?*I) - 0.50908151182035400?
> - 0.090000947697624012?*I)^2 + (((0.61891108886609514? -
> 0.092586296245679601?*I)*s - 0.98781137976673450? +
> 0.11138924672910667?*I)*s - 0.50908151182035400? -
> 0.090000947697624012?*I)/((-(0.47146742486366700? +
> 0.57376175864452606?*I)*s + 0.26292911853237034? + 0.20257307216542353?
> *I)*s + ((0.79619152926825932? + 0.39681742537994036?*I)*s -
> 0.29184858034623629? + 0.68656770475640916?*I)*s -
> 0.40666338022023152? - 1.3892263050186884?*I))/((-
> (0.15155075241697081? + 0.78306247143200070?*I)*s +
> 0.57637682747181885? - 0.29257682595384749?*I)*s +
> (((0.61891108886609514?...
>
> I did not print the result completely, but you can see that it is not
> simplified. You may try to simplify it manually, but there seems to be
> a bug (with sage-4.6.2, at least):
> sage: q.simplify()
> Traceback (most recent call last):
> ...
> TypeError:
>
> There's even no proper error message.
>
> How to avoid accidentally creating a symbolic expression
> ----------------------------------------------------------------------------------
>
> Note one pitfall, though: The symbolic ring "eats" everything, and the
> symbol "I" denotes the complex unit AS AN ELEMENT OF THE SYMBOLIC
> RING.
>
> Hence, when you use the complex unit "I" in your polynomial
> expressions, then you will inevitably end up with symbolic
> expressions, facing the simplification problems exposed above:
> sage: p.parent()
> Univariate Polynomial Ring in s over Complex Interval Field with 53
> bits of precision
> sage: I.parent()
> Symbolic Ring
> sage: (p+I-I).parent()
> Symbolic Ring
>
> Solution: Define I=CIF.gen() (or I=CDF.gen() or I=CC.gen(), depending
> on your coefficients), and then it should just work.
>
> Square roots
> ------------------
>
> Unfortunately, square roots of polynomials are symolic expressions:
> sage: p1.parent()
> Univariate Polynomial Ring in s over Complex Interval Field with 53
> bits of precision
> sage: sqrt(p1).parent()
> Symbolic Ring
>
> Hence, it may be better to work with power series. Their square roots
> are again power series, so that one will not accidentally create
> symbolic expressions:
> sage: PS = K[['s']]
> sage: r1 = PS(p1)
> sage: sqrt(r1)
> 0.0950? - 0.5084?*I + (0.76? - 0.75?*I)*s + (-0.7? + 0.4?*I)*s^2 +
> (1.5? + 0.1?*I)*s^3 + (-3.? - 2.?*I)*s^4 + (1.?e1 + 1.?e1*I)*s^5 + (0.?
> e1 - 3.?e1*I)*s^6 + (-1.?e2 + 1.?e2*I)*s^7 + (1.?e3 - 1.?e2*I)*s^8 +
> (-1.?e3 + 0.?e3*I)*s^9 + (2.?e3 + 1.?e4*I)*s^10 + (-1.?e4 - 1.?
> e4*I)*s^11 + (0.?e5 + 1.?e5*I)*s^12 + (1.?e5 - 1.?e5*I)*s^13 + (-1.?e6
> + 1.?e6*I)*s^14 + (1.?e6 + 0.?e6*I)*s^15 + (-3.?e6 - 2.?e6*I)*s^16 +
> (1.?e7 + 1.?e7*I)*s^17 + (0.?e7 - 4.?e7*I)*s^18 + (-7.?e7 + 1.1?
> e8*I)*s^19 + O(s^20)
> sage: sqrt(r1).parent()
> Power Series Ring in s over Complex Interval Field with 53 bits of
> precision
>
> As you can see from the O(s^20), power series have a limited
> precision. However, if you are lucky, it may help you to deal with the
> square roots.
>
> Good luck,
> Simon
--
To post to this group, send email to sage-support@googlegroups.com
To unsubscribe from this group, send email to
sage-support+unsubscr...@googlegroups.com
For more options, visit this group at
http://groups.google.com/group/sage-support
URL: http://www.sagemath.org
