tracking precision through ndsolve, findfit, regress, etc

*To*: mathgroup at smc.vnet.net*Subject*: [mg61535] tracking precision through ndsolve, findfit, regress, etc*From*: Chris Chiasson <chris.chiasson at gmail.com>*Date*: Sat, 22 Oct 2005 00:35:19 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

For anyone who may have noticed me asking a bunch of arbitrary precision related questions recently, yes - I did just finally figure out how to use precision to track "significant digits" through calculations. Sorry I'm slow... Anyway, I have a few questions: If a distance is measured with a ruler to be 7 and 3/16" inches + or - 0.5/16" inches, how should the precision of the distance number be set? FindRoot and Regress (by default if the inputs don't require high precision) both seem to return numbers that are "machine precision floats". This means they do not carry through the (possibly low) precision (or significance arithmetic or whatever... too many names) of their inputs to their outputs. For example: xample=0.025`1 FindRoot[x==xample,{x,.4}] The following is also encouraging: Interpolation does know how to create InterpolatingFunctions that will track significance *if* the interpolated data as well as the argument to the InterpolatingFunction both have significance numbers to track: data[1]=Sort@Map[Random[Real,#]&,{{1,2},{3,4},{5,6},{7,8}},{2}] data[2]=SetPrecision[data[1],5] Interpolation[data[1]][Max[data[2][[All,1]]]/2] Interpolation[data[2]][Max[data[2][[All,1]]]/2] Unfortunately, NDSolve doesn't seem to be able to create significance tracking InterpolatingFunctions: eqns[2]=x'[t]==2`20 x[t]+4`6&&x[1]==1`15 x[2]/.DSolve[eqns[2],x,t][[1]] x[2]/.NDSolve[eqns[2],x,{t,0,6}][[1]] Anyway, all of this has me thinking. If Mathematica doesn't have an automated solution to the Regress, FindRoot, and NDSolve significance tracking problems - is it possible to track significance at all through "routines" like these? Is there a way to know what precision (number of significant digits) of answer satisfies the original equations which I am trying to solve? BTW, if someone is wondering, my full "equations" through which I am trying to carry significance are as follows in the two commands below: myeqns[1][cNapRatio_,acCap_]= Equal[Times[Power[Log[10],-1],Log[acCap[cNapRatio]]], Plus[Times[0.6107999999999999999`0.9956786262173576,cNapRatio], Times[-2.0360000000000000001`2.9999999999999996, Plus[Times[-0.0061540230743798795`0.9978339382434933, Power[acCap[cNapRatio],-2], Power[Power[acCap[cNapRatio],2],Rational[1,2]]], Times[Power[2,Rational[-1,2]], Power[Plus[Times[2,cNapRatio], Times[0.0410268204958658633`3.3010299956639817, Power[acCap[cNapRatio],-2], Power[Power[acCap[cNapRatio],2],Rational[1,2]]]], Rational[1,2]], Power[Plus[1, Times[Power[2,Rational[-1,2]], Power[Plus[Times[2,cNapRatio], Times[0.0410268204958658633`3.3010299956639817, Power[acCap[cNapRatio],-2], Power[Power[acCap[cNapRatio],2],Rational[1,2]]]], Rational[1,2]]]],-1]]]]]] mysolns[3]= NDSolve[Evaluate[ And[D[myeqns[1][cNapRatio,acCap],cNapRatio], acCap[0]\[Equal](acCap[0]/.FindRoot[ Evaluate[myeqns[1][0,acCap]],{acCap[0],0.6}])]], acCap,{cNapRatio,0,0.5}] (*homotopic continuation - thanks Carl Woll! - (yes I did figure out what that means)*) Thank you for your time, -- Chris Chiasson http://chrischiasson.com/contact/chris_chiasson