Re: Re: Finding unknown parameters using Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg74985] Re: [mg74824] Re: Finding unknown parameters using Mathematica
- From: "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com>
- Date: Fri, 13 Apr 2007 02:09:34 -0400 (EDT)
- References: <461E5A500200000B002AF50E@its-gw-inet57.its.rmit.edu.au>
[Message cross posted on MathGroup]
On 4/12/07, Shafiq Ahmad <shafiq.ahmad at rmit.edu.au> wrote:
> Hi Jean,
>
> Thanks a lot indeed for your quick help and so sorry for my delayed
> response. Actually there was one week public holidays here.
>
> I have tried your approach and your programs runs very well. I do have
> to change increment value for the r1/r2 i.e. {r1, 0.1, 1, 0.15}, {r2,
> 0.1, 1, 0.15}] /. a -> r1 /. b -> r2 in the last before contour plot
> command to avoid error messages for precision.
>
> I would like to understand few points. Pls help.
> 1). When this program runs, first time it takes value for the r1/r2
> single time and get the values for LMN (ref. page 1 of attached file
> pls)
This is what I have understood from your original request.
> 2). When it runs for next (see page 2), it takes single value of r1 and
> keep on incrementing r2 value in the loop and vice versa . This way so
> many values of LMN have been listed. What I was expecting, b1,b2 and p
> will have same values as r1/r2 and r1/r2 should go parallel in
> incrementing the values. What is ur thought ?
ListCountour/DensityPlot requires a rectangular array of heights
(z-values) to be plotted, and the x and y values are assumed to be
evenly spaced. I just made up such an array on the fly only with the
intent to illustrate how to use these functions. Do not take it as
being a solution to your original problem. (The first two requirements
were contradictory.)
> 3.) In contour/ surface 3D plot, I was expecting a plot with 3
> components such as against Z axis -> r2, Y-axis->LMN and X-axis -> r1.
> I'm not sure the graph I'm getting using list contourPlot is using same
> values for these axis.
No. " ListContourPlot[array] by default takes the a and a coordinate
values for each data point to taken to be successive integers starting
at 1. (Online Help)"
> 4). Is there any option in mathematica for getting the surface 3D plots?
ListPlot3D might fits your needs: "ListPlot3D[array] generates a
three$B!>(Bdimensional plot of a surface representing an array of height
values. (Online Help)"
Regards,
Jean-Marc
> Many thanks for your continued help
> Regards
> Shafiq
>
>
>
>
> >>> "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com> 06/04/07 6:26 PM
> >>>
> 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
> >
> >
>
>
>