Solving system of nonlinear equations

Hello all. I want to solve 10 nonlinear equations in 10 variables. I used 'solve' function but it didn't give me a result " it takes too much time and nothing happened". Is there any other way to do that? The equations are generated within the program.

[Star Strider]

Posting the relevant parts of your code will help.

Since you are using ‘solve’ you are obviously using the Symbolic Math Toolbox. How are the equations generated within the program? What are the equations? What do you eventually want to do with them?

[omnia]

 for y=1:10;                          
        taw_w_E=((1-f_ES(y))/taw_w_E_0)^-1;
        taw_G_E=((1-f_ES(y))/taw_G_E_0)^-1;
        taw_E_G=((1-f_GS(y))/taw_E_G_0)^-1;
        gES=const*Nm(y)*ES_deg*(2*f_ES(y)-1)*lorntezES(y)/E_ES(y);
        gGS=const*Nm(y)*GS_deg*(2*f_GS(y)-1)*lorntezGS(y)/E_GS(y);

        Rstim_ES1=gammay*delta_z/(2*ES_deg*Nm(y)) * (gES*(Ss_fw+Ss_bw)+sumterm_ES); %gGS same as gain of sp
        Rstim_GS1=gammay*delta_z/(2*GS_deg*Nm(y)) * (gGS*(Ss_fw+Ss_bw)+sumterm_GS);
        eqn_17=[eqn_17 (rho_k/(NQD_k*2*ES_deg))*(f_wl1/taw_w_E)-f_ES(y)/taw_E_w1(y)+(GS_deg/ES_deg)*(f_GS(y)/taw_G_E)-f_ES(y)/taw_E_G-f_ES(y)/taw_r-Rstim_ES1];
        eqn_18=[eqn_18 (ES_deg/GS_deg)*(f_ES(y)/taw_E_G)-f_GS(y)/taw_G_E-f_GS(y)/taw_r-Rstim_GS1];
    end

    soln=solve(eqn_17(1),eqn_17(2),eqn_17(3),eqn_17(4),eqn_17(5),eqn_18(1),eqn_18(2),eqn_18(3),eqn_18(4),eqn_18(5));

f_ES is a vector of symbols containing 10 symbols, the same for f_GS. All other variables are constants

[Star Strider]

I am still lost. If you do:

eqn_17(1) =  simplify(collect(expand(eqn_17(1))))

and such for every equation in:

soln=solve(eqn_17(1),eqn_17(2),eqn_17(3),eqn_17(4),eqn_17(5),eqn_18(1),eqn_18(2),eqn_18(3),eqn_18(4),eqn_18(5));

what do you get?

It will not be necessary to post them all. Posting those results for ‘eqn_17(1)’ and ‘eqn_18(1)’ will likely give us enough information for now.

[omnia]

I did that. The same problem happened (It takes too much time and no result appeared)

[Walter Roberson]

Algebraically solving 10 non-linear equations in 10 unknowns usually takes a number of days, if you do not run out of memory first. The practical "overnight" limit is typically 5 to 7 simultaneous equations.

[omnia]

to Walter: any suggestions how I can do that?

[Star Strider]

Omnia —

Can you at least post the results of:

      eqn_17(1) =  simplify(collect(expand(eqn_17(1))))

and

      eqn_18(1) =  simplify(collect(expand(eqn_18(1))))

for us to see and comment on? If even running those two statements take too much time and do not produce any results, we probably cannot help you.

[Walter Roberson]

How many distinct symbolic variables are there? If f_ES and F_GS each have 10 variables and the variables are not duplicated between the two, then you have 10 equations in 20 unknowns, and you will not be able to get any solution for that unless you specify which 10 of the variables to solve for. The default would be to try to solve for all the variables in the equations, and that would fail because there are not enough equations for 20 variables. But it would go through all the trouble of trying to solve them in case it found that some of variables canceled out.

[omnia]

to Star:it does not take much time but the expressions are too long. I posted it. The letters you will see are symbols which I solve for the results are

eqn_17(1) = simplify(collect(expand(eqn_17(1))))

116243572939000356925466995496181050988756417015500885967487300260565376098142383/2177451841535983880665681550360714342535490171501907325019394840461312-24106759612069861098695551225927499594865947904174090038632442275946499123708671346269682748552178550535529851103/43107481922375677281777785455889721866265008398046483650146424074809414892368321343246831913348890624*f_ES1+154742504910672534362390528/4629590355721305*f_GS1+10696062763240755492666588545566651870871552/22924621067347399351776919804905*f_GS1*f_ES1, 93129274647902536614837597442612252815992836100743339980078558169111014628930927/1744481186063737167486721053675299996653326609223580508099867270184960-195847093856089006364361094420662118260978494755606202856229994533339797215516721848650859618163/3290821285825280737063640556430009151658106510784804981101685951280027293745435115520*f_ES2+1/2*f_GS2*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES2)-f_ES2*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS2), 103626330145294271399547488616985491803143759292759825718002655074554624519894497/1941112109887430256045440963402253910309178979161816840204687408889856-103352611187346639948378249632190388870641417975316839551018028528483331966147584241340798478019/1727754924111110048732731841372233941756597751011194640559834018976810935588654940160*f_ES3+1/2*f_GS3*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES3)-f_ES3*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS3), 58373828299758524181672916154167152209897980238592151808655183299124894834982973/1093450382333523308371920425280473151189497220792306127667856291135488-220953296391003716827357847410059942870457627330863931535429522687776030286816520365244366808413/3673788080589517484187456074632233658881567029912045270631252603543983570878560468992*f_ES4+1/2*f_GS4*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES4)-f_ES4*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS4), 467662793138478911821560987337953367919454653776277233636551467144233509114019979/8760201622842936781746242469177427822640650126448722261764607318163456-420003102751321346695470874575112574093115874719657380524279107518459792945690928182057590829947/6943727017956516935151655477936956138318706341290652161933110028950528087933717577728*f_ES5+1/2*f_GS5*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES5)-f_ES5*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS5)]

eqn_18(1) = simplify(collect(expand(eqn_18(1)))) 4951760157141521099596496896/4951760157141521*f_ES1-21392125526481510985333177091133303741743104/22924621067347399351776919804905*f_GS1*f_ES1-49834416484629072470448893625577185780599771768551555604709144025064283487553/732393097481956459657749157751051292376114585562437784347229552640*f_GS1+30680057561882358670358695147569069162356107380204523997993/316396502155684688073895940102234179999449896452096, 2*f_ES2*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS2)-f_GS2*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES2)-21284357112504429712330240150093692483987070136032523715989753/17111913822378649269067506399307264442678019262251008*f_GS2+4172443290125780443262733750786428041309050873781515715989753/34223827644757298538135012798614528885356038524502016, 2*f_ES3*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS3)-f_GS3*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES3)-22125439385870456757216457977084860588799301724437220220677455/17138416528350325382144632033093883165385744116088832*f_GS3+4987022857520131375071825943990977423413557608348388220677455/34276833056700650764289264066187766330771488232177664, 2*f_ES4*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS4)-f_GS4*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES4)-3764854543803885230713906370532807563582630288577620767608609/2860819872387000249203626277813416981348911494987776*f_GS4+904034671416884981510280092719390582233718793589844767608609/5721639744774000498407252555626833962697822989975552, 2*f_ES5*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS5)-f_GS5*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES5)-44844049642880379924936339571152042071536339847931206168893117/34382843880587355216597766601334241221602387647528960*f_GS5+10461205762293024708338572969817800849933952200402246168893117/68765687761174710433195533202668482443204775295057920]

[omnia]

to Walter:I have two vectors each of 5 symbols

f_ES =[ f_ES1, f_ES2, f_ES3, f_ES4, f_ES5]

f_GS =[ f_GS1, f_GS2, f_GS3, f_GS4, f_GS5]

then I write 10 equations using these symbols. 'eqn_17' contains 5 equations and 'eqn_18' contains the other 5. The for loop is done 5 times not 10 as I posted before (I am sorry error in typing but it is right in the program)

 for y=1:5;                 
        taw_w_E=((1-f_ES(y))/taw_w_E_0)^-1;
        taw_G_E=((1-f_ES(y))/taw_G_E_0)^-1;
        taw_E_G=((1-f_GS(y))/taw_E_G_0)^-1;
        gES=const*Nm(y)*ES_deg*(2*f_ES(y)-1)*lorntezES(y)/E_ES(y);
        gGS=const*Nm(y)*GS_deg*(2*f_GS(y)-1)*lorntezGS(y)/E_GS(y);

        Rstim_ES1=gammay*delta_z/(2*ES_deg*Nm(y)) * (gES*(Ss_fw+Ss_bw)+sumterm_ES); %gGS same as gain of sp
        Rstim_GS1=gammay*delta_z/(2*GS_deg*Nm(y)) * (gGS*(Ss_fw+Ss_bw)+sumterm_GS);

        eqn_17=[eqn_17 (rho_k/(NQD_k*2*ES_deg))*(f_wl1/taw_w_E)-f_ES(y)/taw_E_w1(y)+(GS_deg/ES_deg)*(f_GS(y)/taw_G_E)-f_ES(y)/taw_E_G-f_ES(y)/taw_r-Rstim_ES1];
        eqn_18=[eqn_18 (ES_deg/GS_deg)*(f_ES(y)/taw_E_G)-f_GS(y)/taw_G_E-f_GS(y)/taw_r-Rstim_GS1];
    end

    soln=solve(eqn_17(1),eqn_17(2),eqn_17(3),eqn_17(4),eqn_17(5),eqn_18(1),eqn_18(2),eqn_18(3),eqn_18(4),eqn_18(5));

[Star Strider]

I have one more request, since I had no idea your equations contained numerical constants:

eqn_17(:) = vpa(simplify(collect(expand(eqn_17(1)))),5)

eqn_18(:) = vpa(simplify(collect(expand(eqn_18(1)))),5)

That should at least make them easier to read and understand.

When I do that with the equations you posted (with the ‘syms’ declaration using the vectors of variables in your answer to Walter), I get:

eqn_17(1) = [ 3.3425e10*f_GS1 - 5.5922e11*f_ES1 + 4.6658e11*f_ES1*f_GS1 + 5.3385e10, f_ES2*(5.0e11*f_GS2 - 5.0e11) - 0.5*f_GS2*(6.6849e10*f_ES2 - 6.6849e10) - 5.9513e10*f_ES2 + 5.3385e10, f_ES3*(5.0e11*f_GS3 - 5.0e11) - 0.5*f_GS3*(6.6849e10*f_ES3 - 6.6849e10) - 5.9819e10*f_ES3 + 5.3385e10, f_ES4*(5.0e11*f_GS4 - 5.0e11) - 0.5*f_GS4*(6.6849e10*f_ES4 - 6.6849e10) - 6.0143e10*f_ES4 + 5.3385e10, f_ES5*(5.0e11*f_GS5 - 5.0e11) - 0.5*f_GS5*(6.6849e10*f_ES5 - 6.6849e10) - 6.0487e10*f_ES5 + 5.3385e10]

eqn_18(1) =[ 1.0e12*f_ES1 - 6.8043e10*f_GS1 - 9.3315e11*f_ES1*f_GS1 + 9.6967e7, f_GS2*(6.6849e10*f_ES2 - 6.6849e10) - 1.2438e9*f_GS2 - 2.0*f_ES2*(5.0e11*f_GS2 - 5.0e11) + 1.2192e8, f_GS3*(6.6849e10*f_ES3 - 6.6849e10) - 1.291e9*f_GS3 - 2.0*f_ES3*(5.0e11*f_GS3 - 5.0e11) + 1.4549e8, f_GS4*(6.6849e10*f_ES4 - 6.6849e10) - 1.316e9*f_GS4 - 2.0*f_ES4*(5.0e11*f_GS4 - 5.0e11) + 1.58e8, f_GS5*(6.6849e10*f_ES5 - 6.6849e10) - 1.3043e9*f_GS5 - 2.0*f_ES5*(5.0e11*f_GS5 - 5.0e11) + 1.5213e8]

However I cannot tell you much more than that the magnitude of your constants (most on the order of 1E+10) could cause numeric problems in the calculation.

[omnia]

To Star

eqn_17(1) = vpa(simplify(collect(expand(eqn_17(1)))),5) [ .53385e11-.55922e12*f_ES1+.33425e11*f_GS1+.46658e12*f_GS1*f_ES1, 93129274647902536614837597442612252815992836100743339980078558169111014628930927/1744481186063737167486721053675299996653326609223580508099867270184960-195847093856089006364361094420662118260978494755606202856229994533339797215516721848650859618163/3290821285825280737063640556430009151658106510784804981101685951280027293745435115520*f_ES2+1/2*f_GS2*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES2)-f_ES2*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS2), 103626330145294271399547488616985491803143759292759825718002655074554624519894497/1941112109887430256045440963402253910309178979161816840204687408889856-103352611187346639948378249632190388870641417975316839551018028528483331966147584241340798478019/1727754924111110048732731841372233941756597751011194640559834018976810935588654940160*f_ES3+1/2*f_GS3*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES3)-f_ES3*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS3), 58373828299758524181672916154167152209897980238592151808655183299124894834982973/1093450382333523308371920425280473151189497220792306127667856291135488-220953296391003716827357847410059942870457627330863931535429522687776030286816520365244366808413/3673788080589517484187456074632233658881567029912045270631252603543983570878560468992*f_ES4+1/2*f_GS4*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES4)-f_ES4*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS4), 467662793138478911821560987337953367919454653776277233636551467144233509114019979/8760201622842936781746242469177427822640650126448722261764607318163456-420003102751321346695470874575112574093115874719657380524279107518459792945690928182057590829947/6943727017956516935151655477936956138318706341290652161933110028950528087933717577728*f_ES5+1/2*f_GS5*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES5)-f_ES5*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS5)]

eqn_18(1) = vpa(simplify(collect(expand(eqn_18(1)))),5) [ .10000e13*f_ES1-.93315e12*f_GS1*f_ES1-.68043e11*f_GS1+.96967e8, 2*f_ES2*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS2)-f_GS2*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES2)-21284357112504429712330240150093692483987070136032523715989753/17111913822378649269067506399307264442678019262251008*f_GS2+4172443290125780443262733750786428041309050873781515715989753/34223827644757298538135012798614528885356038524502016, 2*f_ES3*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS3)-f_GS3*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES3)-22125439385870456757216457977084860588799301724437220220677455/17138416528350325382144632033093883165385744116088832*f_GS3+4987022857520131375071825943990977423413557608348388220677455/34276833056700650764289264066187766330771488232177664, 2*f_ES4*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS4)-f_GS4*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES4)-3764854543803885230713906370532807563582630288577620767608609/2860819872387000249203626277813416981348911494987776*f_GS4+904034671416884981510280092719390582233718793589844767608609/5721639744774000498407252555626833962697822989975552, 2*f_ES5*(2475880078570760549798248448/4951760157141521-2475880078570760549798248448/4951760157141521*f_GS5)-f_GS5*(309485009821345068724781056/4629590355721305-309485009821345068724781056/4629590355721305*f_ES5)-44844049642880379924936339571152042071536339847931206168893117/34382843880587355216597766601334241221602387647528960*f_GS5+10461205762293024708338572969817800849933952200402246168893117/68765687761174710433195533202668482443204775295057920]

[Star Strider]

Or a bit more clearly, since ‘eqn_17(1)’ and ‘eqn_18(1)’ are each actually [1 x 5] symbolic vectors:

eqn_17(1,1) =  3.3425e10*f_GS1 + f_ES1*(4.6658e11*f_GS1 - 5.5922e11) + 5.3385e10

eqn_18(1,1) =  9.6967e7 - f_ES1*(9.3315e11*f_GS1 - 1.0e12) - 6.8043e10*f_GS1

eqn_17(1,2) =  3.3425e10*f_GS2 + f_ES2*(4.6658e11*f_GS2 - 5.5951e11) + 5.3385e10

eqn_18(1,2) =  1.2192e8 - f_ES2*(9.3315e11*f_GS2 - 1.0e12) - 6.8093e10*f_GS2

eqn_17(1,3) =  3.3425e10*f_GS3 + f_ES3*(4.6658e11*f_GS3 - 5.5982e11) + 5.3385e10

eqn_18(1,3) =  1.4549e8 - f_ES3*(9.3315e11*f_GS3 - 1.0e12) - 6.814e10*f_GS3

eqn_17(1,4) =  3.3425e10*f_GS4 + f_ES4*(4.6658e11*f_GS4 - 5.6014e11) + 5.3385e10

eqn_18(1,4) =  1.58e8 - f_ES4*(9.3315e11*f_GS4 - 1.0e12) - 6.8165e10*f_GS4

eqn_17(1,5) =  3.3425e10*f_GS5 + f_ES5*(4.6658e11*f_GS5 - 5.6049e11) + 5.3385e10 

eqn_18(1,5) =  1.5213e8 - f_ES5*(9.3315e11*f_GS5 - 1.0e12) - 6.8154e10*f_GS5

However I am still convinced that the magnitude of many of the constants is causing numeric problems with the solution. There may also be linear dependence problems, since many of the coefficients of different variables appear to be the same.

[omnia]

ok Star. Now it is clear but how can I use fsolve to solve them simultaneously?

when I use different values as a start of my program I get eqn_17(1,1) and eqn_18(1,1) not function of f_ES1 and f_GS1 only but combination of all variables (f_ES1,....,f_ES5,f_GS1,... ,f_GS5). I just start with the simplest case

[Walter Roberson]

In order to use fsolve(), use matlabFunction() to convert the symbolic expressions into function handles. fsolve() accepts function handles on input.

[omnia]

my version of MATLAB does not contain matlabFunction(). I am using version 7 (R14). How can I do that?

[Star Strider]

Use vectorize then use the vectorized expression to create your own anonymous function.

[Walter Roberson]

Did R14 have anonymous functions?

[omnia]

yes it has anonymous function but how can I use it with the system of these nonlinear equations?

[omnia]

I tried to solve the equations by the following commands:

 eqns=[eqn_17; eqn_18];  %10 equations written as colomn
    x0=ones(10,1);       %initial values for the 10 unknowns
    options=optimset('Display','iter');
    [soln,fval]=fsolve(myfun1(eqns),x0,options);

where

function F = myfun1(eqns)

var=findsym(eqns);

F=double(subs(eqns,var,ones(1,10))) ;

but I have the following error

??? Error using ==> optim\private\lsqfcnchk
FUN must be a function, a valid string expression, 
or an inline function object.

Error in ==> fsolve at 169
    funfcn = lsqfcnchk(FUN,'fsolve',length(varargin),funValCheck,gradflag);

Error in ==> try_GS_ES_prob at 41
    [soln,fval]=fsolve(myfun1(eqns),x0,options);

How can I solve it? Any help will be useful

Thanks

[Walter Roberson]

findsym() returns all symbolic variables, so you are substituting ones for all symbolic variables in "eqns" and so your "F" will be a numeric value (a vector likely.) You are not leaving anything to vary to be solved by fsolve()

[Alan Weiss]

You might want to use matlabFunction to convert your symbolic variables to a function handle that will give your objective function. It will be faster than repeatedly using subs. See the example here.

Alan Weiss

MATLAB mathematical toolbox documentation

[Walter Roberson]

The poster is using R14. I don't know if matlabFunction() was available back then?

[omnia]

matlabfunction is not available in my version which I think makes this issue tedious for me.

To Walter: I do not understand what do you mean by "You are not leaving anything to vary to be solved by fsolve()". I think what I wrote is the syntax expression to fsolve. I passed the equations to 'myfun1' (symbolic equations). Then I specified the symbols to be substituted with the initial guess (ones for all the symbols).

Answer 1

[Honglei Chen]

Do you have Optimization Toolbox? If so, you can try fsolve

http://www.mathworks.com/help/toolbox/optim/ug/fsolve.html

[omnia]

I did and I have an error "??? Error using ==> fsolve FSOLVE only accepts inputs of data type double."

[Image Analyst]

So, did you try casting your data to double? If not, what did you think to do when you saw this error?

[omnia]

how can I cast my data to double?? I have 10 unknowns as defined as symbols (can I cast them to double?!)

[Honglei Chen]

Solve is part of Symbolic Computation Toolbox. Since you said the result is not satisfying, I'm suggesting you to try the numerical way to solve it, i.e. fsolve. You may need to redefine your variable and rewrite your equations to fit the syntax.

[omnia]

the problem with me is that fsolve requires to write the equations manually in a separate file. In my program I generate them in a for loop (I posted this part of the program above)

[Walter Roberson]

Use matlabFunction() to convert the symbolic equations into something that fsolve() can deal with.