Grazia Lotti

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

Additive one-dimensional cellular automata are chaotic according to Devaney's definition of chaos

Abstract We study the chaotic behavior of a particular class of dynamical systems: cellular automata. We specialize the definition of chaos given by Devaney for general dynamical systems to the case of cellular automata. A dynamical system ( X , F ) is chaotic according to Devaneys definition of chaos if its transition map F is sensitive to the initial conditions, topologically transitive, and has dense periodic orbits on X . Our main result is the proof that all the additive one-dimensional cellular automata defined on a finite alphabet of prime cardinality are chaotic in the sense of Devaney.

Approximate Solutions for the Bilinear Form Computational Problem

A set of bilinear forms can be evaluated with a multiplicative complexity lower than the rank of the associated tensor by allowing an arbitrarily small error. A topological interpretation of this fact is presented together with the error analysis. A complexity measure is introduced which takes into account the numerical stability of algorithms. Relations are established between the complexities of exact and approximate algorithms.

Algorithm 691: Improving QUADPACK automatic integration routines

Two automatic adaptive integrators from QUADPACK (namely, QAG, and QAGS) are modified by substituting the Gauss-Kronrod rules used for local quadrature with recursive monotone stable (RMS) formulas. Extensive numerical tests, both for one-dimensional and two-dimensional integrals, show that the resulting programs are faster, perform less functional evaluations, and are more suitable

Interpolatory integration formulas for optimal composition

A set of symmetric, closed, interpolatory integration formulas on the interval [-1, 1] with positive weights and increasing degree of precision is introduced. These formulas, called recursive monotone stable (RMS) formulas, allow applying higher order or compound rules without wasting previously computed functional values. An exhaustive search shows the existence of 27 families of RMS formulas, stemming from the simple trapezoidal rule.

Performance analysis of maximum likelihood methods for regularization problems with nonnegativity constraints

In many numerical applications, for instance in image deconvolution, the nonnegativity of the computed solution is required. When a problem of deconvolution is formulated in a statistical frame, the recorded image is seen as the realization of a random process, where the nature of the noise is taken into account. This formulation leads to the maximization of a likelihood function which depends on the statistical property assumed for the noise. In this paper we revisit, under this unifying statistical approach, some iterative methods coupled with suitable strategies for enforcing nonnegativity and other ones which instead naturally embed nonnegativity. For all these methods we carry out a comparative study taking into account several performance indicators. The reconstruction accuracy, the computational cost, the consistency with the discrepancy principle (a common technique for guessing the best regularization parameter) and the sensitivity to this choice are compared in a simulation context, by means of an extensive experimentation on both 1D and 2D problems.

Generalized Cross-Validation applied to Conjugate Gradient for discrete ill-posed problems

In this paper we propose a new method to apply the Generalized Cross-Validation (GCV) as a stopping rule for the Conjugate Gradient (CG). In general, to apply GCV to an iterative method, one must estimate the trace of the so-called influence matrix which appears in the denominator of the GCV function. In the case of CG, unlike what happens with stationary iterative methods, the regularized solution has a nonlinear dependence on the noise which affects the data of the problem. This fact is often pointed out as a cause of poor performance of GCV. To overcome this drawback, our proposal linearizes the dependence by computing the derivatives through iterative formulas. We compare the proposed method with other methods suggested in the literature by an extensive numerical experimentation on both 1D and 2D test problems.

A note on the solution of not balanced banded Toeplitz systems

A direct algorithm is presented for the solution of linear systems having banded Toeplitz coefficient matrix with unbalanced bandwidths. It is derived from the cyclic reduction algorithm, it makes use of techniques based on the displacement rank and it relies on the Morrison–Sherman–Woodbury formula. The algorithm always equals and sometimes outperforms the already known direct ones in terms of asymptotic computational cost. The case where the coefficient matrix is a block banded block Toeplitz matrix in block Hessenberg form is analyzed as well. The algorithm is numerically stable if applied to M-matrices that are point diagonally dominant by columns. Copyright

Solution of infinite linear systems by automatic adaptive iterations

Abstract The problem of approximating the solution of infinite linear systems finitely expressed by a sparse coefficient matrix in block Hessenberg form is considered. The convergence of the solutions of a sequence of truncated problems to the infinite problem solution is investigated. A family of algorithms, some of which are adaptive, is introduced, based on the application of the Gauss–Seidel method to a sequence of truncated problems of increasing size n i with non-increasing tolerance 10 −t i . These algorithms do not require special structural properties of the coefficient matrix and they differ in the way the sequences {n i } and {t i } are generated. The testing has been performed on both infinite problems arising from the discretization of elliptical equations on unbounded domains and stochastic problems arising from queueing theory. Extensive numerical experiments permit the evaluation of the various strategies and suggest that the best trade-off between accuracy and computational cost is reached by some of the adaptive algorithms.

A Divide and Conquer Algorithm for the Superfast Solution of Toeplitz-like Systems

In this paper a new