Solve an algebraic matrix equation

Please I need help to obtain the variables (U, V, W, Th) as the solution of the matrix algebra equation below, say

syms U V W Th
a24 = 0:0.2:5; % which is a step forcing function (in time)
F = rand(4,4)*[U; V; W; Th] - [zeros(4,2), rand(4,2)]*[0; 0; W^2; Th^2] - [zeros(2,26); a24; a24];

Error using symengine (line 59)
Array sizes must match.
Error in sym/privBinaryOp (line 903)
            Csym = mupadmex(op,args{1}.s, args{2}.s, varargin{:});
Error in  -  (line 7)
X = privBinaryOp(A, B, 'symobj::zip', '_subtract');

I get stuck trying to subtract a 4 by 1 symbol affiliated matrix columnwise from a double 4 by 26 matrix, so that I can obtain the variables using

FF = solve(F == 0,[U,V,W,Th])

I need help with this problem please. Thanks in anticipation.

[David Hill]

You need to describe your equation better for F. You cannot multiple a 4x2 with a 4x1.

[Vincent Ike]

No, the matrices in the equation are {(4,4)*(4,1) - (4,4)*(4,1)} - (4,26) = 0. the challenge is to subtract yhe resultant (4,1) from (4,26) columnwise.

[David Hill]

What version of matlab are you using?

rand(4,1)-rand(4,26);%this works and your code above works for the current version

[Walter Roberson]

@David Hill

For this purpose it is marginally important that symbolic expressions are involved.

MATLAB added implicit expansion for numeric values in R2016b.

However, implicit expansion was not added for symbolic expressions until R2017b.

We can deduce that the user is using R2017a or before.

[Vincent Ike]

Yes @walter Roberson I use R2015a

Answer 1

[Walter Roberson]

The operation is permitted in current versions:

syms U V W Th
a24 = 0:0.2:5; % which is a step forcing function (in time)
F = rand(4,4)*[U; V; W; Th] - [zeros(4,2), rand(4,2)]*[0; 0; W^2; Th^2] - [zeros(2,26); a24; a24]

F = 

You did not mark your release or mention your release, so the volunteers are entitled to expect that you are using the newest version.

However, I happen to recognize the issue: in sufficiently old versions of MATLAB, implicit expansion did not exist. Furthermore, in old enough versions, bsxfun() could not be used for symbolic expressions.

In versions that old, you have to repmat()

F = repmat(rand(4,4)*[U; V; W; Th] - [zeros(4,2), rand(4,2)]*[0; 0; W^2; Th^2], 1, 26) - [zeros(2,26); a24; a24]

F = 

[Vincent Ike]

Thanks Walter, thank you everyone who tried to help. I surely need this upgrade.

[Vincent Ike]

@Walter can you do me a favour to obtain the unknowns (U V W Th) when equated to zero, in the above question please?

[Walter Roberson]

syms U V W Th
a24 = 0:0.2:5; % which is a step forcing function (in time)
F = repmat(rand(4,4)*[U; V; W; Th] - [zeros(4,2), rand(4,2)]*[0; 0; W^2; Th^2], 1, 26) - [zeros(2,26); a24; a24]

F = 

size(F)

ans = 1×2

4 26

sol = cell2mat(arrayfun(@(COL) solve(F(:,COL), 'maxdegree', 4), 1:size(F,2), 'uniform', 0))

sol = 1×26 struct array with fields:
    Th
    U
    V
    W

[sol(1).Th, sol(1).U, sol(1).V, sol(1).W]

ans = 

There are exact solutions, but they look pretty useless.

[Vincent Ike]

No it's not useless given my objective, you're actually giving me headway. This figure below is my objective. The only critisim I noticed with the solution right now is that I am having two roots of each variable (U,V,W,Th) per column (instead of one) which doesn't give good result if it must be plotted against the steps (1 to 26).

[Vincent Ike]

here's the code I started infact, I wish you have a look.

[Walter Roberson]

There are four roots per column, not two, if you count the complex solutions. You are working with a quartic.

It would be common to want to filter down to real values.

format long
syms U V W Th
a24 = 0:0.2:5; % which is a step forcing function (in time)
F = repmat(rand(4,4)*[U; V; W; Th] - [zeros(4,2), rand(4,2)]*[0; 0; W^2; Th^2], 1, 26) - [zeros(2,26); a24; a24]

F = 

size(F)

ans = 1×2

4 26

[solTh, solU, solV, solW] = arrayfun(@(COL) solve(F(:,COL), [Th, U, V, W], 'MaxDegree', 4), 1:size(F,2), 'uniform', 0);
Thvals = cell2mat(cellfun(@double, solTh, 'uniform', 0))

Thvals = 
 -0.094682841308648 - 1.169458034215385i  2.086480594346070 + 0.000000000000000i  2.077784406966989 + 0.000000000000000i  2.066795882117770 + 0.000000000000000i  2.053794533719276 + 0.000000000000000i  2.038985728582344 + 0.000000000000000i  2.022522049377702 + 0.000000000000000i  2.004516987533478 + 0.000000000000000i  1.985053995067856 + 0.000000000000000i  1.964192590106933 + 0.000000000000000i  1.941972506033464 + 0.000000000000000i  1.918416480745901 + 0.000000000000000i  1.893532051138880 + 0.000000000000000i  1.867312574149251 + 0.000000000000000i  1.839737600058620 + 0.000000000000000i  1.810772653755723 + 0.000000000000000i  1.780368421188653 + 0.000000000000000i  1.748459280706609 + 0.000000000000000i  1.714961052314520 + 0.000000000000000i  1.679767749597541 + 0.000000000000000i  1.642746990646218 + 0.000000000000000i  1.603733524609510 + 0.000000000000000i  1.562520003400327 + 0.000000000000000i  1.518843566340973 + 0.000000000000000i  1.472365792402213 + 0.000000000000000i  1.422641643933716 + 0.000000000000000i
 -0.094682841308648 + 1.169458034215385i -0.191588462920180 + 0.000000000000000i -0.308693627863736 + 0.000000000000000i -0.383827289614846 + 0.000000000000000i -0.434142419854720 + 0.000000000000000i -0.468377862190450 + 0.000000000000000i -0.491368986603327 + 0.000000000000000i -0.506014476764549 + 0.000000000000000i -0.514166684625613 + 0.000000000000000i -0.517068833132778 + 0.000000000000000i -0.515586704069267 + 0.000000000000000i -0.510339554292225 + 0.000000000000000i -0.501778068300653 + 0.000000000000000i -0.490232746397509 + 0.000000000000000i -0.475944888940360 + 0.000000000000000i -0.459086817629438 + 0.000000000000000i -0.439775100856340 + 0.000000000000000i -0.418078966958619 + 0.000000000000000i -0.394025160701052 + 0.000000000000000i -0.367599905080814 + 0.000000000000000i -0.338748196592134 + 0.000000000000000i -0.307370272150378 + 0.000000000000000i -0.273314637267979 + 0.000000000000000i -0.236366403929786 + 0.000000000000000i -0.196228620500677 + 0.000000000000000i -0.152492309223418 + 0.000000000000000i
  0.000000000000000 + 0.000000000000000i  0.004119107756266 - 1.296399624887958i  0.067019783917584 - 1.418624948607412i  0.110080877217749 - 1.529559144442854i  0.141739116536933 - 1.630183653277971i  0.166261240273264 - 1.722318188701896i  0.185988642082023 - 1.807515406888357i  0.202313918084746 - 1.886977745764290i  0.216121518248089 - 1.961626965218137i  0.228003294982134 - 2.032177207998635i  0.238372272487112 - 2.099190557970499i  0.247526710242373 - 2.163116664678125i  0.255688182050097 - 2.224320763805330i  0.263025259593340 - 2.283103665714060i  0.269668817910081 - 2.339716222355765i  0.275722255406068 - 2.394369966943788i  0.281268513303054 - 2.447245064241453i  0.286375016595215 - 2.498496341645928i  0.291097227662477 - 2.548257929195656i  0.295481251210847 - 2.596646876044579i  0.299565776442169 - 2.643766003083962i  0.303383547239644 - 2.689706177909450i  0.306962490403037 - 2.734548147533245i  0.310326592263617 - 2.778364028616950i  0.313496587518443 - 2.821218529669286i  0.316490506114062 - 2.863169961402136i
  2.092496029555718 + 0.000000000000000i  0.004119107756266 + 1.296399624887958i  0.067019783917584 + 1.418624948607412i  0.110080877217749 + 1.529559144442854i  0.141739116536933 + 1.630183653277971i  0.166261240273264 + 1.722318188701896i  0.185988642082023 + 1.807515406888357i  0.202313918084746 + 1.886977745764290i  0.216121518248089 + 1.961626965218137i  0.228003294982134 + 2.032177207998635i  0.238372272487112 + 2.099190557970499i  0.247526710242373 + 2.163116664678125i  0.255688182050097 + 2.224320763805330i  0.263025259593340 + 2.283103665714060i  0.269668817910081 + 2.339716222355765i  0.275722255406068 + 2.394369966943788i  0.281268513303054 + 2.447245064241453i  0.286375016595215 + 2.498496341645928i  0.291097227662477 + 2.548257929195656i  0.295481251210847 + 2.596646876044579i  0.299565776442169 + 2.643766003083962i  0.303383547239644 + 2.689706177909450i  0.306962490403037 + 2.734548147533245i  0.310326592263617 + 2.778364028616950i  0.313496587518443 + 2.821218529669286i  0.316490506114062 + 2.863169961402136i

Uvals = cell2mat(cellfun(@double, solU, 'uniform', 0))

Uvals = 
 -1.337768754800854 + 0.370281008481964i  4.147886744121340 + 0.000000000000000i  4.203560186062389 + 0.000000000000000i  4.250258520665525 + 0.000000000000000i  4.289236671744841 + 0.000000000000000i  4.321456315964071 + 0.000000000000000i  4.347670628085559 + 0.000000000000000i  4.368479501435571 + 0.000000000000000i  4.384366836852620 + 0.000000000000000i  4.395726450913830 + 0.000000000000000i  4.402880479358474 + 0.000000000000000i  4.406092656438100 + 0.000000000000000i  4.405577976024064 + 0.000000000000000i  4.401509706214511 + 0.000000000000000i  4.394024388753237 + 0.000000000000000i  4.383225226296037 + 0.000000000000000i  4.369184096585521 + 0.000000000000000i  4.351942302360433 + 0.000000000000000i  4.331510046834908 + 0.000000000000000i  4.307864496160035 + 0.000000000000000i  4.280946127701744 + 0.000000000000000i  4.250652829464192 + 0.000000000000000i  4.216830848578836 + 0.000000000000000i  4.179261067482686 + 0.000000000000000i  4.137637978360862 + 0.000000000000000i  4.091536621178077 + 0.000000000000000i
 -1.337768754800854 - 0.370281008481964i  0.151621657240662 + 0.000000000000000i  0.315412697274949 + 0.000000000000000i  0.467299478476283 + 0.000000000000000i  0.605192159902538 + 0.000000000000000i  0.730826828987543 + 0.000000000000000i  0.846241109011629 + 0.000000000000000i  0.953172616120265 + 0.000000000000000i  1.053019876536729 + 0.000000000000000i  1.146902914347910 + 0.000000000000000i  1.235728419650252 + 0.000000000000000i  1.320242309412003 + 0.000000000000000i  1.401069436339662 + 0.000000000000000i  1.478743274374451 + 0.000000000000000i  1.553728349607752 + 0.000000000000000i  1.626437574596235 + 0.000000000000000i  1.697246082881442 + 0.000000000000000i  1.766502754777867 + 0.000000000000000i  1.834540366915902 + 0.000000000000000i  1.901685163978232 + 0.000000000000000i  1.968266631030993 + 0.000000000000000i  2.034628350384807 + 0.000000000000000i  2.101141103746500 + 0.000000000000000i  2.168219934876123 + 0.000000000000000i  2.236347949055212 + 0.000000000000000i  2.306111696526823 + 0.000000000000000i
  0.000000000000000 + 0.000000000000000i -1.579824069165484 + 0.155947166774068i -1.822634446669780 - 0.006667761087284i -2.055005141088642 - 0.138071360027912i -2.276518693858056 - 0.249708586776252i -2.488523987026801 - 0.347748394596868i -2.692416419616214 - 0.435807057388301i -2.889364746362167 - 0.516180905531849i -3.080310180795550 - 0.590423624507290i -3.266009643248373 - 0.659639629270387i -3.447077546638494 - 0.724645073425437i -3.624018716575809 - 0.786062149931840i -3.797253076349249 - 0.844377379631293i -3.967133996978494 - 0.899979289566606i -4.133962012381135 - 0.953183699362246i -4.297995180163404 - 1.004251238526169i -4.459457005967377 - 1.053399815600127i -4.618542581319673 - 1.100813704462502i -4.775423396142555 - 1.146650302181526i -4.930251155852911 - 1.191045246118383i -5.083160841666774 - 1.234116350680520i -5.234273188741532 - 1.275966679170814i -5.383696711496329 - 1.316686971368663i -5.531529373029692 - 1.356357584047785i -5.677859972074953 - 1.395050058324667i -5.822769303735992 - 1.432828397609563i
  4.081554045665995 + 0.000000000000000i -1.579824069165484 - 0.155947166774068i -1.822634446669780 + 0.006667761087284i -2.055005141088642 + 0.138071360027912i -2.276518693858056 + 0.249708586776252i -2.488523987026801 + 0.347748394596868i -2.692416419616214 + 0.435807057388301i -2.889364746362167 + 0.516180905531849i -3.080310180795550 + 0.590423624507290i -3.266009643248373 + 0.659639629270387i -3.447077546638494 + 0.724645073425437i -3.624018716575809 + 0.786062149931840i -3.797253076349249 + 0.844377379631293i -3.967133996978494 + 0.899979289566606i -4.133962012381135 + 0.953183699362246i -4.297995180163404 + 1.004251238526169i -4.459457005967377 + 1.053399815600127i -4.618542581319673 + 1.100813704462502i -4.775423396142555 + 1.146650302181526i -4.930251155852911 + 1.191045246118383i -5.083160841666774 + 1.234116350680520i -5.234273188741532 + 1.275966679170814i -5.383696711496329 + 1.316686971368663i -5.531529373029692 + 1.356357584047785i -5.677859972074953 + 1.395050058324667i -5.822769303735992 + 1.432828397609563i

Vvals = cell2mat(cellfun(@double, solV, 'uniform', 0))

Vvals = 
 -0.504055209573428 + 0.375368805056658i  1.538127719112909 + 0.000000000000000i  1.890511678779503 + 0.000000000000000i  2.239840799349641 + 0.000000000000000i  2.586550305531252 + 0.000000000000000i  2.930975461892485 + 0.000000000000000i  3.273380473029579 + 0.000000000000000i  3.613977359915537 + 0.000000000000000i  3.952938744999917 + 0.000000000000000i  4.290406771478357 + 0.000000000000000i  4.626499475971578 + 0.000000000000000i  4.961315427222944 + 0.000000000000000i  5.294937147196451 + 0.000000000000000i  5.627433650621144 + 0.000000000000000i  5.958862324873501 + 0.000000000000000i  6.289270296695796 + 0.000000000000000i  6.618695379709691 + 0.000000000000000i  6.947166657143294 + 0.000000000000000i  7.274704720846710 + 0.000000000000000i  7.601321554900924 + 0.000000000000000i  7.927020013930278 + 0.000000000000000i  8.251792794379593 + 0.000000000000000i  8.575620717975008 + 0.000000000000000i  8.898470015403628 + 0.000000000000000i  9.220288065124281 + 0.000000000000000i  9.540996600448889 + 0.000000000000000i
 -0.504055209573428 - 0.375368805056658i  0.425952114310920 + 0.000000000000000i  0.841967175397670 + 0.000000000000000i  1.245051555178251 + 0.000000000000000i  1.637772165506094 + 0.000000000000000i  2.022531164515393 + 0.000000000000000i  2.401078554174221 + 0.000000000000000i  2.774665087221919 + 0.000000000000000i  3.144202622804048 + 0.000000000000000i  3.510374041664118 + 0.000000000000000i  3.873704412200672 + 0.000000000000000i  4.234607352502596 + 0.000000000000000i  4.593415923112354 + 0.000000000000000i  4.950403771429729 + 0.000000000000000i  5.305800008380226 + 0.000000000000000i  5.659799970539773 + 0.000000000000000i  6.012573234731134 + 0.000000000000000i  6.364269781274358 + 0.000000000000000i  6.715024917878184 + 0.000000000000000i  7.064963406763741 + 0.000000000000000i  7.414203144181120 + 0.000000000000000i  7.762858705799702 + 0.000000000000000i  8.111045091132313 + 0.000000000000000i  8.458882090336328 + 0.000000000000000i  8.806499901058485 + 0.000000000000000i  9.154047044217846 + 0.000000000000000i
  0.000000000000000 + 0.000000000000000i -0.289213068042489 + 0.316657961026758i -0.067584248920351 + 0.277225925982538i  0.162037330403100 + 0.247762453617693i  0.398150601647183 + 0.223989032114299i  0.639386853460727 + 0.203854096904311i  0.884738982561576 + 0.186250862023458i  1.133475602093557 + 0.170519225798469i  1.385054471259112 + 0.156233909123522i  1.639063078088668 + 0.143104231345594i  1.895179870072590 + 0.130922155241616i  2.153148753794755 + 0.119533255261260i  2.412761938001932 + 0.108819468524092i  2.673848091629708 + 0.098688314897300i  2.936263965527091 + 0.089065870041373i  3.199888328034980 + 0.079892019272787i  3.464617483931162 + 0.071117153491821i  3.730361901441559 + 0.062699808381617i  3.997043630786747 + 0.054604939049523i  4.264594298815672 + 0.046802633892629i  4.532953530091115 + 0.039267139009943i  4.802067688555977 + 0.031976106629563i  5.071888863590774 + 0.024910008045871i  5.342374044773266 + 0.018051669320386i  5.613484444050670 + 0.011385899926830i  5.885184934307497 + 0.004899192691078i
  1.182107457488089 + 0.000000000000000i -0.289213068042489 - 0.316657961026758i -0.067584248920351 - 0.277225925982538i  0.162037330403100 - 0.247762453617693i  0.398150601647183 - 0.223989032114299i  0.639386853460727 - 0.203854096904311i  0.884738982561576 - 0.186250862023458i  1.133475602093557 - 0.170519225798469i  1.385054471259112 - 0.156233909123522i  1.639063078088668 - 0.143104231345594i  1.895179870072590 - 0.130922155241616i  2.153148753794755 - 0.119533255261260i  2.412761938001932 - 0.108819468524092i  2.673848091629708 - 0.098688314897300i  2.936263965527091 - 0.089065870041373i  3.199888328034980 - 0.079892019272787i  3.464617483931162 - 0.071117153491821i  3.730361901441559 - 0.062699808381617i  3.997043630786747 - 0.054604939049523i  4.264594298815672 - 0.046802633892629i  4.532953530091115 - 0.039267139009943i  4.802067688555977 - 0.031976106629563i  5.071888863590774 - 0.024910008045871i  5.342374044773266 - 0.018051669320386i  5.613484444050670 - 0.011385899926830i  5.885184934307497 - 0.004899192691078i

Wvals = cell2mat(cellfun(@double, solW, 'uniform', 0))

Wvals = 
  0.799679476330780 - 0.841095593907892i -1.592380803469869 + 0.000000000000000i -1.674426897140375 + 0.000000000000000i -1.751966864870528 + 0.000000000000000i -1.825652020612525 + 0.000000000000000i -1.895986058491776 + 0.000000000000000i -1.963367746894912 + 0.000000000000000i -2.028118866002349 + 0.000000000000000i -2.090503174160550 + 0.000000000000000i -2.150739680847249 + 0.000000000000000i -2.209012172212263 + 0.000000000000000i -2.265476190489518 + 0.000000000000000i -2.320264233416095 + 0.000000000000000i -2.373489675456903 + 0.000000000000000i -2.425249746225831 + 0.000000000000000i -2.475627792939262 + 0.000000000000000i -2.524694980011617 + 0.000000000000000i -2.572511526093993 + 0.000000000000000i -2.619127537923073 + 0.000000000000000i -2.664583464619840 + 0.000000000000000i -2.708910159496266 + 0.000000000000000i -2.752128491792144 + 0.000000000000000i -2.794248387207062 + 0.000000000000000i -2.835267074342088 + 0.000000000000000i -2.875166136297033 + 0.000000000000000i -2.913906631752097 + 0.000000000000000i
  0.799679476330780 + 0.841095593907892i -0.237720136058116 + 0.000000000000000i -0.444554099309726 + 0.000000000000000i -0.622080124030598 + 0.000000000000000i -0.777863240702378 + 0.000000000000000i -0.917534549677085 + 0.000000000000000i -1.044893165145636 + 0.000000000000000i -1.162544012017425 + 0.000000000000000i -1.272334529393337 + 0.000000000000000i -1.375619727331622 + 0.000000000000000i -1.473423624594866 + 0.000000000000000i -1.566540454409061 + 0.000000000000000i -1.655600263387808 + 0.000000000000000i -1.741112826864168 + 0.000000000000000i -1.823497940488323 + 0.000000000000000i -1.903106899457258 + 0.000000000000000i -1.980238131051339 + 0.000000000000000i -2.055148868114853 + 0.000000000000000i -2.128064105776656 + 0.000000000000000i -2.199183691172421 + 0.000000000000000i -2.268688158280731 + 0.000000000000000i -2.336743784825519 + 0.000000000000000i -2.403507291303494 + 0.000000000000000i -2.469130621645908 + 0.000000000000000i -2.533766365638827 + 0.000000000000000i -2.597574677650533 + 0.000000000000000i
  0.000000000000000 + 0.000000000000000i  0.962249250235563 - 0.770907965007232i  1.106689278696621 - 0.731089165352370i  1.234222274922134 - 0.705220050381949i  1.348956411129022 - 0.686548000601008i  1.453959084556002 - 0.672102297646812i  1.551329236491845 - 0.660393363747881i  1.642530219481457 - 0.650587538929500i  1.728617632248514 - 0.642176854853886i  1.810378484561006 - 0.634830771254546i  1.888416678875136 - 0.628322820203841i  1.963207102920860 - 0.622491315078081i  2.035131028873522 - 0.617216892914531i  2.104500031632106 - 0.612408969851718i  2.171572623828648 - 0.607997196681545i  2.236566126669831 - 0.603925859764615i  2.299665336003049 - 0.600150090784758i  2.361028977575994 - 0.596633227988520i  2.420794602321435 - 0.593344933685396i  2.479082358367701 - 0.590259822202744i  2.535997939360069 - 0.587356440792213i  2.591634918780402 - 0.584616499863304i  2.646076619726849 - 0.582024282742178i  2.699397628465569 - 0.579566186932926i  2.751665031439501 - 0.577230363207401i  2.802939435172886 - 0.575006428500804i
 -1.504961391718419 + 0.000000000000000i  0.962249250235563 + 0.770907965007232i  1.106689278696621 + 0.731089165352370i  1.234222274922134 + 0.705220050381949i  1.348956411129022 + 0.686548000601008i  1.453959084556002 + 0.672102297646812i  1.551329236491845 + 0.660393363747881i  1.642530219481457 + 0.650587538929500i  1.728617632248514 + 0.642176854853886i  1.810378484561006 + 0.634830771254546i  1.888416678875136 + 0.628322820203841i  1.963207102920860 + 0.622491315078081i  2.035131028873522 + 0.617216892914531i  2.104500031632106 + 0.612408969851718i  2.171572623828648 + 0.607997196681545i  2.236566126669831 + 0.603925859764615i  2.299665336003049 + 0.600150090784758i  2.361028977575994 + 0.596633227988520i  2.420794602321435 + 0.593344933685396i  2.479082358367701 + 0.590259822202744i  2.535997939360069 + 0.587356440792213i  2.591634918780402 + 0.584616499863304i  2.646076619726849 + 0.582024282742178i  2.699397628465569 + 0.579566186932926i  2.751665031439501 + 0.577230363207401i  2.802939435172886 + 0.575006428500804i

mask = imag(Thvals) == 0 & imag(Uvals) == 0 & imag(Vvals) == 0 & imag(Wvals) == 0;
[Thvals(mask), Uvals(mask), Vvals(mask), Wvals(mask)]

ans = 52×4

0 0 0 0 2.092496029555718 4.081554045665995 1.182107457488089 -1.504961391718419 2.086480594346070 4.147886744121340 1.538127719112909 -1.592380803469869 -0.191588462920180 0.151621657240662 0.425952114310920 -0.237720136058116 2.077784406966989 4.203560186062389 1.890511678779503 -1.674426897140375 -0.308693627863736 0.315412697274949 0.841967175397670 -0.444554099309726 2.066795882117770 4.250258520665525 2.239840799349641 -1.751966864870528 -0.383827289614846 0.467299478476283 1.245051555178251 -0.622080124030598 2.053794533719276 4.289236671744841 2.586550305531252 -1.825652020612525 -0.434142419854720 0.605192159902538 1.637772165506094 -0.777863240702378

The number of solutions is highly variable. Some of my tests showed as few as two solutions, and some of them showed as high as 84 solutions.

[Vincent Ike]

Thank you Walter, I have got much insight with your responses. Now, I have a good question here, https://www.mathworks.com/matlabcentral/answers/879973-how-to-solve-four-sets-of-ode-having-four-variables. I hope to learn more from you.