Re: Integrate with piecewise function

*To*: mathgroup at smc.vnet.net*Subject*: [mg44132] Re: Integrate with piecewise function*From*: poujadej at yahoo.fr (Jean-Claude Poujade)*Date*: Thu, 23 Oct 2003 07:16:16 -0400 (EDT)*References*: <200310190510.BAA08054@smc.vnet.net> <bn2iko$o0d$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Daniel Lichtblau <danl at wolfram.com> wrote in message news:<bn2iko$o0d$1 at smc.vnet.net>... [...] > Other approaches, already mentioned in this thread, include rewriting > the function in terms of Abs or UnitStep. [...] Among 'other approaches', here is my own approach : To rewrite piecewise constant functions or piecewise linear functions in terms of UnitSteps I wrote the two modules included below. I hope my little DIY will be useful for those who want to replace the Ifs and Whichs by UnitSteps. Regards, --- jcp pieceConstant::usage = "pieceConstant[y0, piecePoints, x] \n returns a piecewise constant function expressed in terms of UnitSteps.\n arguments : \n y0 = leftmost value \n piecePoints = {{x1,y1},...,{xn,yn}} where yi are right values \n at points of discontinuity \n x = variable used to generate output expression \n example : \n pieceConstant[0,{{0,1},{1,0}},x] returns the square 'bump' function \n UnitStep[x] - UnitStep[x-1] "; pieceConstant[y0_,piecePoints_List,x_]:= Module[{lp = Length[piecePoints],f,a,b,eq,vars}, f[u_] := Sum[a[i]UnitStep[u-piecePoints[[i,1]]],{i,1,lp}]+b; eq = Append[f[#[[1]]] == #[[2]]& /@ piecePoints, f[-Infinity] == y0]; vars = Append[Table[a[i],{i,1,lp}],b]; First[f[x]/.Solve[eq,vars]] ]; pieceLinear::usage="pieceLinear[leftSlope0, piecePoints, x] \n returns a piecewise linear function expressed in terms of UnitSteps \n (function may be discontinuous).\n arguments : \n leftSlope0 : leftmost slope \n piecePoints format : \n {{x1, lefty1, righty1, rightSlope1},..., \n {xn, leftyn, rightyn, rightSlopen}} \n x : variable \n example : \n pieceLinear[0,{{0,0,0,1},{1,1,0,1},{2,1,0,1},{3,1,0,0}},x] \n returns the 'sawtooth' function \n x UnitStep[x] - (x-2)UnitStep[x-3] - UnitStep[x-2] - UnitStep[x-1]"; pieceLinear[leftSlope0_, piecePoints_List, x_]:= Module[{lp = Length[piecePoints], deltaFunctions, pieceConstantPoints, pieceDerivative, pieceLinearFunction,k,sol}, (* use delta functions to deal with discontinuities : *) deltaFunctions = (#[[3]]-#[[2]])DiracDelta[x-#[[1]]]& /@ piecePoints; pieceConstantPoints = {#[[1]],#[[4]]}& /@ piecePoints; (* build the derivative, then integrate : *) pieceDerivative = pieceConstant[leftSlope0, pieceConstantPoints,x]+ Plus@@deltaFunctions; pieceLinearFunction = Integrate[pieceDerivative,x]+k; (* compute the integration constant k : *) sol=If[lp == 1, (* specific case if there is only one point : *) Solve[(pieceLinearFunction/.x -> piecePoints[[1,1]]) == piecePoints[[1,3]],k], (* generic case : *) Solve[(pieceLinearFunction /. x -> (piecePoints[[1,1]] + piecePoints[[2,1]])/2) == (piecePoints[[1,3]] + piecePoints[[2,2]])/2,k]]; pieceLinearFunction /. sol[[1]] ];