Shift techniques for Quasi-Birth and Death processes: canonical factorizations and matrix equations

We revisit the shift technique applied to Quasi-Birth and Death (QBD) processes (He, Meini, Rhee, SIAM J. Matrix Anal. Appl., 2001) by bringing the attention to the existence and properties of canonical factorizations. To this regard, we prove new results concerning the solutions of the quadratic matrix equations associated with the QBD. These results find applications to the solution of the Poisson equation for QBDs

On Functions of quasi Toeplitz matrices

Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be a complex valued continuous function, defined for $|z|=1$, such that $\sum_{i=-\infty}^{+\infty}|ia_i|<\infty$. Consider the semi-infinite Toeplitz matrix $T(a)=(t_{i,j})_{i,j\in\mathbb Z^+}$ associated with the symbol $a(z)$ such that $t_{i,j}=a_{j-i}$. A quasi-Toeplitz matrix associated with the continuous symbol $a(z)$ is a matrix of the form $A=T(a)+E$ where $E=(e_{i,j})$, $\sum_{i,j\in\mathbb Z^+}|e_{i,j}|<\infty$, and is called a CQT-matrix. Given a function $f(x)$ and a CQT matrix $M$, we provide conditions under which $f(M)$ is well defined and is a CQT matrix. Moreover, we introduce a parametrization of CQT matrices and algorithms for the computation of $f(M)$. We treat the case where $f(x)$ is assigned in terms of power series and the case where $f(x)$ is defined in terms of a Cauchy integral. This analysis is applied also to finite matrices which can be written as the sum of a Toeplitz matrix and of a low rank correction

Computing the Exponential of Large Block-Triangular Block-Toeplitz Matrices Encountered in Fluid Queues

The Erlangian approximation of Markovian fluid queues leads to the problem of computing the matrix exponential of a subgenerator having a block-triangular, block-Toeplitz structure. To this end, we propose some algorithms which exploit the Toeplitz structure and the properties of generators. Such algorithms allow to compute the exponential of very large matrices, which would otherwise be untreatable with standard methods. We also prove interesting decay properties of the exponential of a generator having a block-triangular, block-Toeplitz structure

General solution of the Poisson equation for Quasi-Birth-and-Death processes

We consider the Poisson equation $(I-P)\boldsymbol{u}=\boldsymbol{g}$, where $P$ is the transition matrix of a Quasi-Birth-and-Death (QBD) process with infinitely many levels, $\bm g$ is a given infinite dimensional vector and $\bm u$ is the unknown. Our main result is to provide the general solution of this equation. To this purpose we use the block tridiagonal and block Toeplitz structure of the matrix $P$ to obtain a set of matrix difference equations, which are solved by constructing suitable resolvent triples

From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms

The problem of reducing an algebraic Riccati equation $XCX-AX-XD+B=0$ to a unilateral quadratic matrix equation (UQME) of the kind $PX^2+QX+R$ is analyzed. New reductions are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm of B.D.O. Anderson [Internat. J. Control, 1978] is nothing else but the cyclic reduction algorithm applied to a suitable UQME. A new algorithm obtained by complementing our reductions with the shrink-and-shift tech- nique of Ramaswami is presented. Finally, faster algorithms which require some non-singularity conditions, are designed. The non-singularity re- striction is relaxed by introducing a suitable similarity transformation of the Hamiltonian

On the tail decay of M/G/1-type Markov renewal processes

The tail decay of M/G/1-type Markov renewal processes is studied. The Markov renewal process is transformed into a Markov chain so that the problem of tail decay is reformulated in terms of the decay of the coefficients of a suitable power series. The latter problem is reduced to analyze the analyticity domain of the power series

Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms

We survey on theoretical properties and algorithms concerning the problem of solving a nonsymmetric algebraic Riccati equation, and we report on some known methods and new algorithmic advances. In particular, some results on the number of positive solutions are proved and a careful convergence analysis of Newton\u27s iteration is carried out in the cases of interest where some singularity conditions are encountered. From this analysis we determine initial approximations which still guarantee the quadratic convergence

SH-wave reflection seismic survey at the Patigno landslide: integration with a previously acquired P-wave seismic profile

Seismic investigation on landslide is hampered by several factors that could prevent the use of the reflection seismic method to characterize the subsurface architecture (Jongmans and Garambois, 2007). Moreover, acquisition and processing of reflection seismic data are more time consuming compared with other geophysical techniques such as refraction seismic and electrical resistivity tomography (ERT), leading inevitably to higher costs. Notwithstanding these difficulties, recently some attempts to delineate the deep slip surface of large landslides have been carried out using P-wave reflection seismic surveys (Apuani et al., 2012; Stucchi and Mazzotti, 2009; Stucchi et al., 2014;). P-wave reflection seismic method is effective in imaging the slip surface at a depth sufficiently greater than the seismic wavelength, whereas, for very shallow horizons, it suffers from the limited resolution that can be obtained by the use of compressional waves. In this regards, SH-waves can be used to overcome this limitation (Deidda and Balia, 2001; Guy, 2006; Pugin et al., 2006,), but they require a specifically-designed energy source for waves generation, geophones measuring horizontal components of particles motion and an accurate choice of acquisition parameters. On the contrary, due to attenuation, the depth of investigation for SHwaves can be lower than for P-waves (Pugin et al., 2006). Therefore the geological understanding of a mass movement can take advantage of a combined use of both these geophysical methodologies. This is the case of the Patigno landslide, a great landslide located in the upper basin of Magra River, in the Northern Appennines, Italy (Fig.1), where a P-wave study carried out in the last years (Stucchi et al., 2014) was able to image the deepest discontinuity of the landslide body at around 40-50 m depth, but no description of the shallower layers can be inferred. Because these surface layers are the slip surfaces of quick reactivation movements of the landslide, an SH high-resolution reflection seismic survey was planned along the previous P-wave profile (Fig.1). This new survey associated to the P-wave investigation allows a more robust description of the landslide body, from the deepest discontinuity up to the very shallow portions of the landslide. This work describes the planning, acquisition and processing of the SH reflection seismic survey, and also gives a possible combined interpretation of both P and SH seismic images