Solve an algebraic matrix equation

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Vincent Ike on 13 Jul 2021
Commented: Vincent Ike on 16 Jul 2021
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)
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.
Walter Roberson on 13 Jul 2021
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 on 14 Jul 2021
Yes @walter Roberson I use R2015a

Walter Roberson on 13 Jul 2021
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 =
Walter Roberson on 15 Jul 2021
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;
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 on 16 Jul 2021
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.

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