# How to solve 3 non-linear equations with 3 unknowns?

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Stefan Krieg on 15 Aug 2022
Commented: Alex Sha on 17 Aug 2022
Good day everyone
I want to solve 3 non-linear equations with 3 unknown variables. I have written a function for this. When execute the function in the command window I run into this warning:
Warning: Unable to solve symbolically. Returning a numeric solution using vpasolve.
> In sym/solve (line 304)
In myFunction (line 26)
ans =
[x*(exp(24424673397975836262400/(252133388810140916979*y)) - 1) - 1748446772768461537/225179981368524800 == 0, z + (28014820978904546331*y*exp(-24424673397975836262400/(252133388810140916979*y)))/(15144266739816366080*x) - 5949100062310601/9007199254740992 == 0, (721279627821056*z)/17034449605697695 + x*(exp((252133388810140916979*((34107876141310053*z)/25961484292674138142652481646100480 + 152237593508774139/6490371073168534535663120411525120))/(850705917302346158658436518579420528640*y)) - 1) + 6011786680494931611/340688992113953900 == 0]
Could anyone please assist me in what I am doing wrong? I need 1 numerical awnser for x,y,z.
Thanks
function F = myFunction (z)
q = 1.60218e-19 %electron discharge
V_oc = 4*44.8 %Open circuit voltage for 4 panels
N_s = 72 %number of cells connected in series
k = 1.38065e-23 %Boltzman constant
T = 298.15 %cell temperature
I_sc = 8.69 %short circuit current
R_sh = 193.6592708 %shunt resistance
dVdI_oc = 1/-1.514044 %Inverse of slope ate open circuit
V_mp = 4*36.6 %Maximum power voltage
I_mp = 8.2 %Maximum power current
I_pv = I_sc
%I_o = x
%n = y
%R_s = z
syms x y z
F(1) = ((x*(exp((q*V_oc)/(y*N_s*k*T))-1))-I_sc+(V_oc/R_sh))==0;
F(2) = (z + (dVdI_oc) + (1/((q*x/(y*N_s*k*T))*exp(q*V_oc/(y*N_s*k*T)))))==0;
F(3) = (x*((exp((q*(V_mp+(I_mp*z)))/y*N_s*k*T))-1)+(V_mp+(I_mp*z))/R_sh + I_mp + I_sc)==0;
sol = solve([F(1),F(2),F(3)],[x,y,z]);
end
Alex Sha on 17 Aug 2022
Hi, I get the results as below:
x: -7.86254507709137
y: -22.0861082560496
z: -416.744522416098

Sam Chak on 15 Aug 2022
Edited: Sam Chak on 15 Aug 2022
You can use vpasolve().
q = 1.60218e-19; %electron discharge
V_oc = 4*44.8; %Open circuit voltage for 4 panels
N_s = 72; %number of cells connected in series
k = 1.38065e-23; %Boltzman constant
T = 298.15; %cell temperature
I_sc = 8.69; %short circuit current
R_sh = 193.6592708; %shunt resistance
dVdI_oc = 1/-1.514044; %Inverse of slope ate open circuit
V_mp = 4*36.6; %Maximum power voltage
I_mp = 8.2; %Maximum power current
I_pv = I_sc;
% I_o = x
% n = y
% R_s = z
syms x y z
F(1) = ((x*(exp((q*V_oc)/(y*N_s*k*T))-1))-I_sc+(V_oc/R_sh))==0;
F(2) = (z + (dVdI_oc) + (1/((q*x/(y*N_s*k*T))*exp(q*V_oc/(y*N_s*k*T)))))==0;
F(3) = (x*((exp((q*(V_mp+(I_mp*z)))/y*N_s*k*T))-1)+(V_mp+(I_mp*z))/R_sh + I_mp + I_sc)==0;
sol = vpasolve([F(1),F(2),F(3)],[x,y,z])
sol = struct with fields:
x: -2.636463254746580127870476337846e+49 y: 7.2714991697885760498634526499995e+48 z: -416.7445224177292689999788029582
Stefan Krieg on 15 Aug 2022
Thank you, much appreciated. This is not the expected solution, I will have look at my equations again.

Walter Roberson on 15 Aug 2022
The numeric solutions being produced are dubious.
Notice how if we increase the number of digits to process at, that the magnitudes of x and y go up a lot, but z stays much the same. Then when we back-substitute, the remainder for the first equation stays much the same, but the remainder for the other two equations changes notably.
Can we do better?
Q = @(v) sym(v); %convert to symbolic number
q = Q(1.60218e-19); %electron discharge
V_oc = 4*Q(44.8); %Open circuit voltage for 4 panels
N_s = Q(72); %number of cells connected in series
k = Q(1.38065e-23); %Boltzman constant
T = Q(298.15); %cell temperature
I_sc = Q(8.69); %short circuit current
R_sh = Q(193.6592708); %shunt resistance
dVdI_oc = 1/Q(-1.514044); %Inverse of slope ate open circuit
V_mp = 4*Q(36.6); %Maximum power voltage
I_mp = Q(8.2); %Maximum power current
I_pv = I_sc;
% I_o = x
% n = y
% R_s = z
syms x y z
F(1,1) = ((x*(exp((q*V_oc)/(y*N_s*k*T))-1))-I_sc+(V_oc/R_sh))==0;
F(2,1) = (z + (dVdI_oc) + (1/((q*x/(y*N_s*k*T))*exp(q*V_oc/(y*N_s*k*T)))))==0;
F(3,1) = (x*((exp((q*(V_mp+(I_mp*z)))/y*N_s*k*T))-1)+(V_mp+(I_mp*z))/R_sh + I_mp + I_sc)==0;
F
F =
%cross-check 50
digits(50)
sol = vpasolve(F,[x,y,z])
sol = struct with fields:
x: 5.778320712828577486739426085443008797321206724901e+66 y: -1.5936901145812319849095567081810637894296223142398e+66 z: -416.74452278539695297294985201961716638096566722981
subs(F, sol)
ans =
%cross-check 100
digits(100)
sol = vpasolve(F,[x,y,z])
sol = struct with fields:
x: 1.778606202685051504194240299371941120912764206302539905506042308878578946691993020069557647352316783e+117 y: -4.905485732079848187309318642813357222474541837281366191242543485143184989714742976625190643174205242e+116 z: -416.747365580410707307368431863945702771674846388716182429050639941886945690525098549691804405869203
subs(F, sol)
ans =
%cross-check 200
digits(200)
sol = vpasolve(F,[x,y,z])
sol = struct with fields:
x: 2.1253277269192755215287267224132413071758900048346594765947120534469673714767739880357567466412505581730887441263769539733683919983979248358548464957150821112414062811924734376764374869229717715842355e+217 y: -5.7914206399459399153919106726969579062403404550768843089398315107147382482227843273334578765531375238518851528617878820724697219915991584359971210064143088616577428137964903429226049413464891654291869e+216 z: -1068.1081891420746454645013840766059447546928609541615260330634049789077827490389047336826713843612196750876654685256342672965304555594108001981875797439687325925188950833936463137625828173999255179196
%cross-check
subs(F, sol)
ans =
%can we do better? What happens if we solve iteratively?
y_partial = solve(F(1), y)
y_partial =
F2 = subs(F(2:end), y, y_partial)
F2 =
z_partial = solve(F2(1), z)
z_partial =
F3 = simplify(subs(F2(2:end), z, z_partial))
F3 =
xsol = vpasolve(F3, x)
xsol =
zsol = subs(z_partial, x, xsol)
zsol =
ysol = subs(y_partial, x, xsol)
ysol =
subs(F, [x, y, z], [xsol, ysol, zsol])
ans =
%that's pretty good!

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