MATLAB: Numerical solution of equations

MATLABnumerical solution

    % 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.

Best Answer

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.

Related Solutions

MATLAB: FSOLVE requires all values returned by functions to be of data type double.

If you want to solve a system of equations symbolically or using symbolic variables, use the solve function instead of fsolve. The fsolve function requires that the output of the function you specify as the first input be double precision numbers, not symbolic expressions.

If you must use fsolve with symbolic variables inside the function, you must convert those symbolic expressions into numbers using double before returning the numbers (not the symbolic expressions) from the function.

As a very simple example, put all this code into a function example373677.m and run it:

function y = example373677
y = fsolve(@mysubfun, 3);
 
function z = mysubfun(q)
x = sym('x');
r = x.^3-2*x-5;
z = double(subs(r, q));

Of course, with solve this is easier.

syms x
y = solve(x.^3-2*x-5)

You may need to use vpa to approximate y numerically.

vpa(y)

One of the solutions returned in y should match the result you received from running the function.

MATLAB: Symbolic Math shows weird behavior – not seen in Wolfram or Octave

It looks like the expected solution can be obtained with some work. Maybe someone else can get there faster.

>> syms x real
>> A = sym([1 -2*cos(x) 0; -sin(x)/cos(x)*sin(x) 1 cos(x); 1 -2*sin(x)/cos(x)*sin(x) 2])
 
A =
 
[                1,            -2*cos(x),      0]
[ -sin(x)^2/cos(x),                    1, cos(x)]
[                1, -(2*sin(x)^2)/cos(x),      2]
>> [V,D]=eig(A)
 
V =
 
[ ((cos(x)^4 + 3*cos(x)^2*sin(x)^2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3) + 4/3))/(cos(x)^4 - 2*sin(x)^6) - (2*(cos(x)^4 + cos(x)^2*sin(x)^4 + cos(x)^2*sin(x)^2))/(cos(x)^4 - 2*sin(x)^6) - (cos(x)^2*sin(x)^2*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3) + 4/3)^2)/(cos(x)^4 - 2*sin(x)^6), - (2*(cos(x)^4 + cos(x)^2*sin(x)^4 + cos(x)^2*sin(x)^2))/(cos(x)^4 - 2*sin(x)^6) - ((cos(x)^4 + 3*cos(x)^2*sin(x)^2)*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 + (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3))/(cos(x)^4 - 2*sin(x)^6) - (cos(x)^2*sin(x)^2*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 + (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3)^2)/(cos(x)^4 - 2*sin(x)^6), - (2*(cos(x)^4 + cos(x)^2*sin(x)^4 + cos(x)^2*sin(x)^2))/(cos(x)^4 - 2*sin(x)^6) - ((cos(x)^4 + 3*cos(x)^2*sin(x)^2)*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 - (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3))/(cos(x)^4 - 2*sin(x)^6) - (cos(x)^2*sin(x)^2*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 - (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3)^2)/(cos(x)^4 - 2*sin(x)^6)]
[      ((3*cos(x)^3 + 2*cos(x)*sin(x)^4)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3) + 4/3))/(2*(cos(x)^4 - 2*sin(x)^6)) - (cos(x)^3 + 2*cos(x)*sin(x)^4 + cos(x)^3*sin(x)^2)/(cos(x)^4 - 2*sin(x)^6) - (cos(x)^3*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3) + 4/3)^2)/(2*(cos(x)^4 - 2*sin(x)^6)),      - (cos(x)^3 + 2*cos(x)*sin(x)^4 + cos(x)^3*sin(x)^2)/(cos(x)^4 - 2*sin(x)^6) - ((3*cos(x)^3 + 2*cos(x)*sin(x)^4)*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 + (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3))/(2*(cos(x)^4 - 2*sin(x)^6)) - (cos(x)^3*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 + (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3)^2)/(2*(cos(x)^4 - 2*sin(x)^6)),      - (cos(x)^3 + 2*cos(x)*sin(x)^4 + cos(x)^3*sin(x)^2)/(cos(x)^4 - 2*sin(x)^6) - ((3*cos(x)^3 + 2*cos(x)*sin(x)^4)*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 - (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3))/(2*(cos(x)^4 - 2*sin(x)^6)) - (cos(x)^3*(1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 - (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 4/3)^2)/(2*(cos(x)^4 - 2*sin(x)^6))]
[                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           1,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         1,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         1]
 
 
D =
 
[ 1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) + (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3) + 4/3,                                                                                                                                                                                                                                                                                                                                                                                           0,                                                                                                                                                                                                                                                                                                                                                                                           0]
[                                                                                                                                                                                     0, 4/3 - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 - (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)),                                                                                                                                                                                                                                                                                                                                                                                           0]
[                                                                                                                                                                                     0,                                                                                                                                                                                                                                                                                                                                                                                           0, 4/3 - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)/2 + (3^(1/2)*(1/(9*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3)) - (((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))*1i)/2 - 1/(18*(((cos(x)^2 + sin(x)^2 - 1/27)^2 - 1/729)^(1/2) - cos(x)^2 - sin(x)^2 + 1/27)^(1/3))]
 
>> D=simplify(diag(D),300)  % unclear why D(1) doesn't simplify to 2 - 1i
 
D =
 
 2 - (2*(-1)^(1/3)*(2*3^(1/2) - 3)*(- 15*3^(1/2) - 26)^(1/3))/3 - 3^(1/2)/3
                                                                          0
                                                                     2 + 1i
 
>> V2=V(:,2);  % pick the eigenvector associated with lamda = 0
>> simplify(imag(V2),100)  % verify the imaginary part is 0
 
ans =
 
 0
 0
 0
 
>> V2=real(V2);  % grab the real part
>> V2=simplify(V2,100)
 
V2 =
 
         - 1/cos(2*x) - 1
 -cos(x)/(2*cos(x)^2 - 1)
                        1
 
 
>> simplify(V2*(1-2*sin(x)^2),100)
 
ans =
 
 -2*cos(x)^2
     -cos(x)
    cos(2*x)

Best Answer

Related Solutions

MATLAB: FSOLVE requires all values returned by functions to be of data type double.

MATLAB: Symbolic Math shows weird behavior – not seen in Wolfram or Octave

Related Question