3 Variables in 3 Equations (nonlinear)

Hi all.

I have to solve the following equations:

• A1*X + A2*Y + A3*Z = A4 (a plane)
• B1*X + B2*Y + B3*Z = B4 (a plane)
• (X-T1)^2 + (Y-T2)^2 + (Z-T3)^2 = D^2 (a sphere)

,where A,B,T,D are known constants, and X Y Z are the solution that I am looking for.

There are exactly 2 possible solutions (intersection points between two planes and a sphere), and I know how to choose the right one (based on the signs of X Y Z).

The main problem is that I have to solve these equations in a for loop (millions of times), where each iteration A and B change, therefore I need to do it as efficiently as possible.

I tried using "fsolve", but it takes much longer than I can allow.

One possible way is to calculate the intersection line between the two planes, and look for a point on that line that satisfies the third equation.

Is there a simpler way you can think of to calculate X Y Z in a loop? Can I somehow get rid of the loops and calculate the entire set of points at once?

Thank you.

Answer 1

[jeewan atwal]

USE Multidimensional Newton-Raphson method to solve non-linear eqs.

[Ilya Tcenov]

Thanks!

Worked great.