How to solve an overdetermined system of equations?

Question

I'm trying to solve an overdetermined system of five equations and four variables, but I'm not being successful in this task. I have the following unknowns, $F,G,K$ and $L$, and I need to solve $F$ and $L$ in terms of $G$ and $K$, that is

$$ F(G,K) \,\, ,\\ L(G,K) \,\, . $$

The system that I'm treating is the following:

$$ F + G = H + J \,\, ,\\ i F \omega -i G \omega =i H \sqrt{\omega ^2-V_0}-i J \sqrt{\omega ^2-V_0} \,\, , \\ H e^{i a \sqrt{\omega ^2-V_0}}+J e^{-i a \sqrt{\omega ^2-V_0}}=K e^{i a \omega }+L e^{-i a \omega } \,\, , \\ i H \sqrt{\omega ^2-V_0} e^{i a \sqrt{\omega ^2-V_0}}-i J \sqrt{\omega ^2-V_0} e^{-i a \sqrt{\omega ^2-V_0}}=i K \omega e^{i a \omega }-i L \omega e^{-i a \omega } \,\, , \\ K e^{i b \omega }+L e^{-i b \omega }=0 \,\, , \\ $$ where the constants $a,b \in \mathbb{R}$. How could I Solve this system?

Thanks in advance!

Answer 1

For each equation list the variables with the ones you want to keep left of a separator (|) and the unwanted variables or constants on the right.

1.(F,G|H,J)

2.(F,G|H,J)

3.(K,L|H,J)

4.(K,L|H,J)

5.(K,L|)

You want (F,G,K|) and (L,G,K|)

Note equation 5 is already in the form (L,K|) or as a function L(K).

To find (F,G,K|) you need equations 1 and/or 2 to get F but with H and J eliminated.

To isolate H and J combine equation 5 with 3 and 4 respectively to get rid of L.

Note equations 3 and 4 are linearly independant in H and J, the +/- sign difference.

5+3 -> 6.(K|H,J) , new equation 6

5+4 -> 7.(K|H,J) , new equation 7

Equations 6 and 7 are 2 linearly independent equations with respect to 2 unknowns H and J so H and J can be isolated.

6+7 -> 8.(K|H) and 9.(K|J) the two new equations 8 and 9.

Substitute 8 and 9 into 1 (or 2) to replace H and J by K to get (F,G,K|).

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Running these instructions in maxima

E1 : F + G - H - J;
tex(%th(1));
E2 : F*w - G*w - H*sqrt(w^2-Vo) + J*sqrt(w^2-Vo);
tex(%th(1));
E3 : H*exp(i*a*sqrt(w^2-Vo)) + J*exp(-i*a*sqrt(w^2-Vo)) - K*exp(i*a*w) - L*exp(-i*a*w);
tex(%th(1));
E4 : H*sqrt(w^2-Vo)*exp(i*a*sqrt(w^2-Vo)) - J*sqrt(w^2-Vo)*exp(-i*a*sqrt(w^2-Vo)) -K*w*exp(i*a*w) + L*w*exp(-i*a*w);
tex(%th(1));
E5 : K*exp(i*b*w) + L*exp(-i*b*w);
tex(%th(1));

L5 : solve(E5,[L]);
tex(%th(1));

E53 : subst(L5,E3);
tex(%th(1));
E54 : subst(L5,E4);
tex(%th(1));

EHJ : solve([E53,E54],[H,J]);
EHJ[1][1];
tex(%th(1));
EHJ[1][2];
tex(%th(1));

EF : subst(EHJ[1],E1);
tex(%th(1));

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Gives the results with equations all equal to zero.

$$-J-H+G+F$$

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$$\sqrt{w^2-{\it Vo}}\,J-\sqrt{w^2-{\it Vo}}\,H-w\,G+w\,F$$

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$$-e^ {- a\,i\,w }\,L-e^{a\,i\,w}\,K+e^ {- a\,i\,\sqrt{w^2-{\it Vo}}}\,J+e^{a\,i\,\sqrt{w^2-{\it Vo}}}\,H$$

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$$w\,e^ {- a\,i\,w }\,L-w\,e^{a\,i\,w}\,K-\sqrt{w^2-{\it Vo}}\,e^ {- a\,i\,\sqrt{w^2-{\it Vo}} }\,J+\sqrt{w^2-{\it Vo}}\,e^{a\,i\,\sqrt{w ^2-{\it Vo}}}\,H$$

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$$e^ {- b\,i\,w }\,L+e^{b\,i\,w}\,K$$

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Answer for L(K):

$$\left[ L=-e^{2\,b\,i\,w}\,K \right] $$

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$$e^{2\,b\,i\,w-a\,i\,w}\,K-e^{a\,i\,w}\,K+e^ {- a\,i\,\sqrt{w^2- {\it Vo}} }\,J+e^{a\,i\,\sqrt{w^2-{\it Vo}}}\,H$$

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$$-w\,e^{2\,b\,i\,w-a\,i\,w}\,K-w\,e^{a\,i\,w}\,K-\sqrt{w^2-{\it Vo}} \,e^ {- a\,i\,\sqrt{w^2-{\it Vo}} }\,J+\sqrt{w^2-{\it Vo}}\,e^{a\,i \,\sqrt{w^2-{\it Vo}}}\,H$$

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$$H=-{{\left(\left(\sqrt{w^2-{\it Vo}}-w\right)\,e^{2\,b\,i\,w}+ \left(-\sqrt{w^2-{\it Vo}}-w\right)\,e^{2\,a\,i\,w}\right)\,e^{-a\,i \,\sqrt{w^2-{\it Vo}}-a\,i\,w}\,K}\over{2\,\sqrt{w^2-{\it Vo}}}}$$

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$$J=-{{e^ {- a\,i\,w }\,\left(\left(\sqrt{w^2-{\it Vo}}+w\right)\,e^{ 2\,b\,i\,w}+\left(w-\sqrt{w^2-{\it Vo}}\right)\,e^{2\,a\,i\,w} \right)\,e^{a\,i\,\sqrt{w^2-{\it Vo}}}\,K}\over{2\,\sqrt{w^2- {\it Vo}}}}$$

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Answer for F(G,K):

$${{\left(\left(\sqrt{w^2-{\it Vo}}+w\right)\,e^{2\,b\,i\,w}+\left(w- \sqrt{w^2-{\it Vo}}\right)\,e^{2\,a\,i\,w}\right)\,e^{a\,i\,\sqrt{w^ 2-{\it Vo}}-a\,i\,w}\,K}\over{2\,\sqrt{w^2-{\it Vo}}}}+{{\left( \left(\sqrt{w^2-{\it Vo}}-w\right)\,e^{2\,b\,i\,w}+\left(-\sqrt{w^2- {\it Vo}}-w\right)\,e^{2\,a\,i\,w}\right)\,e^{-a\,i\,\sqrt{w^2- {\it Vo}}-a\,i\,w}\,K}\over{2\,\sqrt{w^2-{\it Vo}}}}+G+F$$

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The extra equation 2, can be used to eliminate G and create a function F(K).

maxima code:

EF2 : subst(EHJ[1],E2);
tex(%th(1));

EFK : solve([EF,EF2],[G,F]);
tex(%th(1));

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$$-{{\left(\left(\sqrt{w^2-{\it Vo}}+w\right)\,e^{2\,b\,i\,w}+\left(w -\sqrt{w^2-{\it Vo}}\right)\,e^{2\,a\,i\,w}\right)\,e^{a\,i\,\sqrt{w ^2-{\it Vo}}-a\,i\,w}\,K}\over{2}}+{{\left(\left(\sqrt{w^2-{\it Vo}} -w\right)\,e^{2\,b\,i\,w}+\left(-\sqrt{w^2-{\it Vo}}-w\right)\,e^{2 \,a\,i\,w}\right)\,e^{-a\,i\,\sqrt{w^2-{\it Vo}}-a\,i\,w}\,K}\over{2 }}-w\,G+w\,F$$

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$$G=-{{\left(\left(\left(2\,w\,\sqrt{w^2-{\it Vo}}+2\,w ^2-{\it Vo}\right)\,e^{2\,b\,i\,w}+{\it Vo}\,e^{2\,a\,i\,w}\right)\, e^{2\,a\,i\,\sqrt{w^2-{\it Vo}}}+\left(2\,w\,\sqrt{w^2-{\it Vo}}-2\, w^2+{\it Vo}\right)\,e^{2\,b\,i\,w}-{\it Vo}\,e^{2\,a\,i\,w}\right) \,e^{-a\,i\,\sqrt{w^2-{\it Vo}}-a\,i\,w}\,K}\over{4\,w\,\sqrt{w^2- {\it Vo}}}}$$

$$F=-{{\left(\left({\it Vo}\,e^{2\,b\,i\,w}+\left(-2\,w \,\sqrt{w^2-{\it Vo}}+2\,w^2-{\it Vo}\right)\,e^{2\,a\,i\,w}\right) \,e^{2\,a\,i\,\sqrt{w^2-{\it Vo}}}-{\it Vo}\,e^{2\,b\,i\,w}+\left(-2 \,w\,\sqrt{w^2-{\it Vo}}-2\,w^2+{\it Vo}\right)\,e^{2\,a\,i\,w} \right)\,e^{-a\,i\,\sqrt{w^2-{\it Vo}}-a\,i\,w}\,K}\over{4\,w\, \sqrt{w^2-{\it Vo}}}}$$

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Make the computer do the work.

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To get L(G,K):

Solve 3.(K,L|H,J) + 4.(K,L|H,J) to isolate H and J to get equations:

10.(K,L|H) and 11.(K,L|J)

Then combine 1.(F,G|H,J) + 2.(F,G|H,J) to remove F.

12.(G|H,J)

Then substitute H and J from 10 and 11 into 12 to get 13.(G,K,L|)

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The maxima instructions:

EHJ2 : solve([E3,E4],[H,J]);
EHJ2[1][1];
tex(%th(1));
EHJ2[1][2];
tex(%th(1));

EF3 : solve(E1,F);
tex(%th(1));

EnoF : subst(EF3,E2);
tex(%th(1));

EL2 : subst(EHJ2,EnoF);
tex(%th(1));

ELGK : solve(EL2,[L]);
ELGK[1];
tex(%th(1));

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$$H={{e^{-a\,i\,\sqrt{w^2-{\it Vo}}-a\,i\,w}\,\left(\left(\sqrt{w^2- {\it Vo}}-w\right)\,L+\left(\sqrt{w^2-{\it Vo}}+w\right)\,e^{2\,a\,i \,w}\,K\right)}\over{2\,\sqrt{w^2-{\it Vo}}}}$$

$$J={{e^ {- a\,i\,w }\,\left(\left(\sqrt{w^2-{\it Vo}}+w\right)\,e^{a \,i\,\sqrt{w^2-{\it Vo}}}\,L+\left(\sqrt{w^2-{\it Vo}}-w\right)\,e^{ a\,i\,\sqrt{w^2-{\it Vo}}+2\,a\,i\,w}\,K\right)}\over{2\,\sqrt{w^2- {\it Vo}}}}$$

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$$\left[ F=J+H-G \right] $$

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$$w\,\left(J+H-G\right)+\sqrt{w^2-{\it Vo}}\,J-\sqrt{w^2-{\it Vo}}\,H -w\,G$$

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$$w\,\left({{e^ {- a\,i\,w }\,\left(\left(\sqrt{w^2-{\it Vo}}+w \right)\,e^{a\,i\,\sqrt{w^2-{\it Vo}}}\,L+\left(\sqrt{w^2-{\it Vo}}- w\right)\,e^{a\,i\,\sqrt{w^2-{\it Vo}}+2\,a\,i\,w}\,K\right)}\over{2 \,\sqrt{w^2-{\it Vo}}}}+{{e^{-a\,i\,\sqrt{w^2-{\it Vo}}-a\,i\,w}\, \left(\left(\sqrt{w^2-{\it Vo}}-w\right)\,L+\left(\sqrt{w^2-{\it Vo} }+w\right)\,e^{2\,a\,i\,w}\,K\right)}\over{2\,\sqrt{w^2-{\it Vo}}}}- G\right)+{{e^ {- a\,i\,w }\,\left(\left(\sqrt{w^2-{\it Vo}}+w\right) \,e^{a\,i\,\sqrt{w^2-{\it Vo}}}\,L+\left(\sqrt{w^2-{\it Vo}}-w \right)\,e^{a\,i\,\sqrt{w^2-{\it Vo}}+2\,a\,i\,w}\,K\right)}\over{2 }}-{{e^{-a\,i\,\sqrt{w^2-{\it Vo}}-a\,i\,w}\,\left(\left(\sqrt{w^2- {\it Vo}}-w\right)\,L+\left(\sqrt{w^2-{\it Vo}}+w\right)\,e^{2\,a\,i \,w}\,K\right)}\over{2}}-w\,G$$

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Answer L(G,K):

$$L={{\left({\it Vo}\,e^{2\,a\,i\,\sqrt{w^2-{\it Vo}}+2\,a\,i\,w}- {\it Vo}\,e^{2\,a\,i\,w}\right)\,K+4\,w\,\sqrt{w^2-{\it Vo}}\,e^{a\, i\,\sqrt{w^2-{\it Vo}}+a\,i\,w}\,G}\over{\left(2\,w\,\sqrt{w^2- {\it Vo}}+2\,w^2-{\it Vo}\right)\,e^{2\,a\,i\,\sqrt{w^2-{\it Vo}}}+2 \,w\,\sqrt{w^2-{\it Vo}}-2\,w^2+{\it Vo}}}$$

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