I have a set of linear equations, of the form Ax=b . However, the value of b depends on the sign of the solution of x, such that bi=ci+di if xi&gt;0, and bi=ci−di if xi&lt;0. I could solve this by trying each combination of positiveness of x, but that would require solving 2N sets of linear equations, which becomes unfeasible quite quickly. Is this solvable analytically?

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I have a set of linear equations, of the form Ax=b . However, the value of b depends on the sign of the solution of x, such that  bi=ci+di if xi&amp;gt;0, and bi=ci−di if xi&amp;lt;0.  I could solve this by trying each combination of positiveness of x, but that would require solving 2N sets of linear equations, which becomes unfeasible quite quickly.  Is this solvable analytically?

Chettaf04 · Accepted Answer

You can write the system as  Ax=c+Ed  where E is a diagonal matrix whose diagonal elements are ei=±1.  This solves to x=A−1c+A−1Ed. You can solve this while leaving the ei as unknowns. Then you have the xi expressed as linear combinations of the ei, and can quickly run through the various combinations to see which ones have sign(xi)=ei for all i.

I have a set of linear equations, of the form Ax=b . However, the value of b depends on the sign of

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