Schur Flows for Orthogonal Hessenberg Matrices
نویسنده
چکیده
We consider a standard matrix ow on the set of unitary upper Hessenberg matrices with nonnegative subdiagonal elements. The Schur parametrization of this set of matrices leads to ordinary diier-ential equations for the weights and the parameters that are analogous with the Toda ow as identiied with a ow on Jacobi matrices. We derive explicit diierential equations for the ow on the Schur parameters of orthogonal Hessenberg matrices. We also outline an eecient procedure for computing the solution of Jacobi ows and Schur ows.
منابع مشابه
Algorithms for the Geronimus transformation for orthogonal polynomials on the unit circle
Let L̂ be a positive definite bilinear functional on the unit circle defined on Pn, the space of polynomials of degree at most n. Then its Geronimus transformation L is defined by L̂(p, q) = L ( (z − α)p(z), (z − α)q(z) ) for all p, q ∈ Pn, α ∈ C. Given L̂, there are infinitely many such L which can be described by a complex free parameter. The Hessenberg matrix that appears in the recurrence rela...
متن کاملContemporary Mathematics The Schur algorithm for matrices with Hessenberg displacement structure
A Schur-type algorithm is presented for computing recursively the triangular factorization R = LU of a strongly nonsingular n n matrix satisfying a displacement equation RY V R = GH with Hessenberg matrices Y and V and n matrices G, H. If is small compared with n and the matrices Y and V admit fast matrix-vector multiplication, the new algorithm is fast in the sense that it will require less th...
متن کاملAn implicit Q-theorem for Hessenberg-like matrices
The implicit Q-theorem for Hessenberg matrices is a widespread and powerful theorem. It is used in the development of for example implicit QR-algorithms to compute the eigendecomposition of Hessenberg matrices. Moreover it can also be used to prove the essential uniqueness of orthogonal similarity transformations of matrices to Hessenberg form. The theorem is also valid for symmetric tridiagona...
متن کاملOrthonormal Representations for Output System Pairs
A new class of canonical forms is given proposed in which (A,C) is in Hessenberg observer or Schur form and output normal: I − A∗A = C∗C. Here, C is the d × n measurement matrix and A is the advance matrix. The (C,A) stack is expressed as the product of n orthogonal matrices, each of which depends on d parameters. State updates require onlyO(nd) operations and derivatives of the system with res...
متن کاملConstructing a Unitary Hessenberg Matrix from Spectral Data
We consider the numerical construction of a unitary Hessenberg matrix from spectral data using an inverse QR algorithm. Any unitary upper Hessenberg matrix H with nonnegative subdiagonal elements can be represented by 2n ? 1 real parameters. This representation, which we refer to as the Schur parameterization of H; facilitates the development of eecient algorithms for this class of matrices. We...
متن کامل