# Solving system of n equations

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Susan on 10 Apr 2019
Commented: Susan on 10 May 2019
Hi MATLAb guys,
I am stucking at some point and need your help.
Do you know how it is possible to find a relationship between two variables in one equations? for example y = ax+b, (a,b) are given, (x,y) are variables. I would like matlab give me x = (y-b)/a.
I have wrote the following code, but it doesn't work out well. Any idea how to fix it? Thanks in advance
Input: n_W, n_L, W_net1, m_net1, m_net2, beta
Output: W_net2
funX=@(W, W_net1, m_net1) (2*(1 - 2*W))/((1 - 2*W)*(1 + W_net1) + W*W_net1*(1 - (2*W)^m_net1));
funY=@(Z, W_net2, m_net2) (2*(1 - 2*Z))/((1 - 2*Z)*(1 + W_net2) + Z*W_net2*(1 - (2*Z)^m_net2));
funW=@(X,Y,n_W,n_L) (1 - ((1 - X)^(n_W - 1))*(1 - Y)^n_L);
funZ=@(X,Y,n_W,n_L)(1 - ((1 - X)^n_W)*(1 - Y)^(n_L - 1));
funS=@(X,Y,n_W,n_L,beta)(Y*((1 - Y)^(n_L - 1))*(1 - X)^(n_W))-beta;
fun1=@(X,Y,W,Z,n_W,n_L,W_net2, m_net2, W_net1, m_net1, beta) ([funX(W,W_net1,m_net1); funY(Z,W_net2,m_net2) ; funW(X,Y,n_W,n_L); funZ(X,Y,n_W,n_L) ; funS(X,Y,n_W,n_L,beta)]);
n_W = 3;
n_L = 4;
beta = 0.7;
m_net2 = 4;
W_net1 = 8;
m_net1= 4;
fun2=@(P) (P-fun1(P(1),P(2),P(3),P(4),n_W,n_L,beta,m_net1,W_net1,m_net2));
InitialGuess=[0;0;0;0];
fsolve(fun2,InitialGuess)
P.S. The approach is
1) knowing beta, from S ===> relationship between X and Y can be found.
2) from X and W ===> X and Y can be found numerically
3) based on Y, W_net2 can be derived

Walter Roberson on 12 Apr 2019
With this newer set of equations I am able to get down to a long expression in W_net2. The expression is rather long, but turns out to be closed form, all "algebraic" numbers (it involves a cubic and a quartic.) Of the 12 combinations of expressions, 10 of them turn out to be complex valued in the range W_net2 = 0 to 1024. Two of the combinations are real valued -- but only over 0 to slightly less than 4. One of those two has a singularity in that range and so ends up not crossing 0. The other is a bit oddly shaped but does not cross zero in the range. So... there does not appear to be any real-valued solutions.
Susan on 12 Apr 2019
Thanks for the note.
Nice work! I appreciate your time. Just a quick question, is funS founction written correctly? shouldn't be "funS=@(X,Y,n_W,n_L,beta)(Y*((1 - Y)^(n_L - 1))*(1 - X)^(n_W) -beta);"? I mean beta is outside the () in the above equation.
Still I get p2>1 and W_net2<0 even with new equations.
Susan on 12 Apr 2019
Walter, Could you please kindly take a look at my code and tell me why I am not getting the results that you get? Thank you so much in advance.
clear;
clc;
funX=@(X, W, W_net1, m_net1) ( (2*(1 - 2*W)/((1 - 2*W)*(1 + W_net1) + W*W_net1*(1 - (2*W)^m_net1))) - X );
funY=@(Y, Z, W_net2 ,m_net2) ( (2*(1 - 2*Z)/((1 - 2*Z)*(1 + W_net2) + Z*W_net2*(1 - (2*Z)^m_net2))) - Y );
funW=@(W, X, Y, n_W, n_L) ( 1 - ((1 - X)^(n_W - 1))*((1 - Y)^n_L) - W );
funZ=@(Z, X, Y, n_W, n_L) ( 1 - ((1 - X)^n_W)*((1 - Y)^(n_L - 1)) - Z );
funS=@(X, Y, n_W, n_L, Ps) ( Y*((1 - Y)^(n_L - 1))*((1 - X)^n_W) - Ps );
fun1=@(X, Y, W, Z, W_net2, m_net2, n_L, W_net1, m_net1, n_W, Ps) ([funX(X, W, W_net1, m_net1); funY(Y, Z, W_net2, m_net2) ; funW(W, X, Y, n_W, n_L); funZ(Z, X, Y, n_W, n_L) ; funS(X, Y, n_W, n_L, Ps)]);
n_W = 3;
n_L = 4;
Ps = 0.45;
m_net2 = 4;
W_net1 = 16;
m_net1 = 6;
fun2=@(P) (P-fun1(P(1), P(2), P(3), P(4), P(5), m_net2, n_L, W_net1, m_net1, n_W, Ps));
InitialGuess=[0;0;0;0;16];
fsolve(fun2,InitialGuess)
P = sym('p', [5 1]);
fun2(P)

John D'Errico on 10 Apr 2019
syms a x b y
EQ = y == a*x + b;
isolate(EQ,x)
ans =
x == -(b - y)/a

Walter Roberson on 16 Apr 2019
I will not get to this right away; I have something else I need to do tonight.
Susan on 16 Apr 2019
I underestand. Whenever you get time works. I really appreciate your time and help. Thanks again for everything.
Susan on 18 Apr 2019

Susan on 12 Apr 2019
I have got some silly questions. Sorry in advance if they are super simple. Could anyone kindly help me to figure them out?
1) As for a variable that I am looking for, MATLAB gives me
W_net2 = 1665452806855272103430873159617883565696871742382633107303767067803714653521912534085508640394758232435502244395851440048832512/13020781553015561664280552453950494568153170161082649667490158510277680133267873702693313295722419712822602179610396291511542125
How is it possible that I simply get 0.1279 instead of this very long result? Why doesn't MATLAB do the division?
2) When I use solve(), there is a "z" in results. According to my underestanding, by using vpa() I should get rid of z, but I don't. Any idea?

Walter Roberson on 17 Apr 2019
Any relationship between n_L and nL ? Any relationship between n_W and nW ?
Walter Roberson on 17 Apr 2019
funY = @(Y, X) double( subs(solY(1,1), sym('X'), X) - Y);
Susan on 17 Apr 2019
Thank you so much Walter!
And sorry for the mess. Yes, n_L = nL and n_W = nW

Susan on 22 Apr 2019
Hey MATLAB experts,
Can anyone kindly help me to write down the functions in 'solve function' for below equations? Thanks in advance.
ivec = 1 : Nw;
jvec = 1 : Nl;
X = zeros(Nw, K);
Y = zeros(Nl, K);
W = zeros(Nw, K);
Z = zeros(Nl, K);
S = zeros(Nl, K);
for k = 1 : K
for i = ivec
X(i, k) = 2*(1 - 2*W(i,k))/((1 - 2*W(i,k))*(1 + W_net1(i,k)) + W(i,k)*W_net1(i,k)*(1 - (2*W(i,k))^m_net1(i,k)));
ii = setdiff(ivec, i);
tW1 = prod( 1 - X(ii, k) );
tW2 = prod( 1 - Y(jvec, k) );
W(i,k) = 1 - tW1 * tW2;
end
for j = jvec
Y(j, k) = 2*(1 - 2*Z(j,k))/((1 - 2*Z(j,k))*(1 + W_net2(j,k)) + Z(j,k)*W_net2(j,k)*(1 - (2*Z(j,k))^m_net2(j,k)));
i = ivec;
tZ1 = prod(1 - X(i, k));
jj = setdiff(jvec, j);
tZ2 = prod(1 - Y(jj, k));
tZ3 = tZ2 * tZ1;
Z(j,k) = 1 - tZ3;
S(j,k) = Y(j, k) * tZ3;
end
end

Susan on 10 May 2019
Walter,
Could you please kindly help me to solve this issue?
Walter Roberson on 10 May 2019
Sorry, I do no know when I will be well enough to address this.
Susan on 10 May 2019
No worries. Get well soon :)

Susan on 26 Apr 2019
Hi Walter,
In one of the comments above you mentioned that "The tests I am doing effectively recover from NaN, so I know that is not the reason we cannot find a root." Did you do any specific things that the test recover from Nan?
The reason I am asking is that I am using fmincon() to solve two optimization problems (the objective function is the same but I optimize the objfun w.r.t. two different variables), and regardless of what I am selecting the initial values, I ended up with this error
"Error using sqpInterface
Objective function is undefined at initial point. Fmincon cannot continue."