We discuss the solution of regular and singular Sturm-Liouville problems by means of High Order F... more We discuss the solution of regular and singular Sturm-Liouville problems by means of High Order Finite Difference Schemes. We describe a code to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are proposed to emphasize the behaviour of the proposed algorithm.
Bibliometric indexes are customary used in evaluating the impact of scientific research, even tho... more Bibliometric indexes are customary used in evaluating the impact of scientific research, even though it is very well known that in different research areas they may range in very different intervals. Sometimes, this is evident even within a single given field of investigation making very difficult (and inaccurate) the assessment of scientific papers. On the other hand, the problem can be recast in the same framework which has allowed to efficiently cope with the ordering of web-pages, i.e., to formulate the PageRank of Google. For this reason, we call such problem the PaperRank problem, here solved by using a similar approach to that employed by PageRank. The obtained solution, which is mathematically grounded, will be used to compare the usual heuristics of the number of citations with a new one here proposed. Some numerical tests show that the new heuristics is much more reliable than the currently used ones, based on the bare number of citations. Moreover, we show that our model ...
We consider linear systems with coefficient matrices having the ABD or the Bordered ABD (BABD) st... more We consider linear systems with coefficient matrices having the ABD or the Bordered ABD (BABD) structures. These systems arise in the discretization of BVPs for ordinary and partial differential equations with separated and non-separated boundary conditions, respectively. We describe the cyclic reduction algorithm forthe solution of BABD-linear systems which allowed us to write the codes BABDCR and GBABDCR (the latter code is suitable for matrices with a more generic BABD structure). A comparison of the GBABDCR code with respect to the well-known sequential code COLROW on ABD linear systems is then analysed. We report some tests on an OpenMP Fortran 90 parallel version of the GBABDCR code and finally we discuss about the use of GBABDCR inside the BVP code BVP SOLVER.
In this paper we review the parallel solution of sparse linear systems, usually deriving by the d... more In this paper we review the parallel solution of sparse linear systems, usually deriving by the discretization of ODE-IVPs or ODE-BVPs. The approach is based on the concept of parallel factorization of a (block) tridiagonal matrix. This allows to obtain efficient parallel extensions of many known matrix factorizations, and to derive, as a by-product, a unifying approach to the parallel solution of ODEs.
Boundary value methods for the solution of differential-algebraic equations are described. We con... more Boundary value methods for the solution of differential-algebraic equations are described. We consider both initial and boundary value problems and derive an algorithm that does not require additional information from the user, but only the initial or boundary conditions needed in theory to obtain a unique solution.
New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial ... more New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial condition are proposed. They are designed for work on a new kind of a supercomputer – the Infinity Computer, – that is able to deal numerically with finite, infinite and infinitesimal numbers. Due to this fact, the Infinity Computer allows one to calculate the exact derivatives of functions using infinitesimal values of the stepsize. As a consequence, the new methods described in this paper are able to work with the exact values of the derivatives, instead of their approximations.
We focus on the solution of multiparameter spectral problems, and in particular on some strategie... more We focus on the solution of multiparameter spectral problems, and in particular on some strategies to compute coarse approximations of selected eigenparameters depending on the number of oscillations of the associated eigenfunctions. Since the computation of the eigenparameters is crucial in codes for multiparameter problems based on finite differences, we herein present two strategies. The first one is an iterative algorithm computing solutions as limit of a set of decoupled problems (much easier to solve). The second one solves problems depending on a parameter \(\sigma \in [0,1]\), that give back the original problem only when \(\sigma =1\). We compare the strategies by using well known test problems with two and three parameters.
We devise a variable precision floating-point arithmetic by exploiting the framework provided by ... more We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities expressed using the positive and negative finite or infinite powers of the radix $${\textcircled {1}}$$ 1 . The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented.
A well-known drawback of algorithms based on Taylor series formulae is that the explicit calculat... more A well-known drawback of algorithms based on Taylor series formulae is that the explicit calculation of higher order derivatives formally is an over-elaborate task. To avoid the analytical computation of the successive derivatives, numeric and automatic differentiation are usually used. A recent alternative to these techniques is based on the calculation of higher derivatives by using the Infinity Computer—a new computational device allowing one to work numerically with infinities and infinitesimals. Two variants of a one-step multi-point method closely related to the classical Taylor formula of order three are considered. It is shown that the new formula is order three accurate, though requiring only the first two derivatives of y(t) (rather than three if compared with the corresponding Taylor formula of order three). To get numerical evidence of the theoretical results, a few test problems are solved by means of the new methods and the obtained results are compared with the performance of Taylor methods of order up to four.
We report on the progress achieved in the numerical simulation of self-adjoint multiparameter spe... more We report on the progress achieved in the numerical simulation of self-adjoint multiparameter spectral problems for ordinary differential equations. We describe how to obtain a discrete problem by means of High Order Finite Difference Schemes and discuss its numerical solution. Based on this approach, we also define a recursive algorithm to compute approximations of the parameters by means of the solution of a set of problems converging to the original one.
In this monograph, we briefly review the results, concerning the solution of evolutionary equatio... more In this monograph, we briefly review the results, concerning the solution of evolutionary equations by means of block Boundary Value Methods, obtained in the last two years. In particular, the sequential and parallel solution of initial value ODEs is considered. Moreover, the main advantages that it is possible to obtain in the solution of BVPs, DAEs, PDEs and Hamiltonian problems by means of BVMs are also emphasized.
We discuss the solution of regular and singular Sturm-Liouville problems by means of High Order F... more We discuss the solution of regular and singular Sturm-Liouville problems by means of High Order Finite Difference Schemes. We describe a code to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are proposed to emphasize the behaviour of the proposed algorithm.
Bibliometric indexes are customary used in evaluating the impact of scientific research, even tho... more Bibliometric indexes are customary used in evaluating the impact of scientific research, even though it is very well known that in different research areas they may range in very different intervals. Sometimes, this is evident even within a single given field of investigation making very difficult (and inaccurate) the assessment of scientific papers. On the other hand, the problem can be recast in the same framework which has allowed to efficiently cope with the ordering of web-pages, i.e., to formulate the PageRank of Google. For this reason, we call such problem the PaperRank problem, here solved by using a similar approach to that employed by PageRank. The obtained solution, which is mathematically grounded, will be used to compare the usual heuristics of the number of citations with a new one here proposed. Some numerical tests show that the new heuristics is much more reliable than the currently used ones, based on the bare number of citations. Moreover, we show that our model ...
We consider linear systems with coefficient matrices having the ABD or the Bordered ABD (BABD) st... more We consider linear systems with coefficient matrices having the ABD or the Bordered ABD (BABD) structures. These systems arise in the discretization of BVPs for ordinary and partial differential equations with separated and non-separated boundary conditions, respectively. We describe the cyclic reduction algorithm forthe solution of BABD-linear systems which allowed us to write the codes BABDCR and GBABDCR (the latter code is suitable for matrices with a more generic BABD structure). A comparison of the GBABDCR code with respect to the well-known sequential code COLROW on ABD linear systems is then analysed. We report some tests on an OpenMP Fortran 90 parallel version of the GBABDCR code and finally we discuss about the use of GBABDCR inside the BVP code BVP SOLVER.
In this paper we review the parallel solution of sparse linear systems, usually deriving by the d... more In this paper we review the parallel solution of sparse linear systems, usually deriving by the discretization of ODE-IVPs or ODE-BVPs. The approach is based on the concept of parallel factorization of a (block) tridiagonal matrix. This allows to obtain efficient parallel extensions of many known matrix factorizations, and to derive, as a by-product, a unifying approach to the parallel solution of ODEs.
Boundary value methods for the solution of differential-algebraic equations are described. We con... more Boundary value methods for the solution of differential-algebraic equations are described. We consider both initial and boundary value problems and derive an algorithm that does not require additional information from the user, but only the initial or boundary conditions needed in theory to obtain a unique solution.
New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial ... more New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial condition are proposed. They are designed for work on a new kind of a supercomputer – the Infinity Computer, – that is able to deal numerically with finite, infinite and infinitesimal numbers. Due to this fact, the Infinity Computer allows one to calculate the exact derivatives of functions using infinitesimal values of the stepsize. As a consequence, the new methods described in this paper are able to work with the exact values of the derivatives, instead of their approximations.
We focus on the solution of multiparameter spectral problems, and in particular on some strategie... more We focus on the solution of multiparameter spectral problems, and in particular on some strategies to compute coarse approximations of selected eigenparameters depending on the number of oscillations of the associated eigenfunctions. Since the computation of the eigenparameters is crucial in codes for multiparameter problems based on finite differences, we herein present two strategies. The first one is an iterative algorithm computing solutions as limit of a set of decoupled problems (much easier to solve). The second one solves problems depending on a parameter \(\sigma \in [0,1]\), that give back the original problem only when \(\sigma =1\). We compare the strategies by using well known test problems with two and three parameters.
We devise a variable precision floating-point arithmetic by exploiting the framework provided by ... more We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities expressed using the positive and negative finite or infinite powers of the radix $${\textcircled {1}}$$ 1 . The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented.
A well-known drawback of algorithms based on Taylor series formulae is that the explicit calculat... more A well-known drawback of algorithms based on Taylor series formulae is that the explicit calculation of higher order derivatives formally is an over-elaborate task. To avoid the analytical computation of the successive derivatives, numeric and automatic differentiation are usually used. A recent alternative to these techniques is based on the calculation of higher derivatives by using the Infinity Computer—a new computational device allowing one to work numerically with infinities and infinitesimals. Two variants of a one-step multi-point method closely related to the classical Taylor formula of order three are considered. It is shown that the new formula is order three accurate, though requiring only the first two derivatives of y(t) (rather than three if compared with the corresponding Taylor formula of order three). To get numerical evidence of the theoretical results, a few test problems are solved by means of the new methods and the obtained results are compared with the performance of Taylor methods of order up to four.
We report on the progress achieved in the numerical simulation of self-adjoint multiparameter spe... more We report on the progress achieved in the numerical simulation of self-adjoint multiparameter spectral problems for ordinary differential equations. We describe how to obtain a discrete problem by means of High Order Finite Difference Schemes and discuss its numerical solution. Based on this approach, we also define a recursive algorithm to compute approximations of the parameters by means of the solution of a set of problems converging to the original one.
In this monograph, we briefly review the results, concerning the solution of evolutionary equatio... more In this monograph, we briefly review the results, concerning the solution of evolutionary equations by means of block Boundary Value Methods, obtained in the last two years. In particular, the sequential and parallel solution of initial value ODEs is considered. Moreover, the main advantages that it is possible to obtain in the solution of BVPs, DAEs, PDEs and Hamiltonian problems by means of BVMs are also emphasized.
Uploads
Papers by Pierluigi Amodio