Algorithms for multidimensional spectral factorization and sum of squares
Algorithms for multidimensional spectral factorization and sum of squares of polynomial matrices are developed. In the problem of multidimensional spectral factorization for a real \(n\)-variable polynomial matrix \(Z(\xi)\) with the property \(Z^T(-\xi) = Z(\xi)\) another real \(n\)-variable polynomial matrix \(F(\xi)\), the so-called spectral factor, has to be computed such that \(Z(\xi) = F^T(-\xi)F(\xi)\), where \(\xi = (\xi_1,\ldots ,\xi_n)\) denotes an \(n\)-dimensional indeterminate. The problem of sum of squares is the following: For a real \(n\)-variable polynomial matrix \(Z(\xi)\) with \(Z^T(\xi) = Z(\xi)\), \(Z(\xi) \geq 0\) for all \(\xi \in R^n\) a real \(n\)-variable polynomial matrix \(F(\xi)\) has to be determined such that \(Z(\xi) = F^T(\xi)F(\xi)\). It is shown how the problem of multidimensional spectral factorization can be reduced to the factorization of a real symmetric constant matrix. This is achieved by associating a \(2n\)-variable polynomial matrix to the matrix \(Z(\xi)\). Two algorithms for the multidimensional spectral factorization are presented. In the first one, one has to solve a linear matrix inequality and in the second one a linear eigenvalue problem. In the second case also spectral factors which are rational matrices are considered. In the last section the problem of sum of squares is connected with a \(2n\)-variable polynomial matrix and a linear matrix inequality.
polynomial multidimensional spectral factorization
scientific article; zbMATH DE number 5307054