Re: repeated series expansion
Vladimir Skokov <[email protected]> Fri, 16 Jul 2010 16:10:17 +0200
| Newsgroups | gmane.comp.mathematics.axiom.general |
|---|---|
| Message-ID | <[email protected]> |
--===============1649266451== Content-Type: multipart/alternative; boundary=0015175cd7f27ae4e3048b81c815 --0015175cd7f27ae4e3048b81c815 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Dear Ralf, thank you for help! I have got program working. I am sorry for being annoying, but I only start using axiom. Thank you ;) On Fri, Jul 16, 2010 at 2:08 PM, Ralf Hemmecke <[email protected]> wrote: > On 07/16/2010 12:52 PM, Vladimir Skokov wrote: > >> The problem is that my function contains logs, it is not just Taylor >> series. >> > > For instance, I'd like to expand f(x,y)log(x) + g(x,y) at x=3D0, y=3D0. >> So I expect to obtain >> (expansion of f(x,y)) * log(x) + expansion of g(x,y) >> > > Why don't people write exact specifications? The above is meaningless > unless you specify how your output should look like. > > Do you want to have a taylor series in x which has coefficients being > taylor series in y? Or do you want the same with roles of x and y exchang= ed? > Or do you want a *univariate* power series whose coefficients are > (homogeneous) polynomials in x and y with a degree corresponding to the > degree of the power in the taylor series? > > Specify exactly what you want otherwise nobody will be able to help you. > I am not willing to guess your specification. > > > e.g. for one variable "series" does a good job >> series(sin(x)*log(x),x=3D0) >> (5) -> >> (5) >> log(x) 3 log(x) 5 log(x) 7 log(x) 9 log(x) >> 11 >> log(x)x - ------ x + ------ x - ------ x + ------ x - -------- = x >> 6 120 5040 362880 39916800 >> + >> 12 >> O(x ) >> Type: GeneralUnivariatePowerSeries(Expression >> Integer,x,0) >> > > However this is not Taylor series and therefore I cannot extract >> coefficients as you specified. >> > > ???? > > Look at the type. So we have > > (3) -> coefficient(s,1) > > (3) log(x) > Type: Expression(Integer) > > I agree, that this might not be what you expected, but that is the proble= m > with your input. You haven't exactly specified what you wanted. > Believe it or not, Axiom considers "log(x)" as a separate variable. > > More natural would be something like > > U :=3D UnivariateTaylorSeries(Fraction Integer, 'x, 1) > x: U :=3D x::U > log(x) > > (3) > 1 2 1 3 1 4 1 5 1 = 6 > (x - 1) - - (x - 1) + - (x - 1) - - (x - 1) + - (x - 1) - - (x - = 1) > 2 3 4 5 6 > + > 1 7 1 8 1 9 1 10 11 > - (x - 1) - - (x - 1) + - (x - 1) - -- (x - 1) + O((x - 1) ) > 7 8 9 10 > Type: > UnivariateTaylorSeries(Fraction(Integer),x,1) > > Now the error message > > > (6) -> sin x > > >> Error detected within library code: > "sincos: series expansion involves transcendental constants" > > > should be clear. The result is simply not representable, by a series with > just rational coefficients. > > Try > > V :=3D UnivariateTaylorSeries(Expression Integer, 'y, 1) > y: V :=3D y::V > sin(y) > sin(y)*log(y) > > instead. > > It all depends on what you want. AXIOM *is* different from other CAS. > > > Of course the example I have written is oversimplified. I do not know >> the structure of the function, it can contain sin(log(x)) for example. >> > > But you know what you want as a result. > > > Mathematica allows repeated series expansion, however it fails with >> complicated expressions due to some memory limitation. >> > > A Mathematica Series is a *truncated* series. Remove the initial terms an= d > all you are left with is an O(x^10) expression which will not deliver any > more coefficients. AXIOM's series really represent infinite objects. > > Ralf > > > _______________________________________________ > Axiom-math mailing list > [email protected] > http://lists.nongnu.org/mailman/listinfo/axiom-math > --=20 _____________________________________________ Dr. Vladimir Skokov Theory Division GSI Helmholtzzentrum f=FCr Schwerionenforschung GmbH Planckstra=DFe 1 D-64291 Darmstadt ------------------------------------------------------------------------ email: [email protected] phone: +49 6159-71 2751 fax : +49 6159-71 2990 --0015175cd7f27ae4e3048b81c815 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Dear=A0Ralf,=A0<div>thank you for help! I have got program working.=A0</div= ><div>I am sorry for being=A0annoying, but I only start using axiom.=A0</di= v><div>Thank you ;) =A0<br><br><div class=3D"gmail_quote">On Fri, Jul 16, 2= 010 at 2:08 PM, Ralf Hemmecke <span dir=3D"ltr"><<a href=3D"mailto:ralf@= hemmecke.de">[email protected]</a>></span> wrote:<br> <blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p= x #ccc solid;padding-left:1ex;"><div class=3D"im">On 07/16/2010 12:52 PM, V= ladimir Skokov wrote:<br> <blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p= x #ccc solid;padding-left:1ex"> The problem is that my function contains logs, it is not just Taylor series= .<br> </blockquote> <br> <blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p= x #ccc solid;padding-left:1ex"> For instance, I'd like to =A0expand f(x,y)log(x) + g(x,y) at x=3D0, y= =3D0.<br> So I expect to obtain<br> =A0(expansion of f(x,y)) * log(x) + expansion of g(x,y)<br> </blockquote> <br></div> Why don't people write exact specifications? The above is meaningless u= nless you specify how your output should look like.<br> <br> Do you want to have a taylor series in x which has coefficients being taylo= r series in y? Or do you want the same with roles of x and y exchanged? Or = do you want a *univariate* power series whose coefficients are (homogeneous= ) polynomials in x and y with a degree corresponding to the degree of the p= ower in the taylor series?<br> <br> Specify exactly what you want otherwise nobody will be able to help you.<br= > I am not willing to guess your specification.<div class=3D"im"><br> <br> <blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p= x #ccc solid;padding-left:1ex"> e.g. for one variable "series" does a good job<br> series(sin(x)*log(x),x=3D0)<br> (5) -><br> =A0 =A0(5)<br> =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0log(x) =A03 =A0 log(x) =A05 =A0 log(x) =A07= =A0 log(x) =A09 =A0 =A0log(x) =A0 11<br> =A0 =A0 =A0log(x)x - ------ x =A0+ ------ x =A0- ------ x =A0+ ------ x = =A0- -------- x<br> =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 6 =A0 =A0 =A0 =A0 =A0120 =A0 =A0 =A0 = =A05040 =A0 =A0 =A0 362880 =A0 =A0 =A039916800<br> =A0 =A0+<br> =A0 =A0 =A0 =A0 12<br> =A0 =A0 =A0O(x =A0)<br> =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0Type: GeneralUnivariatePowerSeries(= Expression<br> Integer,x,0)<br> </blockquote> <br> <blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p= x #ccc solid;padding-left:1ex"> However this is not Taylor series and therefore I cannot extract<br> coefficients as you specified.<br> </blockquote> <br></div> ????<br> <br> Look at the type. So we have<br> <br> (3) -> coefficient(s,1)<br> <br> =A0 (3) =A0log(x)<br> =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 Type: = Expression(Integer)<br> <br> I agree, that this might not be what you expected, but that is the problem = with your input. You haven't exactly specified what you wanted.<br> Believe it or not, Axiom considers "log(x)" as a separate variabl= e.<br> <br> More natural would be something like<br> <br> U :=3D UnivariateTaylorSeries(Fraction Integer, 'x, 1)<br> x: U :=3D x::U<br> log(x)<br> <br> =A0 (3)<br> =A0 =A0 =A0 =A0 =A0 =A0 =A0 1 =A0 =A0 =A0 =A02 =A0 1 =A0 =A0 =A0 =A03 =A0 = 1 =A0 =A0 =A0 =A04 =A0 1 =A0 =A0 =A0 =A05 =A0 1 =A0 =A0 =A06<br> =A0 =A0 (x - 1) - - (x - 1) =A0+ - (x - 1) =A0- - (x - 1) =A0+ - (x - 1) = =A0- - (x - 1)<br> =A0 =A0 =A0 =A0 =A0 =A0 =A0 2 =A0 =A0 =A0 =A0 =A0 =A03 =A0 =A0 =A0 =A0 =A0= =A04 =A0 =A0 =A0 =A0 =A0 =A05 =A0 =A0 =A0 =A0 =A0 =A06<br> =A0 +<br> =A0 =A0 1 =A0 =A0 =A0 =A07 =A0 1 =A0 =A0 =A0 =A08 =A0 1 =A0 =A0 =A0 =A09 = =A0 =A01 =A0 =A0 =A0 =A010 =A0 =A0 =A0 =A0 =A0 =A011<br> =A0 =A0 - (x - 1) =A0- - (x - 1) =A0+ - (x - 1) =A0- -- (x - 1) =A0 + O((x= - 1) =A0)<br> =A0 =A0 7 =A0 =A0 =A0 =A0 =A0 =A08 =A0 =A0 =A0 =A0 =A0 =A09 =A0 =A0 =A0 = =A0 =A0 =A010<br> =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0Type: UnivariateTaylorS= eries(Fraction(Integer),x,1)<br> <br> Now the error message<br> <br> <br> (6) -> sin x<br> <br> =A0 >> Error detected within library code:<br> =A0 "sincos: series expansion involves transcendental constants"= <br> <br> <br> should be clear. The result is simply not representable, by a series with j= ust rational coefficients.<br> <br> Try<br> <br> V :=3D UnivariateTaylorSeries(Expression Integer, 'y, 1)<br> y: V :=3D y::V<br> sin(y)<br> sin(y)*log(y)<br> <br> instead.<br> <br> It all depends on what you want. AXIOM *is* different from other CAS.<div c= lass=3D"im"><br> <br> <blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p= x #ccc solid;padding-left:1ex"> Of course the example I have written is oversimplified. I do not know<br> the structure of the function, it can contain sin(log(x)) for example.<br> </blockquote> <br></div> But you know what you want as a result.<div class=3D"im"><br> <br> <blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p= x #ccc solid;padding-left:1ex"> Mathematica allows repeated series expansion, however it fails with<br> complicated expressions due to some memory limitation.<br> </blockquote> <br></div> A Mathematica Series is a *truncated* series. Remove the initial terms and = all you are left with is an O(x^10) expression which will not deliver any m= ore coefficients. AXIOM's series really represent infinite objects.<br> <font color=3D"#888888"> <br> Ralf</font><div><div></div><div class=3D"h5"><br> <br> _______________________________________________<br> Axiom-math mailing list<br> <a href=3D"mailto:[email protected]" target=3D"_blank">Axiom-math@nongn= u.org</a><br> <a href=3D"http://lists.nongnu.org/mailman/listinfo/axiom-math" target=3D"_= blank">http://lists.nongnu.org/mailman/listinfo/axiom-math</a><br> </div></div></blockquote></div><br><br clear=3D"all"><br>-- <br><br>_______= ______________________________________<br><br>Dr. Vladimir Skokov<br><br>Th= eory Division<br>GSI Helmholtzzentrum f=FCr Schwerionenforschung GmbH<br>Pl= anckstra=DFe 1<br> D-64291 Darmstadt<br>------------------------------------------------------= ------------------<br>email:=A0=A0 [email protected]<br>phone: +49 6159-71 27= 51<br>fax=A0 :=A0 =A0 +49 6159-71 2990<br> </div> --0015175cd7f27ae4e3048b81c815-- --===============1649266451== Content-Type: text/plain; charset="us-ascii" MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Disposition: inline _______________________________________________ Axiom-math mailing list [email protected] http://lists.nongnu.org/mailman/listinfo/axiom-math --===============1649266451==--