Solving system of 3 non-linear equation with dynamic equality output

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Yokuna on 29 Jun 2022

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Answered: Torsten on 29 Jun 2022

I need to solve the set of equations given below for x1,x2,x3 such that x1+x2+x3==420 (for t=0 to t=0.2), x1+x2+x3==500 (for t=0.2 to t=0.4) and x1+x2+x3==420 (after t=0.4).

I tried to use step function in the code as x1+x2+x3=420*(stepfun(0:0.1:5,1)). But it did not work. Please suggest possible change in the code to incorporate the above change.

syms x1 x2 x3 t

eqns = [

x1 == t*(10*abs((18*x2)/125 - (24*x1)/125 + 111/50)^2*sign((18*x2)/125 - (24*x1)/125 + 111/50) + 10*abs((18*x2)/125 - (24*x1)/125 + 111/50)^(1/2)*sign((18*x2)/125 - (24*x1)/125 + 111/50)) - t*(10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^2*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100) + 10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^(1/2)*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)) + 140

x2 == 2*t*(10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^2*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100) + 10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^(1/2)*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)) - t*(10*abs((18*x2)/125 - (21*x3)/100 + 91/100)^2*sign((18*x2)/125 - (21*x3)/100 + 91/100) + 10*abs((18*x2)/125 - (21*x3)/100 + 91/100)^(1/2)*sign((18*x2)/125 - (21*x3)/100 + 91/100)) - t*(10*abs((18*x2)/125 - (24*x1)/125 + 111/50)^2*sign((18*x2)/125 - (24*x1)/125 + 111/50) + 10*abs((18*x2)/125 - (24*x1)/125 + 111/50)^(1/2)*sign((18*x2)/125 - (24*x1)/125 + 111/50)) + 140

x3 == t*(10*abs((18*x2)/125 - (21*x3)/100 + 91/100)^2*sign((18*x2)/125 - (21*x3)/100 + 91/100) + 10*abs((18*x2)/125 - (21*x3)/100 + 91/100)^(1/2)*sign((18*x2)/125 - (21*x3)/100 + 91/100)) - t*(10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^2*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100) + 10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^(1/2)*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)) + 140

x1+x2+x3==420]

eqns =

E2 = lhs(eqns)-rhs(eqns)

E2 =

F = matlabFunction(E2, 'vars', {[x1,x2,x3], t})

F = function_handle with value:

@(in1,t)[in1(:,1)-t.*(abs(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).^2.*sign(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).*1.0e+1+sqrt(abs(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1)).*sign(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).*1.0e+1)+t.*(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).^2.*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1+sqrt(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2)).*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1)-1.4e+2;in1(:,2)+t.*(abs(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).^2.*sign(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).*1.0e+1+sqrt(abs(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1)).*sign(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).*1.0e+1)+t.*(abs(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).^2.*sign(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).*1.0e+1+sqrt(abs(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2)).*sign(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).*1.0e+1)-t.*(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).^2.*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1+sqrt(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2)).*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1).*2.0-1.4e+2;in1(:,3)-t.*(abs(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).^2.*sign(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).*1.0e+1+sqrt(abs(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2)).*sign(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).*1.0e+1)+t.*(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).^2.*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1+sqrt(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2)).*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1)-1.4e+2;in1(:,1)+in1(:,2)+in1(:,3)-4.2e+2]

T = linspace(0,50,500);

nT = length(T);

sols = zeros(nT, 3);

x0 = [140, 140, 140];

options = optimoptions(@fsolve, 'Algorithm', 'levenberg-marquardt', 'display', 'none');

have_warned = false;

for tidx = 1 : nT

[thissol, ~, exitflag] = fsolve(@(x) F(x,T(tidx)), x0, options);

sols(tidx, :) = thissol;

x0 = thissol;

if exitflag <= 0 & ~have_warned

warning('solution failure code %d starting at time = %g', exitflag, T(tidx));

have_warned = true;

end

plot(T, sols);

legend({'x1', 'x2', 'x3'});

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Answer 1

Torsten on 29 Jun 2022

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syms x1 x2 x3 t A

eqns = [

x1 == t*(10*abs((18*x2)/125 - (24*x1)/125 + 111/50)^2*sign((18*x2)/125 - (24*x1)/125 + 111/50) + 10*abs((18*x2)/125 - (24*x1)/125 + 111/50)^(1/2)*sign((18*x2)/125 - (24*x1)/125 + 111/50)) - t*(10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^2*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100) + 10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^(1/2)*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)) + 140

x2 == 2*t*(10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^2*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100) + 10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^(1/2)*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)) - t*(10*abs((18*x2)/125 - (21*x3)/100 + 91/100)^2*sign((18*x2)/125 - (21*x3)/100 + 91/100) + 10*abs((18*x2)/125 - (21*x3)/100 + 91/100)^(1/2)*sign((18*x2)/125 - (21*x3)/100 + 91/100)) - t*(10*abs((18*x2)/125 - (24*x1)/125 + 111/50)^2*sign((18*x2)/125 - (24*x1)/125 + 111/50) + 10*abs((18*x2)/125 - (24*x1)/125 + 111/50)^(1/2)*sign((18*x2)/125 - (24*x1)/125 + 111/50)) + 140

x3 == t*(10*abs((18*x2)/125 - (21*x3)/100 + 91/100)^2*sign((18*x2)/125 - (21*x3)/100 + 91/100) + 10*abs((18*x2)/125 - (21*x3)/100 + 91/100)^(1/2)*sign((18*x2)/125 - (21*x3)/100 + 91/100)) - t*(10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^2*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100) + 10*abs((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)^(1/2)*sign((24*x1)/125 - (36*x2)/125 + (21*x3)/100 - 313/100)) + 140

x1+x2+x3==A];

E2 = lhs(eqns)-rhs(eqns);

F = matlabFunction(E2, 'vars', {[x1,x2,x3], t, A})

F = function_handle with value:

@(in1,t,A)[in1(:,1)-t.*(abs(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).^2.*sign(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).*1.0e+1+sqrt(abs(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1)).*sign(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).*1.0e+1)+t.*(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).^2.*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1+sqrt(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2)).*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1)-1.4e+2;in1(:,2)+t.*(abs(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).^2.*sign(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).*1.0e+1+sqrt(abs(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1)).*sign(in1(:,1).*(-2.4e+1./1.25e+2)+in1(:,2).*(1.8e+1./1.25e+2)+1.11e+2./5.0e+1).*1.0e+1)+t.*(abs(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).^2.*sign(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).*1.0e+1+sqrt(abs(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2)).*sign(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).*1.0e+1)-t.*(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).^2.*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1+sqrt(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2)).*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1).*2.0-1.4e+2;in1(:,3)-t.*(abs(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).^2.*sign(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).*1.0e+1+sqrt(abs(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2)).*sign(in1(:,2).*(1.8e+1./1.25e+2)-in1(:,3).*(2.1e+1./1.0e+2)+9.1e+1./1.0e+2).*1.0e+1)+t.*(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).^2.*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1+sqrt(abs(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2)).*sign(in1(:,1).*(2.4e+1./1.25e+2)-in1(:,2).*(3.6e+1./1.25e+2)+in1(:,3).*(2.1e+1./1.0e+2)-3.13e+2./1.0e+2).*1.0e+1)-1.4e+2;-A+in1(:,1)+in1(:,2)+in1(:,3)]

T = linspace(0,50,500);

nT = length(T);

sols = zeros(nT, 3);

x0 = [140, 140, 140];

options = optimoptions(@fsolve, 'Algorithm', 'levenberg-marquardt', 'display', 'none');

have_warned = false;

for tidx = 1 : nT

A = (T(tidx)<=0.2)*420 + (T(tidx)>0.2 && T(tidx)<=0.4)*500 + (T(tidx)>0.4)*420;

[thissol, ~, exitflag] = fsolve(@(x) F(x,T(tidx),A), x0, options);

exitflag

sols(tidx, :) = thissol;

x0 = thissol;

if exitflag <= 0 & ~have_warned

warning('solution failure code %d starting at time = %g', exitflag, T(tidx));

have_warned = true;

end

exitflag = 1

exitflag = -2

Warning: solution failure code -2 starting at time = 0.200401

exitflag = -2

exitflag = 1

exitflag = 4

exitflag = 1

exitflag = 4

exitflag = 1

exitflag = 4

plot(T, sols);

legend({'x1', 'x2', 'x3'});

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Solving system of 3 non-linear equation with dynamic equality output

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Solving system of 3 non-linear equation with dynamic equality output

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