By Yuan Y.

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**Sample text**

Likewise, always when the matrix A−1 occurs as an operator acting on some vector or some other matrix, then the expression can be numerically evaluated as we have demonstrated for y. 38 H. van der Vorst For sparse matrices, computational differences may be even much more dramatic. In relevant cases, A−1 may be dense, while L and U are sparse. For instance, if A is a positive deﬁnite tridiagonal matrix, then solving Ax = b via LU decomposition requires only in the order of n arithmetic computations.

Bur. Stand, 49:33–53, 1952. Linear Systems, Eigenvalues, and Projection 45 6. B. N. Parlett. The Symmetric Eigenvalue Problem. , 1980. 7. Y. Saad. Numerical methods for large eigenvalue problems. Manchester University Press, Manchester, UK, 1992. 8. Y. Saad and M. H. Schultz. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. , 7:856–869, 1986. 9. G. W. Stewart. Matrix Algorithms, Vol. II: Eigensystems. SIAM, Philadelphia, 2001. 10. L.

These equations are normally solved by ﬁrst calculating a Schur decomposition for the matrix A. Therefore, ﬁnding the solution of the Lyapunov is quite expensive, the number of operations is at least O(n3 ), where n is the size of the original model. Hence, it is only feasible for small systems. Furthermore, because we arrived at this point using the inverse of an ill-conditioned matrix we have to be careful. B can have very large entries, which will introduce tremendous errors in solving the Lyapunov equation.