wrong solution trying to solve a system of algebric equations

Hi,

trying to solve a system of 8 algebric equations, using the 'solve' function, but apparentely none of the 4 solutions found by Matlab satisfies all the input equations. In particular, if the equation system is:

eqs=[eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8];

and the line where I call 'solve' is:

[a,b,c,d,e,f,g,epeak,parameters,conditions]=solve(eqs,[a,b,c,d,e,f,g,epeak],'ReturnConditions',true);

all the solutions found by Matlab seem to not-satisfy eq4.

To better understand, here it is the code I wrote:

clearvars
clc
syms x a b c d e f g E fc epeak Gf h


% polynomials
cub=a*(x^3)+b*(x^2)+c*x+d;  % cubic curve
quad=e*(x^2)+f*x+g;         % quadratic curve
d_cub=diff(cub,x);          % cubic curve derivate
d_quad=diff(quad,x);        % quadratic curve derivative
d2_cub=diff(d_cub,x);       % cubic curve second derivative
d2_quad=diff(d_quad,x);     % quadratic curve second derivative

% shortcuts
cub_0       =subs(cub,x,0);      	% cubic curve value at 0
cub_fcE     =subs(cub,x,fc/E);    	% cubic curve at fc/E
quad_fcE    =subs(quad,x,fc/E);    	% quadratic curve at fc/E
quad_epeak  =subs(quad,x,epeak);   	% quadratic curve at epeak
d_cub_0     =subs(d_cub,x,0);   	% cubic curve slope at 0
d_cub_fcE   =subs(d_cub,x,fc/E);  	% cubic curve slope at fc/E
d_quad_fcE  =subs(d_quad,x,fc/E); 	% quadratic curve slope at fc/R
d_quad_epeak    =subs(d_quad,x,epeak);  % quadratic curve slope at epeak
d2_cub_fcE      =subs(d2_cub,x,fc/E);  	% cubic curve concavity
d2_quad_fcE     =subs(d2_quad,x,fc/E); 	% quadratic curve concavity

% boundary conditions
eq1=    cub_0==0;           	% cubic value at origin = 0
eq2=    d_cub_0==E;          	% cubic curve slope at origin = E
eq3=    quad_epeak==fc;       	% cubic curve at epeak = fc
eq4=    d_quad_epeak==0;       	% quadratic curve slope at epeak = 0
eq5=    cub_fcE==quad_fcE;      % intersections of curves at fc/E
eq6=    d_cub_fcE==d_quad_fcE;  % curves have same slopes at intersection
eq7=    d2_cub_fcE==d2_quad_fcE;% same concavity at intersection
eq8=    int(cub,x,0,fc/E)+int(quad,x,fc/E,epeak)==(Gf/h);% area undercurves

eqs=[eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8];
[a_,b_,c_,d_,e_,f_,g_,epeak_,params,conds]=solve(eqs,[a,b,c,d,e,f,g,epeak],'ReturnConditions',true);

% test on eq4 = quad_epeak == 0
Nsolutions=size(a_,1);
test_=zeros(2,1);
test_=sym(test_);
for i=1:Nsolutions
    d_quad_epeak=subs(d_quad_epeak,a,a_(i));
    d_quad_epeak=subs(d_quad_epeak,b,b_(i));
    d_quad_epeak=subs(d_quad_epeak,c,c_(i));
    d_quad_epeak=subs(d_quad_epeak,d,d_(i));
    d_quad_epeak=subs(d_quad_epeak,e,e_(i));
    d_quad_epeak=subs(d_quad_epeak,f,f_(i));
    d_quad_epeak=subs(d_quad_epeak,g,g_(i));
    d_quad_epeak=subs(d_quad_epeak,epeak,epeak_(i));
    display(simplify(d_quad_epeak)); % ans = 0, but substituting real values it returns a non-zero value
    
    d_quad_epeak=subs(d_quad_epeak,E,1491); % input value for E
    d_quad_epeak=subs(d_quad_epeak,fc,6.2); % input value for fc
    
    fprintf('test on eq4 returns: %s\n',d_quad_epeak); % it should be zero!
end

As you can see above, eq4 is: d_quad_epeak==0, but substituting the 8 parameters found by Matlab, d_quad_epeak is always about 3.6e-12.

Is it the system I'm trying to solve bad conditioned, or is it an approximation problem, or I did something wrong using the 'solve' function?

[Torsten]

Trust in "solve". The error you get when substituting values for the variables is due to the limited floating point precision.