Transformation Techniques for Toeplitz and Toeplitz-plus-Hankel Matrices Part II. Algorithms
Adam W. Bojańczyk, Georg Heinig · eCommons (Cornell University) · 1996
. In the first part [13] of the paper transformations mappingToeplitz and Toeplitz-plus-Hankel matrices into generalized Cauchy matrices were studied. In this second part fast algorithms for LU-factorization and inversion of generalized Cauchy matrices are discussed. It is shown that the combination of transformation pivoting techniques leads to algorithms for indefinite Toeplitz and Toeplitz-plus-Hankel matrices that are more stable that the classical ones. Special attention is paid to the symmetric and hermitian cases. 1. INTRODUCTION To begin with let us recall the definition of a (generalized) Cauchy matrix. Let c = (c i ) n 1 and d = (d j ) n 1 be fixed n-tuples of complex numbers and A = [a ij ] n 1 a given matrix. Then the Cauchy rank of A with respect to c and d is, by definition, the rank r of the matrix r c;d (A) = [(c i \\Gamma d j )a ij ] n 1 : If r is small compared with the order of the matrix A, then A will be called (generalized) Cauchy matrix. Cauchy matric...