Re: Help on solving simultaneous non-linear equations using FindRoot.

• To: mathgroup at smc.vnet.net
• Subject: [mg28520] Re: [mg28505] Help on solving simultaneous non-linear equations using FindRoot.
• From: "Mark Harder" <harderm at ucs.orst.edu>
• Date: Tue, 24 Apr 2001 01:48:54 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Youyan,
I spent considerable time trying to get your code into acceptable
Mathematica format, but I could not without a lot more effort, which you
could more easily have done yourself, had you learned more Mathematica.  I
have included a notebook with my efforts so far, and the resulting error
messages.
These messages occur because expressions like p[1] are not recognized by
Mathematica as symbols for your variables, but as functions, unless they are
assigned; see the documentation for syntax for functions, etc..  Since I
didn't know what initial values to use, I didn't use the
{variable,min,initial,max} form for variable specifications (also see
below).  Instead I used the secant method specifications; and I had to
substitute some nonzero values (.001) for 0, and 10  for Infinity).

>If I take one endogenous
>variable out of the system and let it to be a given parameter,, and also
>reduce the system into 9 equations, I can get a solution that looks like
>converged,..

I don't think so.  Remove one variable from alist of 12, and you have 11
variables to solve from 9 equations, and I don't think that will a finite
number of solutions, but rather a set of surfaces in some 12-space. You
didn't list p[A] either as a parameter or a variable, so I assigned it to 1
(making it a fixed parameter), which caused Mathematica to complain about 12
equations and 11 variables. Therefore, I made it the 12th variable, with 12
equations.

>but some of the variables have negative values, which are not
>what I expected.

How do you know that negative values are not also solutions?  Did you plug
their values into your system of equations and test them?  FindRoot requires
initial guesses when you give it min..max constraints. Perhaps your choice
of guesses won't converge within the limit of iterations (15 by default); or
else some variable(s) lie outside their constraints. I just noticed that In
your text, you use FindRoot with only initial values, so that no constraints
are given. In this form, FindRoot is free to wander anywhere in Real^12. If
you really did what you show, that is probably why negative values were
returned, possibly when the limit on iterations was reached.

Maybe you will have better luck getting a reply if you translate your code
into Mathematica, and clarify your query.

-mark harder
harderm at ucs.orst.edu

-----Original Message-----
From: Youyan Li <youyanli at acsu.buffalo.edu>
To: mathgroup at smc.vnet.net
Subject: [mg28520] [mg28505] Help on solving simultaneous non-linear equations using
FindRoot.

>Hi,
>
>I am trying to solve a 12-eqaution simultaneous non-linear equations using
>FindRoot function in Mathematica. This system of equations has 9 given
>parameter values and I need to solve for 12 unknown variables. But I have
>the problem in getting the system converged. If I take one endogenous
>variable out of the system and let it to be a given parameter, and also
>reduce the system into 9 equations, I can get a solution that looks like
>converged, but some of the variables have negative values, which are not
>what I expected. I expect all the variables have non-negative values.
>If I specify the range of the variables, usually I cannot get a solution
>at all. I wonder how to solve this problem. Or how to get an acceptable
>solution for this system? Is there any trick of solving large system of
>simultaneous non-linear equations using FindRoot?
>
>Below is my problem that I converted from another program, because I don't
>know how to save a mathematica program into a text file correctly. If you
>could help, would you please take a look at it? Sorry it may look long. I
>am trying to solve functions f[1] to f[12] simultaneously.
>
>
>
>
>> M:=3/5; L:=1.0; c[1]:=1.0; c[2]:=1.0; mu:=2/5; sigma:=5; F[1]:=1.0;
>F[2]:=1.0; T:=3/2;
>
>                               M := 3/5
>
>
>                               L := 1.0
>
>
>                             c[1] := 1.0
>
>
>                             c[2] := 1.0
>
>
>                              mu := 2/5
>
>
>                              sigma := 5
>
>
>                             F[1] := 1.0
>
>
>                             F[2] := 1.0
>
>
>                               T := 3/2
>
>>
>f[1]:=(mu*Y[1]-w[1]*c[1]/p[11])*p[11]^(1-sigma)/(n[1]*p[11]^(1-sigma)+n[2]*
p[21]^(1-sigma))+(mu*Y[2]-w[1]*c[1]*T/p[12])*p[12]^(1-sigma)/(n[1]*p[12]^(1-
sigma)+n[2]*p[22]^(1-sigma))=w[1]*F[1];
>
>                       1.0 w[1]                 1.500000000 w[1]
>            2/5 Y[1] - --------      2/5 Y[2] - ----------------
>                        p[11]                        p[12]
>  f[1] := ------------------------ + ---------------------------
>               4 / n[1]     n[2] \         4 / n[1]     n[2] \
>          p[11]  |------ + ------|    p[12]  |------ + ------|
>                 |     4        4|           |     4        4|
>                 \p[11]    p[21] /           \p[12]    p[22] /
>
>        1.0 w[1]
>
>>
>
>>
>f[2]:=(mu*Y[1]-w[2]*c[2]*T/p[21])*p[21]^(1-sigma)/(n[1]*p[11]^(1-sigma)+n[2
]*p[21]^(1-sigma))+(mu*Y[2]-w[2]*c[2]/p[22])*p[22]^(1-sigma)/(n[1]*p[12]^(1-
sigma)+n[2]*p[22]^(1-sigma))=w[2]*F[2];
>
>                     1.500000000 w[2]                1.0 w[2]
>          2/5 Y[1] - ----------------     2/5 Y[2] - --------
>                          p[21]                       p[22]
>  f[2] := --------------------------- + ------------------------
>                4 / n[1]     n[2] \          4 / n[1]     n[2] \
>           p[21]  |------ + ------|     p[22]  |------ + ------|
>                  |     4        4|            |     4        4|
>                  \p[11]    p[21] /            \p[12]    p[22] /
>
>        1.0 w[2]
>
>>
>f[3]:=n[1]*(c[1]*mu*(Y[1]*p[11]^(-sigma)/(n[1]*p[11]^(1-sigma)+n[2]*p[21]^(
1-sigma))+T*Y[2]*p[12]^(-sigma)/(n[1]*p[12]^(1-sigma)+n[2]*p[22]^(1-sigma)))
+F[1])=M*lambda;
>
>               /                      Y[1]
>  f[3] := n[1] |.4000000000 ------------------------
>               |                 5 / n[1]     n[2] \
>               |            p[11]  |------ + ------|
>               |                   |     4        4|
>               \                   \p[11]    p[21] /
>
>               .6000000000 Y[2]          \
>         + ------------------------ + 1.0| = 3/5 lambda
>                5 / n[1]     n[2] \      |
>           p[12]  |------ + ------|      |
>                  |     4        4|      |
>                  \p[12]    p[22] /      /
>
>>
>f[4]:=n[2]*(c[1]*mu*(T*Y[1]*p[21]^(-sigma)/(n[1]*p[11]^(1-sigma)+n[2]*p[21]
^(1-sigma))+Y[2]*p[22]^(-sigma)/(n[1]*p[12]^(1-sigma)+n[2]*p[22]^(1-sigma)))
+F[2])=M*(1-lambda);
>
>               /                      Y[1]
>  f[4] := n[2] |.6000000000 ------------------------
>               |                 5 / n[1]     n[2] \
>               |            p[21]  |------ + ------|
>               |                   |     4        4|
>               \                   \p[11]    p[21] /
>
>               .4000000000 Y[2]          \
>         + ------------------------ + 1.0| = 3/5 - 3/5 lambda
>                5 / n[1]     n[2] \      |
>           p[22]  |------ + ------|      |
>                  |     4        4|      |
>                  \p[12]    p[22] /      /
>
>>
>f[5]:=(1+(n[1]*p[11]^(1-sigma)+n[2]*p[21]^(1-sigma))/((sigma-1)*(n[1]*p[11]
^(1-sigma)+n[2]*p[21]^(1-sigma)-p[11]^(1-sigma))))*w[1]*c[1]=p[11];
>
>                   /         / n[1]     n[2] \  \
>                   |     1/4 |------ + ------|  |
>                   |         |     4        4|  |
>                   |         \p[11]    p[21] /  |
>       f[5] := 1.0 |1 + ------------------------| w[1] = p[11]
>                   |     n[1]     n[2]      1   |
>                   |    ------ + ------ - ------|
>                   |         4        4        4|
>                   \    p[11]    p[21]    p[11] /
>
>>
>f[6]:=(1+(n[1]*p[12]^(1-sigma)+n[2]*p[22]^(1-sigma))/((sigma-1)*(n[1]*p[12]
^(1-sigma)+n[2]*p[22]^(1-sigma)-p[12]^(1-sigma))))*w[1]*c[1]*T=p[12];
>
>                       /         / n[1]     n[2] \  \
>                       |     1/4 |------ + ------|  |
>                       |         |     4        4|  |
>                       |         \p[12]    p[22] /  |
>   f[6] := 1.500000000 |1 + ------------------------| w[1] = p[12]
>                       |     n[1]     n[2]      1   |
>                       |    ------ + ------ - ------|
>                       |         4        4        4|
>                       \    p[12]    p[22]    p[12] /
>
>>
>f[7]:=(1+(n[1]*p[11]^(1-sigma)+n[2]*p[21]^(1-sigma))/((sigma-1)*(n[1]*p[11]
^(1-sigma)+n[2]*p[21]^(1-sigma)-p[21]^(1-sigma))))*w[2]*c[2]*T=p[21];
>
>                       /         / n[1]     n[2] \  \
>                       |     1/4 |------ + ------|  |
>                       |         |     4        4|  |
>                       |         \p[11]    p[21] /  |
>   f[7] := 1.500000000 |1 + ------------------------| w[2] = p[21]
>                       |     n[1]     n[2]      1   |
>                       |    ------ + ------ - ------|
>                       |         4        4        4|
>                       \    p[11]    p[21]    p[21] /
>
>>
>f[8]:=(1+(n[1]*p[12]^(1-sigma)+n[2]*p[22]^(1-sigma))/((sigma-1)*(n[1]*p[12]
^(1-sigma)+n[2]*p[22]^(1-sigma)-p[22]^(1-sigma))))*w[2]*c[2]=p[22];
>
>                   /         / n[1]     n[2] \  \
>                   |     1/4 |------ + ------|  |
>                   |         |     4        4|  |
>                   |         \p[12]    p[22] /  |
>       f[8] := 1.0 |1 + ------------------------| w[2] = p[22]
>                   |     n[1]     n[2]      1   |
>                   |    ------ + ------ - ------|
>                   |         4        4        4|
>                   \    p[12]    p[22]    p[22] /
>
>> f[9]:=M*lambda*w[1]+(L-M)/2*p[A]=Y[1];
>
>          f[9] := 3/5 lambda w[1] + .2000000000 p[A] = Y[1]
>
>> f[10]:=M*(1-lambda)*w[2]+(L-M)/2*p[A]=Y[2];
>
>       f[10] := 3/5 (1 - lambda) w[2] + .2000000000 p[A] = Y[2]
>
>> f[11]:=(1-mu)*(Y[1]+Y[2])/p[A]=L-M;
>
>                             Y[1] + Y[2]
>                f[11] := 3/5 ----------- = .4000000000
>                                p[A]
>
>>
>f[12]:=w[1]/(n[1]*p[11]^(1-sigma)+n[2]*p[21]^(1-sigma))^(mu/(1-sigma))=w[2]
/(n[1]*p[12]^(1-sigma)+n[2]*p[22]^(1-sigma))^(mu/(1-sigma));
>
>  f[12] :=
>
>             / n[1]     n[2] \(1/10)        / n[1]     n[2] \(1/10)
>        w[1] |------ + ------|       = w[2] |------ + ------|
>             |     4        4|              |     4        4|
>             \p[11]    p[21] /              \p[12]    p[22] /
>
>>
>sol:=fsolve({f[1],f[2],f[3],f[4],f[5],f[6],f[7],f[8],f[9],f[10],f[11],f[12]
},{p[A],p[11],p[12],p[21],p[22],n[1],n[2],w[1],w[2],Y[1],Y[2],lambda},{p[A]=
0..1.5,p[11]=0.5..infinity,p[12]=0.5..infinity,p[21]=0.5..infinity,p[22]=0.5
..infinity,n[1]=0..2,n[2]=0..2,w[1]=0..infinity,w[2]=0..infinity,Y[1]=0..inf
inity,Y[2]=0..infinity,lambda=0..1});
>
>  sol =FindRoot[{
>
>             /              Y[1]      .4000000000 Y[2]      \
>        n[2] |.6000000000 --------- + ---------------- + 1.0| =
>             |                 5              5             |
>             \            p[21]  %1      p[22]  %2          /
>
>        3/5 - 3/5 lambda,
>
>                  1.0 w[1]             1.50 w[1]
>        .4 Y[1] - --------   .4 Y[2] - ---------
>                   p[11]                 p[12]
>        ------------------ + ------------------- = 1.0 w[1],
>                 4                     4
>            p[11]  %1             p[12]  %2
>
>                   1.500000000 w[2]              1.0 w[2]
>        2/5 Y[1] - ----------------   2/5 Y[2] - --------
>                        p[21]                     p[22]
>        --------------------------- + ------------------- = 1.0 w[2],
>                      4                         4
>                 p[21]  %1                 p[22]  %2
>
>             /              Y[1]      .6000000000 Y[2]      \
>        n[1] |.4000000000 --------- + ---------------- + 1.0| =
>             |                 5              5             |
>             \            p[11]  %1      p[12]  %2          /
>
>                        /             1/4 %1         \
>        3/5 lambda, 1.0 |1 + ------------------------| w[1] = p[11],
>                        |     n[1]     n[2]      1   |
>                        |    ------ + ------ - ------|
>                        |         4        4        4|
>                        \    p[11]    p[21]    p[11] /
>
>                    /             1/4 %2         \
>        1.500000000 |1 + ------------------------| w[1] = p[12],
>                    |     n[1]     n[2]      1   |
>                    |    ------ + ------ - ------|
>                    |         4        4        4|
>                    \    p[12]    p[22]    p[12] /
>
>                    /             1/4 %1         \
>        1.500000000 |1 + ------------------------| w[2] = p[21],
>                    |     n[1]     n[2]      1   |
>                    |    ------ + ------ - ------|
>                    |         4        4        4|
>                    \    p[11]    p[21]    p[21] /
>
>            /             1/4 %2         \
>        1.0 |1 + ------------------------| w[2] = p[22],
>            |     n[1]     n[2]      1   |
>            |    ------ + ------ - ------|
>            |         4        4        4|
>            \    p[12]    p[22]    p[22] /
>
>        3/5 lambda w[1] + .2000000000 p[A] = Y[1],
>
>        3/5 (1 - lambda) w[2] + .2000000000 p[A] = Y[2],
>
>            Y[1] + Y[2]                       (1/10)          (1/10)
>        3/5 ----------- = .4000000000, w[1] %1       = w[2] %2      }
>               p[A]
>
>        , {Y[1], 1},{w[1],1} {p[11],1}, {n[1], 0.5},{n[2],0.5}, {p[21],1},
>{Y[2],1}, {p[12],1},{p[22],1},
>
>        {w[2],1},{lambda,0.5}, {p[A],1}]
>
>
>>
>
>
>

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```

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