Properties of positive definite solutions of the equation \(X+A^*X^{-2}A=I\). (English) Zbl 0935.65041

Ivanov, Ivan G.; El-Sayed, Salah M.

Linear Algebra Appl. 279, No. 1-3, 303-316 (1998).

Summary: We discuss some properties of a positive definite solution of the matrix equation \(X+ A^*X^{-2}A= I\). Two effective iterative methods for computing a positive definite solution of this equation are proposed. Necessary and sufficient conditions for the existence of a positive definite solution are derived. Numerical experiments are executed with these methods.

MSC:

65F30	Other matrix algorithms (MSC2010)
65F10	Iterative numerical methods for linear systems
15A24	Matrix equations and identities

Keywords:

numerical experiments; matrix equation; positive definite solution; iterative method

References:

[1]	Anderson, W.N.; Morley, T.D.; Trapp, G.E., Positive solution to \(X = A − Bx\^{}\{−1\} B∗\), Linear algebra appl., 134, 53-62, (1990) · Zbl 0702.15009
[2]	Engwerda, J.C.; Ran Andre, C.M.; Rijkeboer, A.L., Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation \(X + A∗X\^{}\{−1\} A = Q\), Linear algebra appl., 186, 255-275, (1993) · Zbl 0778.15008
[3]	Engwerda, J.C., On the existence of a positive definite solution of the matrix equation X + ATX−1A = I, Linear algebra appl., 194, 91-108, (1993) · Zbl 0798.15013
[4]	Buzbee, B.L.; Golub, G.H.; Nielson, C.W., On direct methods for solving Poisson’s equations, SIAM J. numer. anal., 7, 627-656, (1970) · Zbl 0217.52902
[5]	Housholder, A.S., The theory of matrices in numerical analysis, (1964), Blaisdell New York
[6]	Parodi, M., La localisation des valeurs caracteristiques des matrices et ses applications, (1959), Gauthier-Villars Paris · Zbl 0087.01602
[7]	Petkov, M., On the matrix equations A0xm + A1xm−1 + … + am = 0, (A0λm + A1λm−1 + … + am)x = 0, Ann. sofia university kl. okhridski, 72, 159-164, (1978), (in Bulgarian)
[8]	Wilkinson, J.H., The algebraic eigenvalue problem, (1965), Oxford University Press London · Zbl 0258.65037
[9]	Petkov, M., An iterative method for computing A, (), Sofia
[10]	Salah El-Sayed, M., The study on special matrices and numerical methods for special matrix equations, Ph.D. thesis, (1996), Sofia · Zbl 1034.65502
[11]	Lancaster, P., Theory of matrices, (1969), Academic Press New York · Zbl 0186.05301
[12]	Salah El-Sayed, M., Theorems for the existence and computing of positive definite solutions for two nonlinear matrix equations, () · Zbl 1054.65041