Recursive All Pass Realizations Subject to Tangential Constraints

Given d complex points i and associated directions (of C n ) z i , we develop a recursive algorithm for obtaining a " I 2 \Sigma # -unitary realization fA; B; C; Dg of a \Sigma-unitary transfer matrix U() (and of its inverse) which satisfies U( i )z i = 0; i = 1; :::; d: This algorithm is based on \Sigma-unitary transformations and has O(nd 2 ) complexity. Furthermore we introduce a modification to this algorithm that allows to work in real arithmetic in the case of self conjugate conditions

High Performance Algorithms for Toeplitz and block Toeplitz matrices

this paper we discuss several high performance variants of the classical Schur algorithm algorithms to factor symmetric block Toeplitz matrices. Specifically we discuss routines to factor symmetric positive definite, positive semidefinite and indefinite matrices. Algorithms to obtain the QR factorization of exactly and nearly rank deficient Toeplitz matrices are also discussed. In this paper the classical Schur algorithm for obtaining the Cholesky factorization of symmetric positive definite block Toeplitz matrices [9, 8] is generalized to the block Toeplitz matrix case using a block generalization of the hyperbolic Householder reflectors. The block generalization of the Schur algorithm and various blocking schemes differing in the amount of storage and computational primitives used are described in Section 2. Blocking the hyperbolic Householder transformations allows us to apply these transformations using BLAS 3 primitives rather than the BLAS 2 primitives which are required for plain hyperbolic Householder transformations. On machines with a memory hierarchy this provides us with a faster algorithm. For symmetric indefinite block Toeplitz matrices the Schur algorithm breaks down if the matrix has singular principal minors. A scheme to modify the block Schur algorithm by per

High Performance Algorithms for Toeplitz and block Toeplitz matrices

this paper we discuss several high performance variants of the classical Schur algorithm algorithms to factor symmetric block Toeplitz matrices. Specifically we discuss routines to factor symmetric positive definite, positive semidefinite and indefinite matrices. Algorithms to obtain the QR factorization of exactly and nearly rank deficient Toeplitz matrices are also discussed. In this paper the classical Schur algorithm for obtaining the Cholesky factorization of symmetric positive definite block Toeplitz matrices [9, 8] is generalized to the block Toeplitz matrix case using a block generalization of the hyperbolic Householder reflectors. The block generalization of the Schur algorithm and various blocking schemes differing in the amount of storage and computational primitives used are described in Section 2. Blocking the hyperbolic Householder transformations allows us to apply these transformations using BLAS 3 primitives rather than the BLAS 2 primitives which are required for plain hyperbolic Householder transformations. On machines with a memory hierarchy this provides us with a faster algorithm. For symmetric indefinite block Toeplitz matrices the Schur algorithm breaks down if the matrix has singular principal minors. A scheme to modify the block Schur algorithm by perturbing the generators and obtaining an approximate factorization of the matrix is described in Section 3. The approximate solution is then improved through iterative refinement. The numerical behavior of this method to circumvent the singularities is studied. If an exact factorization of the indefinite block Toeplitz matrix is desired, then one would have to look-ahead over the singular or near singular principal minors. Look-ahead algorithms based on the Levinson algorithm have appeare

High performance algorithms for Toeplitz and block Toeplitz matrices

In this paper, we present several high performance variants of the classical Schur algorithm to factor various Toeplitz matrices. For positive definite block Toeplitz matrices, we show how hyperbolic Householder transformations may be blocked to yield a block Schur algorithm. This algorithm uses BLAS3 primitives and makes efficient use of a memory hierarchy. We present three algorithms for indefinite Toeplitz matrices. Two of these are based on look-ahead strategies and produce an exact factorization of the Toeplitz matrix. The third produces an inexact faetorization via perturbations of singular principal minors. We also present an analysis of the numerical behavior of the third algorithm and derive a bound for the number of iterations to improve the accuracy of the solution. For rank-deficient Toeplitz least-squares problems, we present a variant of the gene-ralized Schur algorithm that avoids breakdown due to an exact rank-deficiency. In the presence of a near rank-deficiency, an approximate rank factorization of the Toeplitz matrix is produced. Finally, we suggest an algorithm to solve the normal equations resulting from a real Toeplitz least-squares problem based on transforming to Cauehy-like matrices. This algorithm exploits both realness and symmetry in the normal equations