how can i solve simultaneous equation using genetic algorithm ?

x =  10*cos(t1) + 5*cos(t1 + t2) 
y = 10*sin(t1) + 5*sin(t1+t2)

i have these two equations and i have known outputs x=10 and y=5, how can i solve them keeping in mind that the outputs are unknown and i want to generate them using genetic algorithms.

[Walter Roberson]

Note: there are multiple solutions for many (x,y) pairs.

[ahmed mohamed]

yes but i have exact known input x=5 , y=10

[Walter Roberson]

There are two solutions for that exact known input.

Answer 1

[John BG]

Hi Ahamed

cos(A)=.5*(exp(j*A)+exp(-j*A))
sin(B)=-j*.5*(exp(j*B)-exp(-j*B))

x = 5*(exp(j*t1)+exp(-j*t1)) + 2.5*(exp(j*(t1+t2))+exp(-j*(t1+t2)))
y = -j*5*(exp(j*t1)-exp(-j*t1)) - j*2.5*(exp(j*(t1+t2))-exp(-j*(t1+t2)))

x = 5*exp(j*t1)+5*exp(-j*t1) + 2.5*(exp(j*t1)*exp(j*t2)+exp(-j*t1)*exp(-j*t2))
y = -j*5*(exp(j*t1)-exp(-j*t1)) - j*2.5*(exp(j*t1)*exp(j*t2)-exp(-j*t1)*exp(-j*t2))

now substitution

u=exp(j*t1)
v=exp(j*t2)

x/5 = u+1/u + .5*(u*v+1/(u*v))
y/(-j*5) = u-1/u - .5*(u*v-1/(u*v))

since you claim to have x and y fixed to constant values

k1=x0/5
k2=y0/(-j*5)

can you solve the u v system?

i also had a look at

x = 5*cos(t1)  + 5*cos(t1) + 5*cos(t1 + t2) 
y = 5*sin(t1) + 5*sin(t1) + 5*sin(t1+t2)

using:

cos(A)+cos(B)=2*cos((A+B)/2)*cos((A-B)/2)
sin(A)+sin(B)=2*sin((A+B)/2)*cos((A-B)/2)

x/5=cos(t1)+2*cos(t1+t2/2)*cos(t2/2)
y/5=sin(t1)+2*sin(t1+t2/2)*cos(t2/2)

and

cos(A+B)=cos(A)*cos(B)-sin(A)*sin(B)
sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

but the u v system seems a good start point, doesn't it?

If t1 and t2 ranges are narrow, perhaps you would like to try Taylor (MacLaurin) approximations for cos() and sin() from http://people.math.sc.edu/girardi/m142/handouts/10sTaylorPolySeries.pdf

If you find this answer of any help solving your question, please click on the thumbs-up vote link,

thanks in advance

John

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[ahmed mohamed]

I have tried the u v method but the code did not work. i  would be thankful if u helped me with the code . Thanks in advance .