Istituto di Scienza e Tecnologie dell'Informazione     
Arioli M., Romani F. Relations between condition numbers and the convergence of the Jacobi Method for real positive definite matrices. In: Numerische Mathematik, vol. 46 pp. 31 - 42. Springer-Verlag, 1985.
Let A be an n x n real symmetric diagonal dominant matrix with positive diagonal part D, and let S^2 = D, and H = SAS. The following relation between the condition number k(H) = ∥H^(-1)∥ ∥H∥ and the spectral radius r of the Jacobi matrix associated to A is proved: (k(H) -1)/(k(H)+ 1) ≦ r ≦ (k(H) -1)/(1 + k(H)/(n -1)). Moreover, relations among k(H), k(A), the condition numbers C(H)=∥∣ H^(-1)∣∣H∣∥, and C(A) are investigated.

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