# solution of 3d nonlinear equation

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x_p, y_p, z_p=(4, 5, 2)

x_1, y_1, z_1=(8, 9, 5)

x_2, y_2, z_2=(2, 5, 1)

x_3, y_3, z_3=(6, 1, 3)

t_1=5.692820*10^-9

t_2=-2.924173*10^-9

t_3=-12.010097*10^-9

c=3.0*10^8

and my three equations are

eqn1 = sqrt((x(s)-x_p)^2+(y(s)-y_p)^2+(z(s)-z_p)^2)-sqrt((x(s)-x_1)^2+(y(s)-y_1)^2+(z(s)-z_2)^2)-(c*t_1)

eqn2 = sqrt((x(s)-x_p)^2+(y(s)-y_p)^2+(z(s)-z_p)^2)-sqrt((x(s)-x_2)^2+(y(s)-y_2)^2+(z(s)-z_2)^2)-(c*t_1)

eqn3 = sqrt((x(s)-x_p)^2+(y(s)-y_p)^2+(z(s)-z_p)^2)-sqrt((x(s)-x_3)^2+(y(s)-y_3)^2+(z(s)-z_3)^2)-(c*t_1)

where only x(s), y(s), z(s) are unknown are rest all are known

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##### 1 Comment

Walter Roberson
on 29 Sep 2020

### Accepted Answer

Walter Roberson
on 29 Sep 2020

x_p = 4; y_p = 5; z_p = 2;

x_1 = 8; y_1 = 9; z_1 = 5;

x_2 = 2; y_2 = 5; z_2 = 1;

x_3 = 6; y_3 = 1; z_3 = 3;

t1 = 5.692820*10^-9;

t2 = -2.924173*10^-9;

t3 = -12.010097*10^-9;

syms xs ys zs %our unknowns

syms c %constant not given in question

eqn1 = sqrt((xs-x_p)^2+(ys-y_p)^2+(zs-z_p)^2)-sqrt((xs-x_1)^2+(ys-y_1)^2+(zs-z_2)^2)-(c*t1);

eqn2 = sqrt((xs-x_p)^2+(ys-y_p)^2+(zs-z_p)^2)-sqrt((xs-x_2)^2+(ys-y_2)^2+(zs-z_2)^2)-(c*t2);

eqn3 = sqrt((xs-x_p)^2+(ys-y_p)^2+(zs-z_p)^2)-sqrt((xs-x_3)^2+(ys-y_3)^2+(zs-z_3)^2)-(c*t3);

sol = solve([eqn1, eqn2, eqn3], [xs, ys, zs]);

disp(sol.xs)

disp(sol.ys)

disp(sol.zs)

the values will be parameterized in c, which you indicate is a known value, but which you did not provide a value for.

There are two solutions for each variable.

You should probably re-substitute the solutions and verify that the values work, as MATLAB warns that some of the solutions produced might no be true solutions.

##### 12 Comments

Walter Roberson
on 30 Sep 2020

That is not 6 possible solutions, that is two solutions with three components each.

possibleSol(:, all(possibleSol>0 & imag(possibleSol)==0, 1))

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