Abstract
A new algorithm for solving the Lyapunov matrix equation X − A∗XA = Q is proposed. The method proceeds by reducing to a special case for which an explicit formula is given. The technique is purely algebraic (i.e. involves no iteration), but does not involve the calculation of the characteristic polynomial of A or reduction to a canonical form. If Q is symmetric and the matrices are of type n × n, the number of multiplications and divisions required is about 4n4. Two simple devices are given whereby the technique can be extended to a wider class of linear matrix equations.
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