Re: Newbie question on FindRoot and NIntergrate
- To: mathgroup at smc.vnet.net
- Subject: [mg80457] Re: Newbie question on FindRoot and NIntergrate
- From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
- Date: Thu, 23 Aug 2007 01:00:19 -0400 (EDT)
- Organization: The Open University, Milton Keynes, UK
- References: <fagu3e$937$1@smc.vnet.net>
Hoa Bui wrote: > I have a series of points: > In[9]:=points > Out[9]={{0.0001,0.359381},{0.0002866,0.403984},{0.000821394,0.454122},{0.00235411,0.510482},{0.00674688,0.573838},{0.0193365,0.645056}} > > Define one of my function as the trapezoidal area created by these > points up to some x: > linNx = Interpolation[points, InterpolationOrder -> 1]; > ff[x_] := Integrate[linNx[s], {s, 0.0001, x}] > > Define the second function as a power law: > gg[x_] := 0.9 x^(1/0.9) > > My third function is the root of the equation: > hh[x_] := FindRoot[ff[y] == gg[x], {y, 0.0001, 0.019}]; > > I can evaluate h at specific values of x, e.g. h[0.001] ({y -> > 0.00107654}), h[0.01] ({y -> 0.0101302}), etc... > > Now say I want to use hh[x] in an integral, > NIntergrate[hh[x],{x, 0.0001, 0.019}] > obviously it doesn't work, and I have tried using Solve instead of > FindRoot but it also did not output a numerical value for the integral > because the inverse function is not in closed form or something.. > > Is there a way for me to compute the integral of hh[x] ? FindRoot returns a list of transformation rules (transformation rules of the form y -> somenumber) which cannot be directly interpreted as a number by NIntegrate. Therefore, you must modified the definition of hh[x] to force it to return a number. To do so, we use the replacement operator /. in the definition below. Also, hh[x] must be called only for numeric value of its argument x. This can be achieved by adding a constraint on x in the LHS of hh as done below. In[1]:= points = {{0.0001, 0.359381}, {0.0002866, 0.403984}, {0.000821394, 0.454122}, {0.00235411, 0.510482}, {0.00674688, 0.573838}, {0.0193365, 0.645056}}; In[2]:= linNx = Interpolation[points, InterpolationOrder -> 1]; ff[x_] := Integrate[linNx[s], {s, 0.0001, x}] In[4]:= gg[x_] := 0.9 x^(1/0.9) In[5]:= hh[x_?NumberQ] := y /. FindRoot[ff[y] == gg[x], {y, 0.0001, 0.019}]; In[6]:= hh[0.001] Out[6]= 0.00107654 In[7]:= hh[0.01] Out[7]= 0.0101302 In[8]:= NIntegrate[hh[x], {x, 0.0001, 0.019}] Out[8]= 0.000183073 -- Jean-Marc