# Re: taylor expansions with real vectors

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## Re: taylor expansions with real vectors

 Dear list, I have a big deal concerning the development of a Taylor expansion. require(Matrix) e1 <- as.vector(1:5) e2 <- as.vector(6:10) in order to obtain all the combinations between these two vectors following a Taylor expansion (or more simply through a Maclaurin series) for real numbers. We have f(x) = f(0) + f'(0)(x-0) + f''(0)(x-0)^2/2! + … + f^(k)(0)(x-0)^k/k! with f(x) = e1 + e2 for Taylor expansion (r = 1)         + 1/2!*e1^2 + 1/2!*e2^2 + 1/2!*e1*e2 for Taylor expansion (r = 2) excluding e2*e1         + 1/3!*e1^3 + 1/3!*e1^2*e2 + 1/3!*e2^2*e1 + 1/3!*e2^3 for Taylor expansion (r = 3) excluding e2*e1^2 and e1*e2^2        ... I already write the number of possible combinations as : x <- as.vector(0) for (r in 1:r){x[r] <- 2*(sum(choose(2*q+r-1,r))-sum(choose(q+r-1,r)))}# q: number of lag of e1 and e2; r: order of taylor expansion nstar   <- sum(x) # N* number of total combinations   How to write f(x) in a general framework? Quid of this framework when e1 and e2 are completed with their lags if q > 1? Your help or advice would be greatly appreciated Bilel -- View this message in context: http://r.789695.n4.nabble.com/RE-taylor-expansions-with-real-vectors-tp4636886.htmlSent from the R help mailing list archive at Nabble.com. ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: taylor expansions with real vectors

 On Wed, Jul 18, 2012 at 9:47 AM, bilelsan <[hidden email]> wrote: > Dear list, > > I have a big deal concerning the development of a Taylor expansion. > > require(Matrix) > e1 <- as.vector(1:5) > e2 <- as.vector(6:10) > > in order to obtain all the combinations between these two vectors following > a Taylor expansion (or more simply through a Maclaurin series) for real > numbers. > We have f(x) = f(0) + f'(0)(x-0) + f''(0)(x-0)^2/2! + … + f^(k)(0)(x-0)^k/k! > with > f(x) = e1 + e2 for Taylor expansion (r = 1) >         + 1/2!*e1^2 + 1/2!*e2^2 + 1/2!*e1*e2 for Taylor expansion (r = 2) > excluding e2*e1 >         + 1/3!*e1^3 + 1/3!*e1^2*e2 + 1/3!*e2^2*e1 + 1/3!*e2^3 for Taylor > expansion (r = 3) excluding e2*e1^2 and e1*e2^2 >        ... > I already write the number of possible combinations as : > x <- as.vector(0) > for (r in 1:r){x[r] <- 2*(sum(choose(2*q+r-1,r))-sum(choose(q+r-1,r)))}# q: > number of lag of e1 and e2; r: order of taylor expansion > nstar   <- sum(x) # N* number of total combinations > > How to write f(x) in a general framework? > Quid of this framework when e1 and e2 are completed with their lags if q > > 1? > Your help or advice would be greatly appreciated > See the section on Taylor expansions in the Ryacas package vignette. Depending on what you want to do that may or may not be relevant. -- Statistics & Software Consulting GKX Group, GKX Associates Inc. tel: 1-877-GKX-GROUP email: ggrothendieck at gmail.com ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: taylor expansions with real vectors

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## Re: taylor expansions with real vectors

 On Wed, Jul 18, 2012 at 06:02:27PM -0700, bilelsan wrote: > Leave the Taylor expansion aside, how is it possible to compute with [R]: > f(e) = e1 + e2 #for r = 1 > + 1/2!*e1^2 + 1/2!*e2^2 + 1/2!*e1*e2 #for r = 2, excluding e2*e1 > + 1/3!*e1^3 + 1/3!*e1^2*e2 + 1/3!*e2^2*e1 + 1/3!*e2^3 #for r = 3, excluding > e2*e1^2 and e1*e2^2 > + ... #for r = k > In other words, I am trying to figure out how to compute all the possible > combinations as exposed above. Hi. For a general r, do you mean the following sum of products?   1/r! (e1^r + e1^(r-1) e2 + ... e1 e2^(r-1) + e2^r) If this is correct, then try   f <- 0   for (r in 1:k) {      f <- f + 1/factorial(r) * sum(e1^(r:0)*e2^(0:r))   } Hope this helps. Petr Savicky. ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: taylor expansions with real vectors

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