[Dept. of Hydroacoust. and Seismol., Natl. Defence Res. Establishment FOA 260), S-172 90 Sundbyberg, Sweden]
The wave field is decomposed into its frequency--wave-number components. Compound matrices for solid layers provide a convenient way of computing the boundary values at a fluid--solid interface [M. B. Porter and E. L. Reiss, J. Acoust. Soc. Am. 77, 1760--1767 1985)], with loss-of-precision control. A certain vector is propagated through a sequence of multiplications with compound matrices, one for each layer. It is shown that computations of this kind can be performed more efficiently if each compound matrix is decomposed as a product of sparse matrices that are applied in sequence. Two kinds of compound-matrix factorizations are proposed. In connection with dispersion computations, our first factorization gives a method that is related to the ``fast form'' of Knopoff's method [F. Schwab et al., Bull. Seismol. Soc. Am. 74, 1555--1578 1984)]. This algorithm is slightly more efficient, however, and its range of applicability is wider. The second compound-matrix factorization gives a method that is significantly faster than the ``fast form'' of Knopoff's method. Very few arithmetic operations are needed. It also provides a good basis for analyzing the numerical performance of compound-matrix propagation. Finally, it is shown how propagator-matrix factorization can be used to enhance the efficiency for multi-frequency computations and computation of full wave-fields, by wave-number integration or modal synthesis.