Solving lengthy nonlinear complex equations in complex variables

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ubtc
ubtc 2020-10-28
评论： Alex Sha 2020-10-30
Hello Everyone!
I am trying to solve a set of 4 complex nonlinear equations in 4 complex variables, containing lengthy algebraic expressions with large complex numbers. I have tried vpasolve, solve, fsolve but all get stuck due to voluminous computations involved & no result is appeared. The expressions/equations cannot be shortened or moulded. Please provide suggestions to solve this problem. The code is as under:
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>> syms Y1s Y3s Y4s Y7s
>> e1=(220000/(sqrt(3)))/(((1.8e17*(Y3s*(4.4e63 - 2.0e63i) + Y4s*(6.9e63 - 3.1e63i) + Y7s*(9.4e63 - 4.2e63i) + Y3s*Y4s*(7.8e61 + 1.4e62i) + Y3s*Y7s*(1.1e62 + 1.9e62i) + Y4s*Y7s*(1.6e62 + 2.9e62i) - Y3s*Y4s*Y7s*(5.5e60 - 3.9e60i) - 7.0e64 - 2.1e65i))/(- Y1s*(1.2e82 + 3.7e82i) - Y3s*(1.4e82 + 4.1e82i) - Y4s*(1.5e82 + 4.3e82i) - Y7s*(2.0e82 + 5.5e82i) + Y1s*Y3s*(7.7e80 - 3.4e80i) + Y1s*Y4s*(1.2e81 - 5.4e80i) + Y3s*Y4s*(1.4e81 - 6.2e80i) + Y1s*Y7s*(1.7e81 - 7.4e80i) + Y3s*Y7s*(1.9e81 - 8.6e80i) + Y4s*Y7s*(2.0e81 - 9.2e80i) + Y1s*Y3s*Y4s*(1.4e79 + 2.4e79i) + Y1s*Y3s*Y7s*(1.9e79 + 3.3e79i) + Y1s*Y4s*Y7s*(2.9e79 + 5.1e79i) + Y3s*Y4s*Y7s*(3.4e79 + 5.9e79i) - Y1s*Y3s*Y4s*Y7s*(9.7e77 - 6.9e77i) - 9.7e83 + 2.7e83i))*484)-((3.9046-44.6295*i)*1000);
>> e3=(220000/(sqrt(3)))/(((4.4e16*(Y1s*(1.7e64 - 7.8e63i) + Y4s*(3.2e64 - 1.4e64i) + Y7s*(4.3e64 - 2.0e64i) + Y1s*Y4s*(3.1e62 + 5.4e62i) + Y1s*Y7s*(4.3e62 + 7.5e62i) + Y4s*Y7s*(7.7e62 + 1.3e63i) - Y1s*Y4s*Y7s*(2.2e61 - 1.6e61i) - 3.2e65 - 9.3e65i))/(- Y1s*(1.2e82 + 3.7e82i) - Y3s*(1.4e82 + 4.1e82i) - Y4s*(1.5e82 + 4.3e82i) - Y7s*(2.0e82 + 5.5e82i) + Y1s*Y3s*(7.7e80 - 3.4e80i) + Y1s*Y4s*(1.2e81 - 5.4e80i) + Y3s*Y4s*(1.4e81 - 6.2e80i) + Y1s*Y7s*(1.7e81 - 7.4e80i) + Y3s*Y7s*(1.9e81 - 8.6e80i) + Y4s*Y7s*(2.0e81 - 9.2e80i) + Y1s*Y3s*Y4s*(1.4e79 + 2.4e79i) + Y1s*Y3s*Y7s*(1.9e79 + 3.3e79i) + Y1s*Y4s*Y7s*(2.9e79 + 5.1e79i) + Y3s*Y4s*Y7s*(3.4e79 + 5.9e79i) - Y1s*Y3s*Y4s*Y7s*(9.7e77 - 6.9e77i) - 9.7e83 + 2.7e83i))*484)-((3.4339-39.25*i)*1000);
>> e4=(220000/(sqrt(3)))/(((5.0e3*(Y1s*(1.2e77 + 5.5e75i) + Y3s*(1.4e77 + 5.7e75i) + Y7s*(2.0e77 + 7.0e75i) + Y1s*Y3s*(1.4e74 + 2.5e75i) + Y1s*Y7s*(2.9e74 + 5.2e75i) + Y3s*Y7s*(3.7e74 + 6.1e75i) - Y1s*Y3s*Y7s*(1.1e74 - 1.7e73i) + 5.3e77 - 4.0e78i))/(Y1s*(2.4e81 - 1.7e82i) + Y3s*(2.6e81 - 1.9e82i) + Y4s*(2.7e81 - 2.0e82i) + Y7s*(3.1e81 - 2.6e82i) + Y1s*Y3s*(3.8e80 + 1.6e79i) + Y1s*Y4s*(5.9e80 + 2.8e79i) + Y3s*Y4s*(6.8e80 + 2.8e79i) + Y1s*Y7s*(8.1e80 + 3.4e79i) + Y3s*Y7s*(9.3e80 + 3.4e79i) + Y4s*Y7s*(9.9e80 + 3.5e79i) + Y1s*Y3s*Y4s*(6.9e77 + 1.2e79i) + Y1s*Y3s*Y7s*(1.0e78 + 1.7e79i) + Y1s*Y4s*Y7s*(1.5e78 + 2.6e79i) + Y3s*Y4s*Y7s*(1.8e78 + 3.0e79i) - Y1s*Y3s*Y4s*Y7s*(5.3e77 - 8.5e76i) - 4.4e83 - 8.5e82i))*484)-((1.5165-17.3338*i)*1000);
>> e7=(220000/(sqrt(3)))/(((1.2e4*(Y1s*(1.9e77 + 7.3e76i) + Y3s*(2.2e77 + 8.2e76i) + Y4s*(2.3e77 + 8.7e76i) - Y1s*Y3s*(1.1e75 - 4.2e75i) - Y1s*Y4s*(1.7e75 - 6.4e75i) - Y3s*Y4s*(2.0e75 - 7.4e75i) - Y1s*Y3s*Y4s*(1.3e74 + 2.2e73i) + 2.8e78 - 6.0e78i))/(Y1s*(2.5e82 - 4.9e82i) + Y3s*(2.7e82 - 5.5e82i) + Y4s*(2.8e82 - 5.8e82i) + Y7s*(3.5e82 - 7.5e82i) + Y1s*Y3s*(1.1e81 + 4.2e80i) + Y1s*Y4s*(1.7e81 + 6.7e80i) + Y3s*Y4s*(2.0e81 + 7.7e80i) + Y1s*Y7s*(2.4e81 + 9.2e80i) + Y3s*Y7s*(2.7e81 + 1.0e81i) + Y4s*Y7s*(2.9e81 + 1.1e81i) - Y1s*Y3s*Y4s*(1.0e79 - 3.8e79i) - Y1s*Y3s*Y7s*(1.4e79 - 5.2e79i) - Y1s*Y4s*Y7s*(2.2e79 - 8.0e79i) - Y3s*Y4s*Y7s*(2.5e79 - 9.3e79i) - Y1s*Y3s*Y4s*Y7s*(1.7e78 + 2.7e77i) - 1.2e84 - 7.0e83i))*484)-((4.8371-55.2888*i)*1000);
>> result = solve(e1,e3,e4,e7,Y1s,Y3s,Y4s,Y7s)
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Star Strider
Star Strider 2020-10-29
My pleasure!
I would definitely recommend re-coding these in a way that fsolve can use them. (The fzero function is not an option here.) If you give it complex initial estimates, it will use those and return complex results. If you give it real initial estimates, it will return real results (unless the results can only be complex). It may be necesary to iterate several times with different random initial estimates over a wide range of values, positive and negative, to estimate all of them, assuming they can be uniquely estimated. I would normally suggest plotting it, however with so many parameters, that is not an option. It appears to be first-order (linear) with respect to each parameter (I did not examine the equations closely), so it is likely not necessary to be concerned about complex-conjugate roots.
ubtc
ubtc 2020-10-29
Thanks for your suggestions! "fsolve" happens to solve the equations, whereas "vpasolve" remains stuck.

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回答（2 个）

Alex Sha
Alex Sha 2020-10-29
Try to find numerical solutions as below;
y1s.realpart: 11.0169967890388
y1s.imagpart: -125.141575853451
y3s.realpart: 7.81970377305325
y3s.imagpart: -106.356946529629
y4s.realpart: 3.07796056296378
y4s.imagpart: -36.8035908733627
y7s.realpart: 15.4211136064375
y7s.imagpart: -178.419654796614
  2 个评论
ubtc
ubtc 2020-10-29
Thanks!
Will you please elaborate that whether you solved independently for real & imaginary parts or you split up solution in real & imaginary parts? And what method did you use to get more precision?
Alex Sha
Alex Sha 2020-10-30
Hi, ubtc, the results are obtained by 1stOpt, a math software other than Matlab, much easy for using, and without requiring the guess of initial-start valus for each parameter, the total code like below:
ComplexPar Y1s,Y3s,Y4s,Y7s;
ComplexStr = i;
Function
(220000/(sqrt(3)))/(((1.8e17*(Y3s*(4.4e63 - 2.0e63*i) + Y4s*(6.9e63 - 3.1e63*i) + Y7s*(9.4e63 - 4.2e63*i) + Y3s*Y4s*(7.8e61 + 1.4e62*i) + Y3s*Y7s*(1.1e62 + 1.9e62*i) + Y4s*Y7s*(1.6e62 + 2.9e62*i) - Y3s*Y4s*Y7s*(5.5e60 - 3.9e60*i) - 7.0e64 - 2.1e65*i))/(- Y1s*(1.2e82 + 3.7e82*i) - Y3s*(1.4e82 + 4.1e82*i) - Y4s*(1.5e82 + 4.3e82*i) - Y7s*(2.0e82 + 5.5e82*i) + Y1s*Y3s*(7.7e80 - 3.4e80*i) + Y1s*Y4s*(1.2e81 - 5.4e80*i) + Y3s*Y4s*(1.4e81 - 6.2e80*i) + Y1s*Y7s*(1.7e81 - 7.4e80*i) + Y3s*Y7s*(1.9e81 - 8.6e80*i) + Y4s*Y7s*(2.0e81 - 9.2e80*i) + Y1s*Y3s*Y4s*(1.4e79 + 2.4e79*i) + Y1s*Y3s*Y7s*(1.9e79 + 3.3e79*i) + Y1s*Y4s*Y7s*(2.9e79 + 5.1e79*i) + Y3s*Y4s*Y7s*(3.4e79 + 5.9e79*i) - Y1s*Y3s*Y4s*Y7s*(9.7e77 - 6.9e77*i) - 9.7e83 + 2.7e83*i))*484)-((3.9046-44.6295*i)*1000);
(220000/(sqrt(3)))/(((4.4e16*(Y1s*(1.7e64 - 7.8e63*i) + Y4s*(3.2e64 - 1.4e64*i) + Y7s*(4.3e64 - 2.0e64*i) + Y1s*Y4s*(3.1e62 + 5.4e62*i) + Y1s*Y7s*(4.3e62 + 7.5e62*i) + Y4s*Y7s*(7.7e62 + 1.3e63*i) - Y1s*Y4s*Y7s*(2.2e61 - 1.6e61*i) - 3.2e65 - 9.3e65*i))/(- Y1s*(1.2e82 + 3.7e82*i) - Y3s*(1.4e82 + 4.1e82*i) - Y4s*(1.5e82 + 4.3e82*i) - Y7s*(2.0e82 + 5.5e82*i) + Y1s*Y3s*(7.7e80 - 3.4e80*i) + Y1s*Y4s*(1.2e81 - 5.4e80*i) + Y3s*Y4s*(1.4e81 - 6.2e80*i) + Y1s*Y7s*(1.7e81 - 7.4e80*i) + Y3s*Y7s*(1.9e81 - 8.6e80*i) + Y4s*Y7s*(2.0e81 - 9.2e80*i) + Y1s*Y3s*Y4s*(1.4e79 + 2.4e79*i) + Y1s*Y3s*Y7s*(1.9e79 + 3.3e79*i) + Y1s*Y4s*Y7s*(2.9e79 + 5.1e79*i) + Y3s*Y4s*Y7s*(3.4e79 + 5.9e79*i) - Y1s*Y3s*Y4s*Y7s*(9.7e77 - 6.9e77*i) - 9.7e83 + 2.7e83*i))*484)-((3.4339-39.25*i)*1000);
(220000/(sqrt(3)))/(((5.0e3*(Y1s*(1.2e77 + 5.5e75*i) + Y3s*(1.4e77 + 5.7e75*i) + Y7s*(2.0e77 + 7.0e75*i) + Y1s*Y3s*(1.4e74 + 2.5e75*i) + Y1s*Y7s*(2.9e74 + 5.2e75*i) + Y3s*Y7s*(3.7e74 + 6.1e75*i) - Y1s*Y3s*Y7s*(1.1e74 - 1.7e73*i) + 5.3e77 - 4.0e78*i))/(Y1s*(2.4e81 - 1.7e82*i) + Y3s*(2.6e81 - 1.9e82*i) + Y4s*(2.7e81 - 2.0e82*i) + Y7s*(3.1e81 - 2.6e82*i) + Y1s*Y3s*(3.8e80 + 1.6e79*i) + Y1s*Y4s*(5.9e80 + 2.8e79*i) + Y3s*Y4s*(6.8e80 + 2.8e79*i) + Y1s*Y7s*(8.1e80 + 3.4e79*i) + Y3s*Y7s*(9.3e80 + 3.4e79*i) + Y4s*Y7s*(9.9e80 + 3.5e79*i) + Y1s*Y3s*Y4s*(6.9e77 + 1.2e79*i) + Y1s*Y3s*Y7s*(1.0e78 + 1.7e79*i) + Y1s*Y4s*Y7s*(1.5e78 + 2.6e79*i) + Y3s*Y4s*Y7s*(1.8e78 + 3.0e79*i) - Y1s*Y3s*Y4s*Y7s*(5.3e77 - 8.5e76*i) - 4.4e83 - 8.5e82*i))*484)-((1.5165-17.3338*i)*1000);
(220000/(sqrt(3)))/(((1.2e4*(Y1s*(1.9e77 + 7.3e76*i) + Y3s*(2.2e77 + 8.2e76*i) + Y4s*(2.3e77 + 8.7e76*i) - Y1s*Y3s*(1.1e75 - 4.2e75*i) - Y1s*Y4s*(1.7e75 - 6.4e75*i) - Y3s*Y4s*(2.0e75 - 7.4e75*i) - Y1s*Y3s*Y4s*(1.3e74 + 2.2e73*i) + 2.8e78 - 6.0e78*i))/(Y1s*(2.5e82 - 4.9e82*i) + Y3s*(2.7e82 - 5.5e82*i) + Y4s*(2.8e82 - 5.8e82*i) + Y7s*(3.5e82 - 7.5e82*i) + Y1s*Y3s*(1.1e81 + 4.2e80*i) + Y1s*Y4s*(1.7e81 + 6.7e80*i) + Y3s*Y4s*(2.0e81 + 7.7e80*i) + Y1s*Y7s*(2.4e81 + 9.2e80*i) + Y3s*Y7s*(2.7e81 + 1.0e81*i) + Y4s*Y7s*(2.9e81 + 1.1e81*i) - Y1s*Y3s*Y4s*(1.0e79 - 3.8e79*i) - Y1s*Y3s*Y7s*(1.4e79 - 5.2e79*i) - Y1s*Y4s*Y7s*(2.2e79 - 8.0e79*i) - Y3s*Y4s*Y7s*(2.5e79 - 9.3e79*i) - Y1s*Y3s*Y4s*Y7s*(1.7e78 + 2.7e77*i) - 1.2e84 - 7.0e83*i))*484)-((4.8371-55.2888*i)*1000);
The outcome will be:
y1s.realpart: 11.0169967890388
y1s.imagpart: -125.141575853451
y3s.realpart: 7.81970377305327
y3s.imagpart: -106.356946529629
y4s.realpart: 3.07796056296379
y4s.imagpart: -36.8035908733627
y7s.realpart: 15.4211136064375
y7s.imagpart: -178.419654796614

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ubtc
ubtc 2020-10-29
Thank you, everyone!
I have solved the problem by using fsolve. (I had not tried this properly earliear). The answer is as following:-
ans =
1.0e+02 *
0.1102 - 1.2514i 0.0782 - 1.0636i 0.0308 - 0.3680i 0.1542 - 1.7842i
It is same as pointed out by Mr. Alex Sha above but only limited to four decimal places. However, as you can see in original equations, I have simplified the equations using simplify & Variable-precision arithmetic (up to 2 decimal points)
eq = simplify(eq,'Steps',10)
eq = vpa(eq, 2)
I shall definitely try to solve without using "vpa" via "fsolve" method. However, Is there some more accurate way other than "fsolve", to solve such lengthy nonlinear equations in complex variables (more than 8 equations in same no. of variables) with more precision without using "vpa"?

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