Thanks, Ted. I meant to post this previous reply to the list:
On 10-07-14 07:43, David Bickel wrote:
Hi Ted,
Here are my translations to "integrate"-like syntax of a conflation of
examples from the following Mathematica pages:
x <- 2.232; sum(f = function(n) {x^n/factorial(n)}, lower = 0, upper =
Inf, numeric.method = "HypergeometricTermZeilberger",
verify.convergence = TRUE)
Maybe there is an R function that will either add one term at a time
until some specified tolerance level is met or that will add a
specified number of terms but throw an error if the specified
tolerance level is exceeded.
Best regards,
David
http://reference.wolfram.com/mathematica/tutorial/SummationOfSeries.html
http://reference.wolfram.com/mathematica/ref/Sum.html
List members, Ted's helpful reply follows.
David
On 10-07-14 08:48, (Ted Harding) wrote:
Hi David,
I think you should post this (and your previous reply) also to
R-help, since there may be some people who know of something
relevant. I'm not aware of any R function which operates on
the lines of the "Mathematica-like" example you gave in the
previous one.
R is primarily oriented towards statistical computation, and
mainly implements technical mathematical computations so far
as they are needed for statistical purposes. Mathematica, on
the other hand, is directly oriented to general mathematical
computation.
One issue (which is illustrated by the "Mathematica" example)
is that the convergence criterion which should be applied for
a given series will depend on what that series is. For example,
in a series of alternating positive and negative terms, which
successively decrease to zero in absolute value, the convergence
criterion is easy: the error made in truncating the summation
at a given term is less than the absolute value of the next term
(and, also, convergence is guaranteed for any such series).
On the other hand, the criterion to applied for the sum of a
series like sum[from 0 to infinity]( 1/(n^alpha) ) (alpha> 1)
is not so obvious, and may require a theoretical study specific
to the particular series (indeed, some series have given rise
to research articles studying their convergence which have been
published over many years, their convergence may occupy whole
chapters of textbooks on mathematical analysis, and the precise
nature of their convergence could still be an ongoing problem).
With best wishes,
Ted.
On 14-Jul-10 12:23:06, David Bickel wrote:
More generally, can R approximate the limit of a specified function as
an argument diverges? If so, I might be able to use a partial sum as
that specified function.
David
On 14/07/2010 7:18 AM, (Ted Harding) wrote:
On 14-Jul-10 10:57:02, David Bickel wrote:
What are some reliable R functions that can compute the value of a
convergent series?
David
Please give an example of a definition (as you would specify it
to an R function) of a "convergent series" that you want to compute!
Your query is about as general as could possibly be, and cannot
be presented to R as it stands, since a convergent series is an
infinite sequence of terms which could be anything so long as the
series converges.
There are already many R functions -- such as sin(), exp() -- which
compute values for specific convergent series, though often by
special methods which do not work through the series of terms.
Presumably a general series would be specified in terms of some
rule which defines the nth term in terms of the (n-1)th, and
possibly preceding, terms. Once such a specification has been
given, an R function can be written. Are you thinking of something
on the lines of
sersum<- function(x,fun1,fun)
where fun1(x) would be a user-supplied function which computes
the first term as a function of x, and fun(x,n,tn) a user-supplied
function which computes term n in terms of tn=term (n-1), and n,
for n>1?
E.g. for the exponential series,
fun1<- function(x) 1
fun<- function(x,n,tn) tn*x/n
Ted.
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Fax-to-email: +44 (0)870 094 0861
Date: 14-Jul-10 Time: 13:48:29
------------------------------ XFMail ------------------------------
--
David R. Bickel, PhD
Associate Professor
Ottawa Institute of Systems Biology
Biochem., Micro. & I. Department
Mathematics & Statistics Department
University of Ottawa
451 Smyth Road
Ottawa, Ontario K1H 8M5
http://www.statomics.com
Office Tel: (613) 562-5800 ext. 8670
Office Fax: (613) 562-5185
Office Room: RGN 4510F
Lab Tel.: (613) 562-5800 ext. 8304
Lab Room: RGN 4501T
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