MATLAB Answers

How to solve a transcendental equation?

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Ahsan Noor
Ahsan Noor on 16 Jan 2020
Commented: John D'Errico on 22 Jan 2020
Could someone please help me find the solution solution of the following Transcendental equation
epsd=1;
epm=-1.0169 - 0.0250i;
zeta=-134-0.016i;
sg=sqrt(k^2-epsm);
ac=sqrt(k^2-epsd);
ga=sqrt(k^2-zeta);
%and the actual equation solved for 'k' is the follwoing:
k^2*(epsd-epsm)=ga*(al*epsm+sg*epsd)
%The equation has complex roots.

  1 Comment

John D'Errico
John D'Errico on 22 Jan 2020
How is this transcendental? NOT.
By the way, a big problem in your code is you never defined epsm. You did define epm, which I assume is supposed to be the same thing. But then you try to use epsm.

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Answers (1)

Guru Mohanty
Guru Mohanty on 22 Jan 2020
Hi, I understand you are trying to solve this equation having complex roots. You can solve the equation using solvefunction. Here is the code for it.
clc;
clear all;
syms k
epsd=1;
epm=-1.0169 - 0.0250i;
zeta=-134-0.016i;
sg=sqrt(k^2-epsm);
ac=sqrt(k^2-epsd);
ga=sqrt(k^2-zeta);
% and the actual equation solved for 'k' is the follwoing:
eqn = k^2*(epsd-epsm) == ga*(ac*epsm+sg*epsd);
%The equation has complex roots.
Sol = double(solve(eqn,k));
disp(Sol);

  3 Comments

Walter Roberson
Walter Roberson on 22 Jan 2020
It is odd that vpasolve() returns an empty solution.
Guru Mohanty
Guru Mohanty on 22 Jan 2020
I have checked with vpasolve and it is returning the same solution. You may try it using the following code.
clc;
clear all;
syms k
epsd=1;
epsm=-1.0169 - 0.0250i;
zeta=-134-0.016i;
sg=sqrt(k^2-epsm);
ac=sqrt(k^2-epsd);
ga=sqrt(k^2-zeta);
% and the actual equation solved for 'k' is the follwoing:
eqn = k^2*(epsd-epsm) == ga*(ac*epsm+sg*epsd);
%The equation has complex roots.
Sol = double(solve(eqn,k));
Sol1=double(vpasolve(eqn,k));
disp(Sol);
disp(Sol1);
Walter Roberson
Walter Roberson on 22 Jan 2020
Notice that the original code defines epsd and epm but not epsm, but uses epsm anyhow.
It just so happens that epsm is a function in the Mapping Toolbox that returns a value of 1e-6 .
When that (accidental) value is used in eqn, solve() can find four roots for the equation, but vpasolve() returns empty.
eqn =
(9007190247541737*k^2)/9007199254740992 == (k^2 + 134 + 2i/125)^(1/2)*((k^2 - 1)^(1/2)/1000000 + (k^2 - 1/1000000)^(1/2))
>> solve(eqn)
ans =
-(4*root(z^8 + z^6*(834454315777359667686173382578062569804530196775819264995357092773326925608159487/358138990351727082983582815083491551894474553273290039871750000 + 348727176742777174630250231036071447379804163125721955440400056568445403136i/1253486466231044790442539852792220431630660936456515139551125) + z^4*(391359845386070448888706838714665986023487930089577485026236049546130268088207431035018827349227/5013945864924179161770159411168881726522643745826060558204500000000000 + 365074490565570817424189483780559028445600816138306420570226898955525860019295613878272i/19585726034860074850664685199878444244229077132133049055486328125) + z^2*(388438881466780657720765702614211601147238780002224632671430663479495981379013233177116409/2506972932462089580885079705584440863261321872913030279102250000000000000000 + 11365997441581558821082008143008366080734161299451741824482885801329342805966848i/306026969294688669541635706248100691316079330189578891491973876953125) + (794444088335024803982924562035202394153874285989604449373894552359019645159961116409/10027891729848358323540318822337763453045287491652121116409000000000000000000000000 + 12827783001197039242204012145810813792023975258553917324748232056635392i/683095913604215780226865415732367614544819933458881454223155975341796875), z, 5)^2 + 1/250000)^(1/2)/2
(4*root(z^8 + z^6*(834454315777359667686173382578062569804530196775819264995357092773326925608159487/358138990351727082983582815083491551894474553273290039871750000 + 348727176742777174630250231036071447379804163125721955440400056568445403136i/1253486466231044790442539852792220431630660936456515139551125) + z^4*(391359845386070448888706838714665986023487930089577485026236049546130268088207431035018827349227/5013945864924179161770159411168881726522643745826060558204500000000000 + 365074490565570817424189483780559028445600816138306420570226898955525860019295613878272i/19585726034860074850664685199878444244229077132133049055486328125) + z^2*(388438881466780657720765702614211601147238780002224632671430663479495981379013233177116409/2506972932462089580885079705584440863261321872913030279102250000000000000000 + 11365997441581558821082008143008366080734161299451741824482885801329342805966848i/306026969294688669541635706248100691316079330189578891491973876953125) + (794444088335024803982924562035202394153874285989604449373894552359019645159961116409/10027891729848358323540318822337763453045287491652121116409000000000000000000000000 + 12827783001197039242204012145810813792023975258553917324748232056635392i/683095913604215780226865415732367614544819933458881454223155975341796875), z, 5)^2 + 1/250000)^(1/2)/2
-(4*root(z^8 + z^6*(834454315777359667686173382578062569804530196775819264995357092773326925608159487/358138990351727082983582815083491551894474553273290039871750000 + 348727176742777174630250231036071447379804163125721955440400056568445403136i/1253486466231044790442539852792220431630660936456515139551125) + z^4*(391359845386070448888706838714665986023487930089577485026236049546130268088207431035018827349227/5013945864924179161770159411168881726522643745826060558204500000000000 + 365074490565570817424189483780559028445600816138306420570226898955525860019295613878272i/19585726034860074850664685199878444244229077132133049055486328125) + z^2*(388438881466780657720765702614211601147238780002224632671430663479495981379013233177116409/2506972932462089580885079705584440863261321872913030279102250000000000000000 + 11365997441581558821082008143008366080734161299451741824482885801329342805966848i/306026969294688669541635706248100691316079330189578891491973876953125) + (794444088335024803982924562035202394153874285989604449373894552359019645159961116409/10027891729848358323540318822337763453045287491652121116409000000000000000000000000 + 12827783001197039242204012145810813792023975258553917324748232056635392i/683095913604215780226865415732367614544819933458881454223155975341796875), z, 6)^2 + 1/250000)^(1/2)/2
(4*root(z^8 + z^6*(834454315777359667686173382578062569804530196775819264995357092773326925608159487/358138990351727082983582815083491551894474553273290039871750000 + 348727176742777174630250231036071447379804163125721955440400056568445403136i/1253486466231044790442539852792220431630660936456515139551125) + z^4*(391359845386070448888706838714665986023487930089577485026236049546130268088207431035018827349227/5013945864924179161770159411168881726522643745826060558204500000000000 + 365074490565570817424189483780559028445600816138306420570226898955525860019295613878272i/19585726034860074850664685199878444244229077132133049055486328125) + z^2*(388438881466780657720765702614211601147238780002224632671430663479495981379013233177116409/2506972932462089580885079705584440863261321872913030279102250000000000000000 + 11365997441581558821082008143008366080734161299451741824482885801329342805966848i/306026969294688669541635706248100691316079330189578891491973876953125) + (794444088335024803982924562035202394153874285989604449373894552359019645159961116409/10027891729848358323540318822337763453045287491652121116409000000000000000000000000 + 12827783001197039242204012145810813792023975258553917324748232056635392i/683095913604215780226865415732367614544819933458881454223155975341796875), z, 6)^2 + 1/250000)^(1/2)/2
>> vpasolve(eqn)
ans =
Empty sym: 0-by-1

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