Re: Finding unknown parameters using Mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg74824] Re: Finding unknown parameters using Mathematica*From*: "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com>*Date*: Fri, 6 Apr 2007 04:26:52 -0400 (EDT)*References*: <4614D43E0200000B002AF26F@its-gw-inet57.its.rmit.edu.au>

On 4/5/07, Shafiq Ahmad <shafiq.ahmad at rmit.edu.au> wrote: > Hi Jean-Marc, > Thanks for your good response. I tried to run your codes but could not. > Could you please send me again with input formate and appreciate if you > could add comments with your codes. Looks to me that you have better > idea than me to get values of all unknown parameters and also to get the > value of LMN from main equaition and plot result in 3D for LMN, r1 and > r2. > > Really appreciate your quick response. > Reagrds > Shafiq > > > >>> Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com> 05/04/07 5:59 AM >>> > Shafiq Ahmad wrote: > > Hi Danial, Andres Guzmanand and Group members, > > > > Thanks indeed for a good response from Danial and Andres. This is my > > first experience to find such a cooperative group. > > > > I did try to solve these non-linear equations doing some > modifications. > > I put 2 parameters r1 and r2 equal to 1 and solved rest of the > equations > > for 3 parameters using FindRoot function (as mentioned by Andres). > > > > Below are the codes in the input form. Thanks for Danial to give me > good > > advice and now really easy for every one to understand codes in the > > input formate. Next I'm trying to make these codes as generalized in > the > > following manner and strongly appreciate if any one can help me in > this > > regard. > > > > 1): I need to use a loop function for r1 and r2 values with an > increment > > of 0.1 and range for 0 to 1, it should come up with an array or list > of > > 10 values for b1, b2 and p. I did try it many ways, but mathematica > > could not accept it. Are you sure you want 10 values only? (This request is inconsistent with request #3.) I have defined LMN as a function of 5 parameters/variables. The "loop" is done by the function Table over the variables (r1 and r2, with r2 = r1 according to what you stated above), which also returns a lists of transformation rules for the unknown parameters (b1, b2, p). Join adds the rules for the r1 and r2 to the results returned by FindRoot. The last line (pts = ...) compute the corresponding values of LMN. Note that I have assumed that the undefined function L in your code is indeed the function LMN. Moreover, the code I wrote is almost a literal translation of what I have understood of your requirements into Mathematica programming language. LMN[r1_, r2_, b1_, b2_, p_] := Module[{n = Length[x1]}, n*Log[p] + n*Log[p + 1] + n*Log[b1] + n*Log[r1] + n*Log[b2] + n*Log[r2] + (b1 - 1)*Sum[Log[x1[[j]]], {j, 1, n}] + (b2 - 1)* Sum[Log[x2[[j]]], {j, 1, n}] - (p + 2)*Sum[Log[ 1 + r1*x1[[j]]^b1 + r2*x2[[j]]^b2], {j, 1, n}]] x1 = {1, 2, 3, 4}; x2 = {1.7, 3.8, 4.9, 4.6}; sols = Table[Block[{r2, b1, b2, p, Eqn1, Eqn2, Eqn3}, r2 = r1; Eqn1 = D[LMN[r1, r2, b1, b2, p], b1] == 0; Eqn2 = D[LMN[r1, r2, b1, b2, p], b2] == 0; Eqn3 = D[LMN[r1, r2, b1, b2, p], p] == 0; Join[{a -> r1, b -> r2}, FindRoot[ {Eqn1, Eqn2, Eqn3}, {b1, 10}, {b2, 10}, {p, 10}]]], {r1, 0.1, 1, 0.1}] /. a -> r1 /. b -> r2 pts = LMN[r1, r2, b1, b2, p] /. sols > > 2): After having values for all 5 parameters (r1,r2,b1,b2 and p), I > need > > to put all these values in the main equation " LMN" to get another > list > > or array for LMN values . Individually I used all parameter values, I > > got LMN value but mathematica don't accept if I give all values in the > > form of list or array and do it all one time. > > > > 3): After having LMN values, I would like plot a 3D plot or 3D contour > > plot / surface plot with array / or list values for r1,r2 and LMN. If you have a one-dimensional list of 10 values you cannot use any of the contour/density/surface plots. Below is a version of the code returning a square array (10 by 10) that can be used with, say, ListContourPlot. LMN[r1_, r2_, b1_, b2_, p_] := Module[{n = Length[x1]}, n*Log[p] + n*Log[p + 1] + n*Log[b1] + n*Log[r1] + n*Log[b2] + n*Log[r2] + (b1 - 1)*Sum[Log[x1[[j]]], {j, 1, n}] + (b2 - 1)* Sum[Log[x2[[j]]], {j, 1, n}] - (p + 2)*Sum[Log[ 1 + r1*x1[[j]]^b1 + r2*x2[[j]]^b2], {j, 1, n}]] x1 = {1, 2, 3, 4}; x2 = {1.7, 3.8, 4.9, 4.6}; sols = Table[Block[{b1, b2, p, Eqn1, Eqn2, Eqn3}, Eqn1 = D[LMN[r1, r2, b1, b2, p], b1] == 0; Eqn2 = D[LMN[r1, r2, b1, b2, p], b2] == 0; Eqn3 = D[LMN[r1, r2, b1, b2, p], p] == 0; Join[{a -> r1, b -> r2}, FindRoot[{Eqn1, Eqn2, Eqn3}, {b1, 10}, {b2, 10}, {p, 10}]]], {r1, 0.1, 1, 0.1}, {r2, 0.1, 1, 0.1}] /. a -> r1 /. b -> r2; pts = LMN[r1, r2, b1, b2, p] /. sols; ListContourPlot[pts]; Regards, Jean-Marc > > Strongly appreciate if some one can share his/her thoughts. > > > > Regards > > Shafiq > > > > =================================================== > > > > n = 4 > > > > x1 = {1, 2, 3, 4} > > x2 = {1.7, 3.8, 4.9, 4.6} > > > > r1 = 1 > > r2 = 1 > > > > LMN = n*Log[p] + n*Log[p + 1] + n*Log[b1] + n*Log[r1] + n*Log[b2] + > > n*Log[r2] + > > (b1 - 1)*Sum[Log[x1[[j]]], {j, 1, n}] + > > (b2 - 1)*Sum[Log[x2[[j]]], {j, 1, n}] - > > (p + 2)*Sum[Log[1 + r1*x1[[j]]^b1 + r2*x2[[j]]^b2], {j, 1, n}] > > > > Eqn1 = D[L, b1] == 0 > > > > Eqn2 = D[L, b2] == 0 > > > > Eqn3 = D[L, p] == 0 > > > > FindRoot[{Eqn1, Eqn2, Eqn3}, {b1, 10}, {b2, 10}, {p, 10}] > > > > ================================================ > > Regards > > Shafiq > [snip] > > Please, find hereunder an example of what you could do to achieve what > you want, assuming that I have correctly understood your request. (Note > that the following code is not an example of good programming practice. > Quick and dirty, but it works!) > > In[1]:= > LMN[r1_, r2_, b1_, b2_, p_] := > Module[{n = Length[x1]}, n*Log[p] + n*Log[p + 1] + > n*Log[b1] + n*Log[r1] + n*Log[b2] + n*Log[r2] + > (b1 - 1)*Sum[Log[x1[[j]]], {j, 1, n}] + > (b2 - 1)*Sum[Log[x2[[j]]], {j, 1, n}] - > (p + 2)*Sum[Log[1 + r1*x1[[j]]^b1 + > r2*x2[[j]]^b2], {j, 1, n}]] > > In[2]:= > x1 = {1, 2, 3, 4}; > x2 = {1.7, 3.8, 4.9, 4.6}; > > In[4]:= > sols = Table[Block[{r2, b1, b2, p, Eqn1, Eqn2, Eqn3}, > r2 = r1; Eqn1 = D[LMN[r1, r2, b1, b2, p], b1] == > 0; Eqn2 = D[LMN[r1, r2, b1, b2, p], b2] == 0; > Eqn3 = D[LMN[r1, r2, b1, b2, p], p] == 0; > Join[{a -> r1, b -> r2}, FindRoot[ > {Eqn1, Eqn2, Eqn3}, {b1, 10}, {b2, 10}, > {p, 10}]]], {r1, 0.1, 1, 0.1}] /. a -> r1 /. > b -> r2 > > Out[4]= > {{r1 -> 0.1, r2 -> 0.1, b1 -> 5.341935448717984, > b2 -> 3.9436875174176804, p -> 0.3721885234758986}, > {r1 -> 0.2, r2 -> 0.2, b1 -> 5.534042581697728, > b2 -> 4.045832415596042, p -> 0.2940686067033152}, > {r1 -> 0.30000000000000004, > r2 -> 0.30000000000000004, b1 -> 5.626397092321994, > b2 -> 4.095035940496871, p -> 0.26139996011190647}, > {r1 -> 0.4, r2 -> 0.4, b1 -> 5.697719904063341, > b2 -> 4.136114771126181, p -> 0.24147559059244234}, > {r1 -> 0.5, r2 -> 0.5, b1 -> 5.7580442518190855, > b2 -> 4.172731896134758, p -> 0.22753002319002713}, > {r1 -> 0.6000000000000001, r2 -> 0.6000000000000001, > b1 -> 5.810601737463024, b2 -> 4.20572325658841, > p -> 0.2170152505036831}, > {r1 -> 0.7000000000000001, r2 -> 0.7000000000000001, > b1 -> 5.8571172402134515, b2 -> 4.235590112750038, > p -> 0.20870033683174416}, {r1 -> 0.8, r2 -> 0.8, > b1 -> 5.898742384194647, b2 -> 4.262751009402106, > p -> 0.20190091294950313}, {r1 -> 0.9, r2 -> 0.9, > b1 -> 5.9363209375865305, b2 -> 4.287567369366186, > p -> 0.19619991348677382}, {r1 -> 1., r2 -> 1., > b1 -> 5.970499008585786, b2 -> 4.310347981142145, > p -> 0.1913259981578475}} > > In[5]:= > pts = LMN[r1, r2, b1, b2, p] /. sols > > Out[5]= > {-12.86601076530539, -13.020764326190708, > -13.17718328097974, -13.311047643795543, > -13.424702323411111, -13.522321810909304, > -13.607333751744086, -13.682314918879541, > -13.749187133239175, -13.809399854995021} > > In[6]:= > sols = Table[Block[{b1, b2, p, Eqn1, Eqn2, Eqn3}, > Eqn1 = D[LMN[r1, r2, b1, b2, p], b1] == 0; > Eqn2 = D[LMN[r1, r2, b1, b2, p], b2] == 0; > Eqn3 = D[LMN[r1, r2, b1, b2, p], p] == 0; > Join[{a -> r1, b -> r2}, FindRoot[ > {Eqn1, Eqn2, Eqn3}, {b1, 10}, {b2, 10}, > {p, 10}]]], {r1, 0.1, 1, 0.1}, > {r2, 0.1, 1, 0.1}] /. a -> r1 /. b -> r2; > pts = LMN[r1, r2, b1, b2, p] /. sols; > ListContourPlot[pts]; > > [...graphic deleted...] > > Hope this helps, > Jean-Marc > >