Re: Replacing terms and expanding one at a time

*To*: mathgroup at christensen.cybernetics.net*Subject*: [mg1836] Re: [mg1804] Replacing terms and expanding one at a time*From*: adams (Adam Strzebonski)*Date*: Mon, 7 Aug 1995 20:24:26 -0400

You can use a series for replacement. Then only the coefficents at "significant" powers of y are computed: In[1]:= e = x^10; rep = {x-> a1 y + a2 y^2 + a3 y^3 + a4 y^4}; In[2]:= (e2 = Normal[Series[ e /. rep, {y,0,15}]]);//Timing Out[2]= {6.05 Second, Null} In[3]:= rep1 = {x -> Series[a1 y + a2 y^2 + a3 y^3 + a4 y^4, {y,0,6}]}; (* The lowest order "unsignificant" term in x^10 will be here (a1 y)^9*O[y]^7 = O[y]^16 *) In[4]:= (e3 = Normal[e/.rep1]);//Timing Out[4]= {0.0833333 Second, Null} In[5]:= e2-e3 Out[5]= 0 Adam Strzebonski WRI Begin forwarded message: >From: Stephen Corcoran <corcoran at news.ox.ac.uk> >Subject: [mg1804] Replacing terms and expanding one at a time >Organization: Oxford University Suppose I have an expression like e = x^10, where I want to replace x by something like rep = {x-> a1 y + a2 y^2 + a3 y^3 + a4 y^4} I then want to expand e, keeping terms up to say , order 15. I can do this by using: e2 = Normal[Series[ e /. rep,{y,0,15}]] Presumably, however, this is a relatively inefficient way of proceeding as it involves manipulation of the product of 10 4th degree polynomials. Is there a way to replace one of the x's at a time, and then do the expansions,i.e. something like: e2 = x^9 (a1 y + a2 y^2 + a3 y^3 + a4 y^4) e3 = x^8 (a1^2 y^2 + ..... + a4^2 y^8) e4 = x^7 (a1^3 y^3 + ..... + a4^3 y^12) e5 = x^6 (a1^4 y^4 + ..... + 4 a3 a4^3 y^15) .. and so on ? If so, is there any better in terms of speed and/or memory usage? Is this more or less what Mathematica does anyway? Thanks. ---------------------------------------------------------------------- --- Stephen Corcoran, email: corcoran at stats.ox.ac.uk (internet) Dept. of Statistics, corcoran at uk.ac.ox.stats (janet) University of Oxford, 1, South Parks Road phone: (01865) 272879 OXFORD, OX1 3TG fax: (01865) 272595 ---------------------------------------------------------------------- ----