Carmine Di Fiore

Matrix algebras in Quasi-Newton methods for unconstrained minimization

Summary. In this paper a new class of quasi-Newton methods, named ℒQN, is introduced in order to solve unconstrained minimization problems. The novel approach, which generalizes classical BFGS methods, is based on a Hessian updating formula involving an algebra ℒ of matrices simultaneously diagonalized by a fast unitary transform. The complexity per step of ℒQN methods is O(n log n), thereby improving considerably BFGS computational efficiency. Moreover, since ℒQNs iterative scheme utilizes single-indexed arrays, only O(n) memory allocations are required. Global convergence properties are investigated. In particular a global convergence result is obtained under suitable assumptions on f. Numerical experiences [7] confirm that ℒQN methods are particularly recommended for large scale problems.

MATRIX DECOMPOSITIONS USING DISPLACEMENT RANK AND CLASSES OF COMMUTATIVE MATRIX ALGEBRAS

Using the notion of displacement rank, we look for a unifying approach to representations of a matrix A as sums of products of matrices belonging to commutative matrix algebras. These representations are then considered in case A is the inverse of a Toeplitz or a Toeplitz plus Hankel matrix. Some well-known decomposition formulas for A (Gohberg-Semencul or Kailath et al., Gader, Bini-Pan, and Gohberg-Olshevsky) turn out to be special cases of the above representations. New formulas for A in terms of algebras of symmetric matrices are studied, and their computational aspects are discussed.

Matrix algebras in optimal preconditioning

The theory and the practice of optimal preconditioning in solving a linear system by iterative processes is founded on some theoretical facts understandable in terms of a class V of spaces of matrices including diagonal algebras and group matrix algebras. The V-structure lets us extend some known crucial results of preconditioning theory and obtain some useful information on the computability and on the efficiency of new preconditioners. Three preconditioners not yet considered in literature, belonging to three corresponding algebras of V, are analyzed in detail. Some experimental results are included.

On a set of matrix algebras related to discrete Hartley-type transforms

A set of fast real transforms including the well known Hartley transform is fully investigated. Mixed radix splitting properties of Hartley-type transforms are examined in detail. The matrix algebras diagonalized by the Hartley-type matrices are expressed in terms of circulant and (−1)-circulant matrices.

Low-complexity minimization algorithms

SUMMARY Structured matrix algebras L and a generalized BFGS-type iterative scheme have been recently investigated to introduce low-complexity quasi-Newton methods, named LQN, for solving general (non-structured) minimization problems. In this paper we introduce the L k QN methods, which exploit

Matrix displacement decompositions and applications to Toeplitz linear systems

Using the approach of Bozzo, Di Fiore, and Zellini, new matrix displacement decomposition formulas are introduced. It is shown how an arbitrary square matrix A can be expressed as sums of products of Hessenberg algebra matrices and high level (block) matrices whose submatrices are Hessenberg algebra matrices and have variable sizes. In most cases these block factors are block-diagonal matrices. Then these formulas are used in sequential and parallel solution of Toeplitz systems.

Hartley-type algebras in displacement and optimization strategies

Abstract The Hartley-type ( Ht ) algebras are used to face efficiently the solution of structured linear systems and to define low complexity methods for solving general (nonstructured) nonlinear problems. Displacement formulas for the inverse of a symmetric Toeplitz matrix in terms of Ht transforms are compared with the well known Ammar–Gader formula. The L QN unconstrained optimization methods, which define Hessian approximations by updating n × n matrices from an algebra L , can be implemented for L =Ht with an O( n ) amount of memory allocations and O( n log n ) arithmetic operations per step. The L QN methods with the lowest experimental rate of convergence are shown to be linearly convergent.

Euler-Richardson method preconditioned by weakly stochastic matrix algebras: a potential contribution to Pagerank computation

Let S be a column stochastic matrix with at least one full row. Then S describes a Pagerank-like random walk since the computation of the Perron vector x of S can be tackled by solving a suitable M-matrix linear system Mx = y, where M = I − τ A, A is a column stochastic matrix and τ is a positive coefficient smaller than one. The Pagerank centrality index on graphs is a relevant example where these two formulations appear. Previous investigations have shown that the Euler- Richardson (ER) method can be considered in order to approach the Pagerank computation problem by means of preconditioning strategies. In this work, it is observed indeed that the classical power method can be embedded into the ER scheme, through a suitable simple preconditioner. Therefore, a new preconditioner is proposed based on fast Householder transformations and the concept of low complexity weakly stochastic algebras, which gives rise to an effective alternative to the power method for large-scale sparse problems. Detailed mathematical reasonings for this choice are given and the convergence properties discussed. Numerical tests performed on real-world datasets are presented, showing the advantages given by the use of the proposed Householder-Richardson method.