I have an equation system of the form Aix + Biy + Ciz = Di, where (x,y,z) is a unit vector, and (Ai, Bi, Ci, Di) are sets of measurements from a noisy system (with typically 3-5 independant readings).My first intuition to solve this problem was to pose this as an overdetermined linear equation system AX = B where X = (x,y,z), and to solve for X. However, with that approach, I have no way to enforce that the solution for vector X is a unit vector.Is there an elegant (or standard) solution to that problem, or should I simply dive into non-linear equation solving solutions?

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Ashly Kaufman · Accepted Answer

Unfortunately, the problem is almost certainly innately non-linear, since your solution space is inherently so. Pragmatically, the best approach is probably to treat your overdetermined linear system as a minimization-problem, least-squares style; this gives you a (positive definite) function to be minimized. Find the (exact) minimum value in             R        3  , project that down to       S    2   (i.e., normalize your vector X) to find the closest point on the sphere, and use that as a starting point for some descent-based method (i.e., find the component of the gradient of your goodness-of-fit function tangent to the sphere, add that to your current point on       S    2   and renormalize down to the sphere again, and keep doing that until you meet whatever convergence criterion you want to use. There&#039;s no guarantee that this will work if your data is particularly ill-conditioned, but in practice it&#039;s likely to be the most straightforward method.

Ayana Waller · Accepted Answer

This problem can be solved using singular value decomposition. The original problem can be stated as  m  i  n  ‖  A  x  −  b      ‖    2    ,  ‖  x      ‖    2    =  1By substituting      A    ′    =  [  A  ;  −  b  ]  ,      x    ′    =  w  x  ;  wand choosing   w  =            ‖      x              ‖        2                    ‖      x      ;      1              ‖        2             we get   ‖      x    ′        ‖    2    =  w  ‖  x  ;  1      ‖    2    =  ‖  x      ‖    2  , so the problem becomes  m  i  n  ‖      A    ′        x    ′        ‖    2    ,  ‖      x    ′        ‖    2    =  1which is same as in this question: Minimizing   ‖  A  x      ‖    2   subject to   ‖  x      ‖    2    =  1 It can be solved by decomposing       A    ′   into   U  Σ      V    T   and then taking the last column of   V, which corresponds to the smallest singular value of       A    ′  .Finally   x is calculated by dividing       x    ′   by the last element, taking elements of       x    ′   except the last one and then normalizing it.

I have an equation system of the form Aix + Biy + Ciz = Di, where (x,y,z) is a unit vector, and (Ai,

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