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Method for Approximating solutions to systems of non-linear equations question.

  1. Nov 19, 2005 #1
    I was wondering if any one could double check my work to see if the following method is correct. I will also check it for in detail, but just want to get some fresh eyes on the problem to see if I've missed something.
    Suppose you have a system of three non-linear polynomial equations in three unknowns that have a solution.
    f(x,y,z) = a1x^n1 + b1y^m1 + c1z^p1 = K1
    g(x,y,z)= a2x^n2 + b2y^m2 +c2z^p2 = K2
    g(x,y,z)= a3x^n3 + b3y^m3 +c3z^p3 = K3
    A numerical method for honing in on the solution is as follows:
    1) Choose an initial starting point (xSubn,ySubn,zSubn), and find the tangent planes to the three surfaces above. The intersection of three planes yields a single point (x,y,z).
    2) Solve the system of linear equations that describe the intersection of the three tangent planes to acquire a the next point
    3) Find the new tangent plane to each of the surfaces at the point
    (xSub(n+1),ySub(n+1),zSub(n+1)), and solve those systems of equations for the next point in the sequence by repeating step 2.
    4) If you keep repeating your steps above, either the (x,y,z) point will hone in to a solution of the equations, or will diverge from the solutions to the equations.
    I am guessing that if the point (xSub(n+1),ySub(n+1),zSub(n+1)), is closer to a solution of the equations, than (xSubn,ySubn,zSubn), that the process above will converge to the solution.
    I am also guessing that if the point (xSub(n+1),ySub(n+1),zSub(n+1)) is farther from the solution, that is a "poorer guess," than the point (xSubn,ySubn,zSubn), that the process above will converge to the solution.
    All of the steps above assume that there is a solution to the given set of 3 equations in 3 unknowns in question.
    Is this at all correct, or is it totally out of the ball park?
    Edwin G. Schasteen
    Last edited: Nov 19, 2005
  2. jcsd
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