Solving 5 very complicated equations with 5 unknowns

Hi, I'm trying to derive a zoom lens and this equation is what i came across. Currently, I'm trying to find 5 variables with 5 equations(feq1, feq2, feq3, feq4, feq5).The equations were generated as shown below.

syms f1 f2 f3 f4 fe p2 p3;
fl = @(f1, f2, d) 1/(1/f1 + 1/f2 - d/(f1 * f2)); %focal length formula
bfl = @(f1, f2, d) (f2 * (d - f1)) / (d - (f1 + f2)); %back focal length formula
f_34 = fl(f4, f3, p3);
bfl_34 = bfl(f4, f3, p3);
f_234 = fl(f_34, f2, bfl_34+p2-p3);
bfl_234 = bfl(f_34, f2, bfl_34 + (p2-p3));
f_1234 = fl(f_234, f1, bfl_234 + (80-p2));
bfl_1234 = bfl(f_1234, fe, bfl_234 + (80 - p2));
f_1234e = fl(f_1234, fe, bfl_1234 + (125 - 80)) % The equation of the lens system

f_1234e = 

%% substituting p2 and p3 in my equation
feq1 = subs(subs(f_1234e, p2, 63.8654),p3, 44.6468) == 179.0893;
feq2 = subs(subs(f_1234e, p2, 57.8781),p3, 43.4521) == 166.8919;
feq3 = subs(subs(f_1234e, p2, 52.4784),p3, 41.9933) == 151.9181;
feq4 = subs(subs(f_1234e, p2, 49.7862),p3, 40.6357) == 143.6190;
feq5 = subs(subs(f_1234e, p2, 44.6669),p3, 36.5343) == 122.4775;

Which is very complicated.

I've aleady tried solve() and vpasolve(), but their outputs were empty.

Attempt #1:

S = vpasolve([feq1, feq2, feq3, feq4, feq5],[f1 f2 f3 f4 fe]);

Attempt #2:

S = solve([feq1, feq2, feq3, feq4, feq5]);

Please tell me if i've got something wrong or other ways to solve this hard question.

Thanks!

Answer 1

[Torsten]

syms f1 f2 f3 f4 fe p2 p3
fl = @(f1, f2, d) 1/(1/f1 + 1/f2 - d/(f1 * f2)); %focal length formula
bfl = @(f1, f2, d) (f2 * (d - f1)) / (d - (f1 + f2)); %back focal length formula
f_34 = fl(f4, f3, p3);
bfl_34 = bfl(f4, f3, p3);
f_234 = fl(f_34, f2, bfl_34+p2-p3);
bfl_234 = bfl(f_34, f2, bfl_34 + (p2-p3));
f_1234 = fl(f_234, f1, bfl_234 + (80-p2));
bfl_1234 = bfl(f_1234, fe, bfl_234 + (80 - p2));
f_1234e = fl(f_1234, fe, bfl_1234 + (125 - 80)); % The equation of the lens system
%% substituting p2 and p3 in my equation
feq1 = subs(subs(f_1234e, p2, 63.8654),p3, 44.6468) == 179.0893;
feq2 = subs(subs(f_1234e, p2, 57.8781),p3, 43.4521) == 166.8919;
feq3 = subs(subs(f_1234e, p2, 52.4784),p3, 41.9933) == 151.9181;
feq4 = subs(subs(f_1234e, p2, 49.7862),p3, 40.6357) == 143.6190;
feq5 = subs(subs(f_1234e, p2, 44.6669),p3, 36.5343) == 122.4775;
f = [feq1;feq2;feq3;feq4;feq5];
f = lhs(f)-rhs(f);
f = matlabFunction(f);
F = @(x)f(x(1),x(2),x(3),x(4),x(5));
f0 = [68;35;-14;96;186];
sol = fsolve(F,f0)

Equation solved, inaccuracy possible.

The vector of function values is near zero, as measured by the value
of the function tolerance. However, the last step was ineffective.

sol = 5×1

68.5354 35.7632 -14.0363 96.2849 186.3001

[Bradley Chen]

Thanks a lot! I really appreciate it.

Answer 2

[Bradley Chen]

ohh! I got the answer! I have the problem solving on other's pc, but when i tried it on my laptop,

it worked!

It took a long time tho.

syms f1 f2 f3 f4 fe p2 p3;
fl = @(f1, f2, d) 1/(1/f1 + 1/f2 - d/(f1 * f2)); %focal length formula
bfl = @(f1, f2, d) (f2 * (d - f1)) / (d - (f1 + f2)); %back focal length formula
f_34 = fl(f4, f3, p3);
bfl_34 = bfl(f4, f3, p3);
f_234 = fl(f_34, f2, bfl_34+p2-p3);
bfl_234 = bfl(f_34, f2, bfl_34 + (p2-p3));
f_1234 = fl(f_234, f1, bfl_234 + (80-p2));
bfl_1234 = bfl(f_1234, fe, bfl_234 + (80 - p2));
f_1234e = fl(f_1234, fe, bfl_1234 + (125 - 80)) % The equation of the lens system
%% substituting p2 and p3 in my equation
feq1 = subs(subs(f_1234e, p2, 63.8654),p3, 44.6468) == 179.0893;
feq2 = subs(subs(f_1234e, p2, 57.8781),p3, 43.4521) == 166.8919;
feq3 = subs(subs(f_1234e, p2, 52.4784),p3, 41.9933) == 151.9181;
feq4 = subs(subs(f_1234e, p2, 49.7862),p3, 40.6357) == 143.6190;
feq5 = subs(subs(f_1234e, p2, 44.6669),p3, 36.5343) == 122.4775;
%% solving
S = solve([feq1, feq2, feq3, feq4, feq5]);

S is the answer.

[Torsten]

I suspect there will be many more "answers" depending on the initial guesses for the unknowns. At least "fsolve" showed this behaviour when using it for your problem.

[Bradley Chen]

@Torsten, thanks a lot!

However, i'm not able to figure out how to use fsolve.

Would it be possible to show me your code?