Re: Finding unknown parameters using Mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg74766] Re: Finding unknown parameters using Mathematica*From*: Daniel Huber <dh at metrohm.ch>*Date*: Wed, 4 Apr 2007 04:06:58 -0400 (EDT)*References*: <461390240200000B002AF1AB@its-gw-inet57.its.rmit.edu.au>

Hi Shafiq, If L has no value, your code can not work. Consider: D[L,b1]==0. If L has no value the drivative will be zero and the statement gives True. To make a list from 0.1..1: Table[x,{x,0.1,1,0.1}] To calculate an expression (LMN) for different sets of parameters we make an example: Assume that pars={a,b} are the parameters and vals are a list of different sets of values: vals={{0,0},{0,1},{1,1}} and expr= b a^2 is the expression you wish to evaluate for different sets of parameters.. We then create a list of rules: rules= MapThread[Rule,{pars,#}]&/@valsset and applies the rules to expr: expr /. rules. This gives a list of values of expr for different values of the parameters. To make a 3D Plot we need a rectangular array of height (expr) values. We can get automatically this if we replace in the foregoing the values of vars by an rectangular arra, e.g.: vars={{{0,0},{0,1}},{{1,0},{1,1}}. The foregoing procedure then returns an arry of values of expr, e.g.: {{0,0},{0,1}}. This is what ListPlot3D or ListContourPlot wants as input. Daniel 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. > > 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. > > 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 > > > > >>>> dh <dh at metrohm.ch> 03/04/07 12:36 AM >>> >>>> > Hi Shafiq. > there is something wrong in your code. What should:"=E2=88=91" mean? I > think "Sum". Note that it is better if you change your code to InputForm > > before posting it. This is done by selecting and then menu: Cell/Convert > > To/InputForm. > Further, your set up of the equations is wrong. What you wrote is: > Short Form of drivative == explicite Form of derivative > (D[L,r1]==...), what shlould give True. Note that the form: D[L,r1] is > only used in differential equations. In your case you would write: > eqns={ explicite form of derivative == 0, ...} > Solve[eqns,vars] > Daniel > > Shafiq Ahmad wrote: > >> Dear group members, >> >> I'm very new to mathematica and trying to solve a set of non-linear >> > system = > >> of equations to find the unknown parameters for a bivariate >> > distribution. = > >> I've 5 unknown parameters (i.e. b1,b2,r1,r2,p) and 5 set of equations. >> > I = > >> tried to get the general solution , but could not. I don't know how to >> > = > >> solve these nonlinear equations to get the unknown parameters. And >> > also = > >> not aware how to give initial value in the solve function or any other >> > = > >> function (e.g all parameters b1,b2,r1,r2,p if I give initial boundary >> > = > >> value =1). >> In the below codes, x1 and x2 are 2 variables (e.g. data from 2 >> > quality = > >> characteristics; stress and strain etc. etc.) >> >> Any comments / suggestions how to solve these equation for >> > b1,b2,r1,r2,p = > >> where as putting intial value for all these unknown parameters =1 >> >> Ahmad S. >> >> > > > > -- Daniel Huber Metrohm Ltd. Oberdorfstr. 68 CH-9100 Herisau Tel. +41 71 353 8585, Fax +41 71 353 8907 E-Mail:<mailto:dh at metrohm.ch> Internet:<http://www.metrohm.ch>