Milvio Capovani

Spectral and computational properties of band symmetric toeplitz matrices

Abstract We are investigating spectral properties of band symmetric Toeplitz matrices (BST matrices). By giving a suitable representation of a BST matrix, we achieve separation results and multiplicity conditions for the eigenvalues of a 7°r 5-diagonal BST matrix and also structural properties of the eigenvectors. We give eigenvalue bounds for a 2k 1-diagonal BST matrix and also necessary and sufficient conditions for positive definiteness which are easy to check. The same conditions apply in the case of block BST matrices, either with full blocks or with BST blocks. We exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a BST matrix, either for sequential or for parallel computations. Two algorithms, based on the bisection technique and Newtons method, are shown to be very fast for computing the eigenvalues of a 7− or 5-diagonal BST-matrix.

O ( n 2.7799 ) complexity for n × n approximate matrix multiplication

The problem of multiplying (m X n) X (n X k) matrices (the (m, n, k) problem) is an interesting case of the bilinear forms computation theory. We call EC-algorit’hms (Exactly-Computing) those solving exactly the problem in the real arithmetic. They are usually estimated by the number of multiplications required. The state of the art is the O(n2*7g51) ECalgorithm given by [3] for the (n, n, n) problem. We introduce here a new class of algorithms approximating the result with arbitrary precision. We call them’ APA-algorithms (Arbitrary-Precision-Approximating). A bilinear EC-algorithm to compute the set of bilinear forms

Tensor rank and border rank of band Toeplitz matrices

Let

Fast parallel and sequential computations and spectral properties concerning band Toeplitz matrices

\mathcal{B}_{n,h,k}

Metodi Numerici per l'Algebra Lineare.

be the class of

Lower bounds of the complexity of linear algebras

n \times n

Su alcune questioni di complessita' computazionale numerica

band Toeplitz matrices

Introduzione alla Matematica Computazionale

A = (a_{i,j} )

Complessità computazionale numerica

such that

Vibrations of rotors: The effect of lubricant damping

a_{i,j} = 0