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A theorem of the alternatives for the equation Ax+B|x|=b. (English) Zbl 1070.15002
The main purpose of this work is to present a theorem of the alternatives for a nonlinear equation of the type Ax+B|x|=b, where A,B n×n . Equations of this form arise quite naturally in solving systems of interval linear equations. The theorem says that given two matrices A,D n×n , D0, then exactly one of the two alternatives holds: a) for each B n×n , with |B|D, where |B|=(|b ij |), and for each b n the above equation has a unique solution, and b) there exist λ[0,1] and a ±1-vector y, such that the equation Ax+λdiag(y)D|x|=0 has a nontrivial solution. The alternative of (b) can also be reformulated in two equivalent ways. Finally, it is shown that for given rational data A and D, the problem of determining which one of the alternatives (a), (b) holds is NP-hard.
15A06Linear equations (linear algebra)
15A18Eigenvalues, singular values, and eigenvectors
65H10Systems of nonlinear equations (numerical methods)
65H17Eigenvalue and bifurcation problems of nonlinear algebraic equations (numerical methods)
65G30Interval and finite arithmetic