Re: "Incremental Integration"?

*To*: mathgroup at smc.vnet.net*Subject*: [mg31788] Re: [mg31774] "Incremental Integration"?*From*: BobHanlon at aol.com*Date*: Sun, 2 Dec 2001 04:25:04 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

In a message dated 2001/12/1 4:30:41 AM, siegman at stanford.edu writes: >Given a rather messy function f[x] , I want to make a Table of its integral > >from x0 up to a set of equally spaced points xn = x0 + n*dx with maximum >value >xFinal -- in other words > > g[x_] := NIntegrate[f[xi], {xi, x0, x}] > > Table[ {x, g[x]}, {x, x0, xFinal, dx}] > >Is Mathematica smart enough to calculate each successive value using the > >previous value plus integration just over the increment in x to the next >value? > >If not, what would be a clean and readable and also reasonably efficient >way to >code this? > >[Note: For me anyway, there's some emphasis on the "clean and readable" >part -- >avoiding esoteric Mathematica syntax, not taking terseness as the ultimate >goal.] > f[x_] := x^2; x0=0; xFinal=10; dx=2; rng = Range[x0, xFinal, dx]; Transpose[{rng, FoldList[#1+#2&,0, NIntegrate[f[x],{x,#[[1]],#[[2]]}]& /@ Partition[rng,2,1]]}] If readability is taken as the primary metric for code, then you will likely end up using predominantly procedural programming and not make effective use of functional programming. If a complex expressions cannot be read easily, build the expression step-by-step and convert all of the steps (input and output) into a merged text cell to keep with the code. This should be done in as general a form as is practical. For example, evaluate, merge, and convert to text the following Clear[x0,f] rng = {x0,x1,x2,x3}; Partition[%,2,1] Integrate[f[x], {x,#[[1]],#[[2]]}]& /@ % FoldList[#1+#2&,0,%] (* NOTE: For NIntegrate the numbers are folded and each integral is evaluated once. *) {rng,%} Transpose[%] Bob Hanlon Chantilly, VA USA