Solving system of nonlinear equations

Question

omnia 2012-8-2

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/45141-solving-system-of-nonlinear-equations

编辑： John Kelly 2015-3-2

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.

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omnia 2012-8-6

编辑：omnia 2012-8-6

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 2012-8-6

在 MATLAB Online 中打开

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 2012-8-6

编辑：Star Strider 2012-8-6

在 MATLAB Online 中打开

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 2012-8-6

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 2012-8-6

编辑：Star Strider 2012-8-6

在 MATLAB Online 中打开

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.

Walter Roberson 2012-8-9

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

omnia 2012-8-11

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).

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Answer 1

Honglei Chen 2012-8-2

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链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/45141-solving-system-of-nonlinear-equations#answer_55288

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

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

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omnia 2012-8-5

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)