solve 3 nonlinear equations with 3 unkonws with vpasolve....warning show up "seems to be affected by some numeric instability."

Hello everyone...I have problem to solve three system of non linear equations with 3 unknowns as shown below

where phai = 32 and alpha = 20 and Bim = 100

when I solve it, I found the solutions with warning

Warning: Solution '[c1 = 0.56333314518518598688984408662648, c2 =

0.8936045102316401179555243045528, c3 = 0.98968113734245934645180689276312]' seems to

be affected by some numeric instability. Inserting this solution into equation

'1168228461484486686*c1 - 23000232427283236*c2 + 7693275550769427*c3 -

920009297091329440*c1*c2 + 307731022030777080*c1*c3 - 9200092970913294400*c1^2*c2 +

3077310220307770800*c1^2*c3 + 612278275105588400*c1^2 + 6122782751055884000*c1^3 = 0'

produces the residue '0.00000022275530153006534874837556159388'.

> In symengine

In mupadengine/evalin_internal

In mupadengine/feval_internal

In sym/vpasolve (line 172)

In Problem_2_442105849 (line 81)

I use this code

%##########################################################################
% Problem(2)(c) ==> f(c) = c/(1+alpha*c)^2, alpha = 20, phai^2 = 32, Bim = 100
% Subtitute f(c) = c/(1+alpha*c)^2 into Equations as follow:
% Eq.9  (13.59530877 + 1/(1+alpha*c1)^2)*c1 - 20.42831009*c2                       + 6.833001321*c3      = 0
% Eq.10 14.57168991*c1                      - (91.40469119 + 1/(1+alpha*c2)^2)*c2  - 76.83300129*c3      = 0
% Eq.3  0.9482702526c1                      - 14.948270256c2                       + (14+Bim)c3    - Bim = 0
% Solution
% Inputs
alpha = 20;
phai2 = 32^2;
Eq9 = phai2*(c1/(1+alpha*c1)^2) + 13.59530877*c1   - 20.42831009*c2    + 6.833001321*c3;
Eq10 = 14.57168991*c1  - phai2*(c2/(1+alpha*c2)^2) - 91.40469119*c2  + 76.83300129*c3;
Eq6 = 0.9482702526*c1 - 14.948270256*c2 + (14+Bim)*c3 - Bim; 
sol_3 = vpasolve([Eq9 == 0, Eq10==0, Eq6==0],[c1,c2,c3]);                   % Using vpasolve to solve numerically
% Display the solutions
fprintf("Solutions (c) as follow:\n");
k=0;
for i=1:length(sol_3.c1)
    if isreal(sol_3.c1(i))
        k=k+1;
        fprintf("|| Solution %d\n",k);
        fprintf("|| c1 = %.4f  c2 = %.4f  c3 = %.4f\n",sol_3.c1(i),sol_3.c2(i),sol_3.c3(i));
    end
end
for i=1:length(sol_3.c1)
    if ~isreal(sol_3.c1(i))
        k=k+1;
        fprintf("|| Solution %d\n",k);
        fprintf("|| c1 = %.4f%+.4fi  c2 = %.4f%+.4fi  c3 = %.4f%+.4fi\n",real(sol_3.c1(i)),imag(sol_3.c1(i)),real(sol_3.c2(i)),imag(sol_3.c2(i)),real(sol_3.c3(i)),imag(sol_3.c3(i)));
    end
end
fprintf("||=======================================================================||\n");
%##########################################################################

Answer 1

[Chidvi Modala]

Hi Abdullah,

The vpa commands, like vpasolve, try to return an answer accurate to the digits() setting. Here that fails and the system issues a warning to indicate that. The error message tells you how accurate the system managed to calculate a solution to. The solutions appear to be fairly accurate and good enough. You will be able to complete your work without trouble by simply turning off the warnings,

warning('off', 'symbolic:numeric:NumericalInstability')

Before doing so, you may check the returned results are good enough. For example in the warning,

Warning: Solution '[c1 = - 0.0051621569086099834898074490891583 - 0.00048822629555725762835974056914787i, c2 = -

0.032429710190290267356308550056465 - 0.039482775202317809077679764419658i, c3 = 0.87298357059287256401584217486686 -

0.0051731247693730497577646047617452i]' seems to be affected by some numeric instability. Inserting this solution into equation

'1168228461484486686*c1 - 23000232427283236*c2 + 7693275550769427*c3 - 920009297091329440*c1*c2 + 307731022030777080*c1*c3 -

9200092970913294400*c1^2*c2 + 3077310220307770800*c1^2*c3 + 612278275105588400*c1^2 + 6122782751055884000*c1^3 = 0' produces the

residue '- 0.00000065744475774907684818232970828457 - 0.000000015950832025180537446276284353478i'.

The system has a solution that is accurate to 7 digits (You can look at the residue and count the zeros). If you require a more accurate result then you can increase digits by inserting, for example,

digits(45)

into the script before vpasolve. This will not remove the warnings though.