Geodesic
Efficient computation of enclosures for the exact solvents of a quadratic matrix equation
The electronic journal of linear algebra, Tome 20 (2010), pp. 519-536.

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Summary: None of the usual floating point numerical techniques available for solving the quadratic matrix equation AX 2 + BX + C = 0 with square matrices A, B, C and X, can provide an exact solution; they can just obtain approximations to an exact solution. We use interval arithmetic to compute an interval matrix which contains an exact solution to this quadratic matrix equation, where we aim at obtaining narrow intervals for each entry. We propose a residual version of a modified Krawczyk operator which has a cubic computational complexity, provided that A is nonsingular and Xand X + A - 1 Bare diagonalizable. For the case that A is singular or nearly singular, but B is nonsingular we provide an enclosure method analogous to a functional iteration method. Numerical examples have also been given.
Classification : 65G20, 65F30
Keywords: quadratic matrix equation, matrix square root, interval analysis, krawczyk operator, automatic result verification 
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     author = {Hashemi, Behnam and Dehghan, Mehdi},
     title = {Efficient computation of enclosures for the exact solvents of a quadratic matrix equation},
     journal = {The electronic journal of linear algebra},
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     volume = {20},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ELA_2010__20__a16/}
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Hashemi, Behnam; Dehghan, Mehdi. Efficient computation of enclosures for the exact solvents of a quadratic matrix equation. The electronic journal of linear algebra, Tome 20 (2010), pp. 519-536. http://geodesic.mathdoc.fr/item/ELA_2010__20__a16/