Rectangular co-solutions of polynomial matrix equations and applications

Model reduction in H2 using matrix solutions of polynomial equations

A method is given for solving an optimal H2 approximation problem for SISO linear time-invariant stable systems. The method guarantees that the global optimum is found. It is based on constructive algebra, but compared with earlier results, the method has much smaller time and memory requirements, and can therefore be applied to systems of significantly higher McMillan degree. The use of Buchbe...

Symmetric matrix polynomial equations

For problems of linear control system synthesis, an apparatus of polynomial equations (for single-variable case) and of matrix polynomial equations (for multivariable case) was successfully developed in recent times, cf. [1]. In connection with quadratic criteria, we are led to equations of special type, containing an operation of conjugation ah-* a* representing a(s) i-> a( — s) for continuous...

Polynomial solutions of differential equations

A new approach for investigating polynomial solutions of differential equations is proposed. It is based on elementary linear algebra. Any differential operator of the form L(y) = k=N ∑ k=0 ak(x)y, where ak is a polynomial of degree ≤ k, over an infinite ground field F has all eigenvalues in F in the space of polynomials of degree at most n, for all n. If these eigenvalues are distinct, then th...

Least Squares Solutions of Inconsistent Fuzzy Linear Matrix Equations

Fast Rectangular Matrix Multiplication and Applications

First we study asymptotically fast algorithms for rectangular matrix multiplication. We begin with new algorithms for multiplication of an n_n matrix by an n_n matrix in arithmetic time O(n), |=3.333953..., which is less by 0.041 than the previous record 3.375477... . Then we present fast multiplication algorithms for matrix pairs of arbitrary dimensions, estimate the asymptotic running time as...