# Restricting solutions using lsqnonlin with complex unknowns

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So, I'm trying to solve a system of equations that has complex numbers, the thing is that it is supossed to be a 12x12, however I write it as a 9 unknowns with 12 equations ever since three of my unknowns
l1,L2,L3
Are complex numbers, where the real and imaginary part are two sepparate unknowns, i'm trying to solve my system with the following code:
fun = @(x)[x(1)*(exp(1i*alfa(1))-1)+x(2)*(exp(1i*x(4))-1)-x(3)*(exp(1i*rho(1))-1),x(1)*(exp(1i*alfa(2))-1)+x(2)*(exp(1i*x(5))-1)-x(3)*(exp(1i*rho(2))-1),x(1)*(exp(1i*alfa(3))-1)+x(2)*(exp(1i*x(6))-1)-x(3)*(exp(1i*rho(3))-1),x(1)*(exp(1i*alfa(4))-1)+x(2)*(exp(1i*x(7))-1)-x(3)*(exp(1i*rho(4))-1),x(1)*(exp(1i*alfa(5))-1)+x(2)*(exp(1i*x(8))-1)-x(3)*(exp(1i*rho(5))-1),x(1)*(exp(1i*alfa(6))-1)+x(2)*(exp(1i*x(9))-1)-x(3)*(exp(1i*rho(6))-1)];
[x,fval] = lsqnonlin(fun,[40+20i,40+20i,40+40i,1,1,1,1,1,1])
L1=x(1)
l2=x(2)
L3=x(3)
phi1=x(4)
phi2=x(5)
phi3=x(6)
phi4=x(7)
phi5=x(8)
phi6=x(9)
I need to restrict the solutions to only those who have phi1, phi2, phi3, phi4, phi5, phi6 stricly as real numbers, without any complex part
Full code below
%Introducción de puntos y cambio de eje de referencia
%Los puntos se escogieron de acuerdo al trabajo del mecanismo de 10 barras
clc
clear all
L=82.4;
L2=42.7;
l1=39.7;
Px=[6.56044 -4.12471 -19.6017 -18.2934 -1.8923 21.1196 17.6706];
Py=[0.995643 0.00147957 1.14159 2.07962 6.30745 11.3302 4.55826];
Theta1=[1.8073522328820310935623781790315, 1.6640119075452171534709629667332, 1.6070200513956322755772098198635, 1.6722276935006778576202574721229, 2.0499843737905114408996166747857, 2.5269236792605256431572972964069, 2.104035221140493127739748316949];
Theta2=[1.5061633838536617569715490239474, 1.3818662717816540137480189935275, 1.0670342257290320137842410556993, 1.039607130354159250776488333576, 1.0892436455770986051727755147433, 1.206386181298577448316370700124, 1.5032866660028009663082593137064];
for j=1:6
alfa(j)=(Theta1(j+1)-Theta1(j)); %Alfa representa el cambio de una posición en grados a otra
end
%Angulos de entrada de la manivela para generar una unica rotacion de 360
%grados, que se distribuye de forma uniforme para la generación de función
gamma=[40 70 100 180 240 360 40];
for j=1:6
rho(j)=(gamma(j+1)-gamma(j))*(pi/180);
end
%Sistemas de ecuaciones
fun = @(x)[x(1)*(exp(1i*alfa(1))-1)+x(2)*(exp(1i*x(4))-1)-x(3)*(exp(1i*rho(1))-1),x(1)*(exp(1i*alfa(2))-1)+x(2)*(exp(1i*x(5))-1)-x(3)*(exp(1i*rho(2))-1),x(1)*(exp(1i*alfa(3))-1)+x(2)*(exp(1i*x(6))-1)-x(3)*(exp(1i*rho(3))-1),x(1)*(exp(1i*alfa(4))-1)+x(2)*(exp(1i*x(7))-1)-x(3)*(exp(1i*rho(4))-1),x(1)*(exp(1i*alfa(5))-1)+x(2)*(exp(1i*x(8))-1)-x(3)*(exp(1i*rho(5))-1),x(1)*(exp(1i*alfa(6))-1)+x(2)*(exp(1i*x(9))-1)-x(3)*(exp(1i*rho(6))-1)];
[x,fval] = lsqnonlin(fun,[40+20i,40+20i,40+40i,1,1,1,1,1,1])
L1=x(1)
l2=x(2)
L3=x(3)
phi1=x(4)
phi2=x(5)
phi3=x(6)
phi4=x(7)
phi5=x(8)
phi6=x(9)

Matt J on 25 Jun 2022
Edited: Matt J on 25 Jun 2022
l1,L2,L3 are complex numbers, where the real and imaginary part are two sepparate unknowns,
If so, then you have 12 unknowns, not 9. Write your function in terms of 12 unknowns so that lsqnonlin knows that:
fun = @(P) residualFunction(P,rho,alfa);
[P,fval] = lsqnonlin(fun,[40,20,40,20,40,40,1,1,1,1,1,1]);
L=complex(P(1:3), P(4:6));
phi=P(7:end);
function F=residualFunction(P,rho,alfa)
x(1:3)=complex(P(1:2:6), P(2:2:6));
x(4:9)=P(7:end);
expr= [x(1)*(exp(1i*alfa(1))-1)+x(2)*(exp(1i*x(4))-1)-x(3)*(exp(1i*rho(1))-1),x(1)*(exp(1i*alfa(2))-1)+x(2)*(exp(1i*x(5))-1)-x(3)*(exp(1i*rho(2))-1),x(1)*(exp(1i*alfa(3))-1)+x(2)*(exp(1i*x(6))-1)-x(3)*(exp(1i*rho(3))-1),x(1)*(exp(1i*alfa(4))-1)+x(2)*(exp(1i*x(7))-1)-x(3)*(exp(1i*rho(4))-1),x(1)*(exp(1i*alfa(5))-1)+x(2)*(exp(1i*x(8))-1)-x(3)*(exp(1i*rho(5))-1),x(1)*(exp(1i*alfa(6))-1)+x(2)*(exp(1i*x(9))-1)-x(3)*(exp(1i*rho(6))-1)];
F=[ real(expr(:)),imag(expr(:)) ];
end
Thanks, tried to adjust the constraints to what makes sense in my problem, I'm going to check if it works or not.

R2022a

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