numerical solution of equations

    % clc
clear all
L_r=2.4e-6;
L_m=11e-6;
C_r=1.3e-6;
R_L=533.33;
f_s=95e+3;
n=0.067;
V_in=20;
R_ac=n^2*R_L*8/pi^2;
f_r=1/(2*pi*(C_r*L_r)^0.5);
omega_r=1/((C_r*L_r)^0.5);
l=L_r/L_m;
%l=0.341;
Z_0=(L_r/C_r)^0.5;
Q=Z_0/R_ac;
F=f_s/f_r;
den_M=(1+l-l/(F^2))^2+Q^2*(F-1/F)^2;
M=1/(den_M)^0.5;
R_0=(L_r/C_r)^0.5;
gama=pi/F;
r_L=R_L*n^2/R_0;
options=optimset('Display','on',...
    'Algorithm','trust-region-reflective','LargeScale','off','MaxFunEvals',20000);   
     x0=[-1, -1, 0.001,1.3, M];
     % Make a starting guess at the solution
   func_CCM=@(x)myfun_CCM(x,l,F,r_L);
     [x,fval] = fsolve(func_CCM,x0,options);

Can anyone please help me to solve these equations numerically? it seems that it is too sensitive to initial conditions.

[Torsten]

But "fsolve" is a numerical solver ...

Best wishes

Torsten.

[Walter Roberson]

We do not appear to have the code for myfun_CCM

[Torsten]

It's appended.

Best wishes

Torsten.

Answer 1

[Walter Roberson]

My tests suggest that there are an infinite number of solutions.

If I add in the constraint that x(1) and x(2) are both negative, the solution space becomes much cleaner, but it appears that there may be a line of solutions, with the line being straight in some combinations of parameters and curved in other combinations, and possibly circular in some combinations.

For future reference, the least-squares residue for the combination of expressions is

((18446744073709551616*17394581254853885892832513553193^(1/2))/236684080876592592551267357340200625 + sin(x3)*(1/x5 - x2 + 1) + x1*cos(x3))^2 + ((36893488147419103232*17394581254853885892832513553193^(1/2))/236684080876592592551267357340200625 + 2*sin(x3)*(1/x5 - x2 + 1) + 2*x1*cos(x3))^2 + (x1 + sin(x4 - (147573952589676412928*17394581254853885892832513553193^(1/2))/206560652401389894977386098444166875)*(1/x5 + cos(x3 - x4)*(x1*sin(x3) - cos(x3)*(1/x5 - x2 + 1) + 2) - sin(x3 - x4)*(sin(x3)*(1/x5 - x2 + 1) + x1*cos(x3))) + cos(x4 - (147573952589676412928*17394581254853885892832513553193^(1/2))/206560652401389894977386098444166875)*(cos(x3 - x4)*(sin(x3)*(1/x5 - x2 + 1) + x1*cos(x3)) + sin(x3 - x4)*(x1*sin(x3) - cos(x3)*(1/x5 - x2 + 1) + 2)))^2 + (x2 + cos(x4 - (147573952589676412928*17394581254853885892832513553193^(1/2))/206560652401389894977386098444166875)*(1/x5 + cos(x3 - x4)*(x1*sin(x3) - cos(x3)*(1/x5 - x2 + 1) + 2) - sin(x3 - x4)*(sin(x3)*(1/x5 - x2 + 1) + x1*cos(x3))) - sin(x4 - (147573952589676412928*17394581254853885892832513553193^(1/2))/206560652401389894977386098444166875)*(cos(x3 - x4)*(sin(x3)*(1/x5 - x2 + 1) + x1*cos(x3)) + sin(x3 - x4)*(x1*sin(x3) - cos(x3)*(1/x5 - x2 + 1) + 2)) - 1)^2 + ((4548824903*2046358809243513^(1/2)*3777893186295716^(1/2)*17394581254853885892832513553193^(1/2)*((944473296573929*(x4 - (147573952589676412928*17394581254853885892832513553193^(1/2))/206560652401389894977386098444166875)^2)/8657671885261016 + (944473296573929*(x3 - x4)^2)/8657671885261016 - x3*((944473296573929*x3)/4328835942630508 + sin(x3)*(1/x5 - x2 + 1) + x1*cos(x3)) - (x4 - (147573952589676412928*17394581254853885892832513553193^(1/2))/206560652401389894977386098444166875)*((944473296573929*x4)/4328835942630508 - (944473296573929*x3)/4328835942630508 + sin(x3)*(1/x5 - x2 + 1) + x1*cos(x3)) - (cos(x3 - x4) - 1)*(x1*sin(x3) - cos(x3)*(1/x5 - x2 + 1) + 2) - cos(x4 - (147573952589676412928*17394581254853885892832513553193^(1/2))/206560652401389894977386098444166875)*(1/x5 + cos(x3 - x4)*(x1*sin(x3) - cos(x3)*(1/x5 - x2 + 1) + 2) - sin(x3 - x4)*(sin(x3)*(1/x5 - x2 + 1) + x1*cos(x3))) - (sin(x3)*(1/x5 - x2 + 1) + x1*cos(x3))*(x3 - x4) + sin(x4 - (147573952589676412928*17394581254853885892832513553193^(1/2))/206560652401389894977386098444166875)*(cos(x3 - x4)*(sin(x3)*(1/x5 - x2 + 1) + x1*cos(x3)) + sin(x3 - x4)*(x1*sin(x3) - cos(x3)*(1/x5 - x2 + 1) + 2)) - (cos(x3) - 1)*(1/x5 - x2 + 1) + x1*sin(x3) + (944473296573929*x3^2)/8657671885261016 + sin(x3 - x4)*(sin(x3)*(1/x5 - x2 + 1) + x1*cos(x3))))/89202980794122488529539561397346631680000 + 1)^2

That is, the solutions to the equations are at the places where the value of the above expression is 0 to within round-off error.

The following points all have residues less than 1E-31:

-0.7893647991479240122 -0.07848194316452314356 0.2116665368024348848 3.368786592896710452 0.954455687562678956
-0.7342868372668747146 -0.1043149569124650483 0.1857827717154188485 3.316837355667334286 0.9590097329406017668
-0.7141410122343361255 -0.1132428863355284254 0.1763740381986312611 3.297941832531899209 0.960505275770085043
-0.7657521078494986533 -0.08981468491900718165 0.2005405401791646725 3.346462465130042752 0.9564920761787284453
-0.6545274048516261933 -0.1380641831582223866 0.1487093547457488951 3.242346378048229116 0.964406054329632112
-0.673402048965733524 -0.1304611011568412027 0.1574404088488735587 3.259898327875586421 0.9632551956473980326
-0.7885658988038088957 -0.07887175531296626574 0.2112893753354327664 3.368029973071825722 0.9545266613594469707
-0.6313265821814251222 -0.1470895072137880233 0.1380117592601543541 3.220833785484086498 0.9657149274626131152
-0.6340799795889885404 -0.1460367794731131375 0.1392793404585129791 3.223383279139203417 0.9655656647739427889
-0.7977538716778466155 -0.07436144836990092599 0.2156301412872833057 3.376737307572778235 0.9537015922654279443
-0.6203079422825569234 -0.1512530968793995567 0.132944372571758368 3.210640594577584217 0.9662959603635126182
-0.5891111202857871598 -0.1626167636765204483 0.1186422157795321286 3.181861714243269024 0.9678004153941159871
-0.7621971392857225247 -0.09148719120250312087 0.1988693214509161256 3.343108412113263928 0.9567877009437142366
-0.7573998274798614538 -0.09373029243595867865 0.1966156436224273385 3.338585071103084445 0.95718210886302868
-0.6759919663640672205 -0.1293994550296596602 0.1586404582852840583 3.262310353742346791 0.9630912213066177285
-0.5311255232197079623 -0.1820947997211517511 0.09222987957828468475 3.128677224820066716 0.9700512940513597027
-0.1718182982997054076 -0.2575868934821719525 -0.0671464685328081734 2.806722519066860766 0.9689597447593119028
-0.2916654246821250851 -0.2407989633614635028 -0.01472712274347974526 2.912811460171774502 0.9721115933024442324
-0.7412294326058586069 -0.101174132197184824 0.1890323645364604954 3.323362021083449136 0.95847335685781343
-0.06553453868040369501 -0.2656912380179828892 -0.1131059015254947397 2.71354224926287646 0.9639346742965682058
-0.757354024815799054 -0.0937516318148517297 0.1965941351824513905 3.338541899886751541 0.9571858494603404655

Notice there is a fair bit of variation in the parameters.

Residue as low as 1E-31 is essentially "magic" compared to the fact that most of your input coefficients only have two significant figures, with R_L being the stand-out at 5 significant figures. If we assume, for example, that L_r=2.4e-6 means L_r is in the range 2.4e-6 +/- 0.05E-6 then we should expect changes in the residue that are relatively large compared to these residues: the lines of solution would be expected to move noticeably.

[safi58]

Thanks Walter. I think the equations are too sensitive to the initial conditions and also the numerical values of the parameters.

[Walter Roberson]

I was just doing some cross-tests and noticed that with the residue I posted above, the derivative with respect to x1 or x5 (possibly others) has a singularity near x5 = 0; I will need to recheck that.