Carlos Amaral Hölbig

An accurate and efficient selfverifying solver for systems with banded coefficient matrix

Publisher Summary This chapter discuss a self verifying solver for systems of linear equations Ax = b with banded matrices A and the future adaptation of the algorithms to cluster computers. An implementation of an algorithm to compute efficiently component-wise good enclosures for the solution of a sparse linear system on typical cluster computers are discussed. Implementation works with point, as well as interval data (data afflicted with tolerances). The algorithm is implemented using C-XSC library (a C++ class library for extended scientific computing). The chapter discusses software for validated numerics in high performance environments, using C-XSC in connection with the MPICH library. Actually, solver for linear system with banded matrices runs on two different clusters: ALICE 1 at the University of Wuppertal and LabTeC 2 at UFRGS. The preliminary tests with matrix-matrix multiplication show that the C-XSC library needs to be optimized in several ways to be efficient in a high performance environment (up to now the main goal of C-XSC was functionality and portability, not speed).

Selfverifying Solvers for Linear Systems of Equations in C-XSC

In this paper we discuss the implementation of selfverifying solvers for systems of linear equations Ax = b with dense and banded matrices A and the future adaptation these solvers to high performance environments. The algorithms were implemented using C-XSC (a C++ class library for extended scientific computing). We discuss, too, the integration between C-XSC and MPI libraries on cluster computers. The main topics of our research are the development of software tools for Validated Numerics in High Performance Environments using C-XSC and MPI, the optimization of C-XSC and its use on cluster computers and the application these software tools to real life problems [5].

Parallel Environment with High Accuracy for Resolution of Numerical Problems

In this paper we describe a high performance environment, like cluster computers, with high accuracy obtained by use of C-XSC library. The C-XSC library is a (free) C++ class library for scientific computing for the development of numerical algorithms delivering highly accurate and automatically verified results by use of the interval arithmetic. These calculus in high accuracy must be available for some basic arithmetic operations, mainly the operations that accomplish the summation and dot product. Because of these aspects, we wish to use the high performance through a cluster environment where we have several nodes executing tasks or calculus. The communication will be done by message passing using the MPI communication library. To obtain the high accuracy in this environment extensions or changes in the parallel programs had done to guarantee that the quality of final result done on cluster, where several nodes collaborate for the final result of the calculus, maintain the same result quality obtained in one sequential high accuracy environment. To validate the environment developed in this work we done basic tests about the dot product, the matrix multiplications, the implementation of interval solvers for banded and dense matrices and the implementation of some numeric methods to solve linear systems with the high accuracy characteristic (some of the methods implemented are used in real life applications like hydrodynamic, agriculture and power electric systems). With these tests we done analysis and comparisons about the performance and accuracy obtained with and without the use of C-XSC library in sequential and parallel programs. With the implementation of these routines and methods will be open a large research field about the study of real life applications that need during their resolution (or in part of their resolution) to calculate arithmetic operations with more accuracy than the accuracy obtained by the traditional computational tools. Our software run on labtec (UFRGS) and Colorado (UPF) clusters. Nowadays we are working in the implementation of parallel versions of programs to solve linear systems (without and with high accuracy) and the optimization of C-XSC library on cluster computers.

Statistical Correction of the Result of Weather Forecast by Applying the Model Output Calibration

This work describes the simulation models calibration method called Model Output Calibration. In order to verify its effectiveness, presents the application of the MOC in weather forecast correction generated by the Eta 15Km model at CPTEC/INPE. Eta is a regional model for numerical weather prediction. The results of the statistical correction of Eta forecast were positive, with satisfactory improvements in the variables tested (temperature and relative humidity). The use of this approach shows the possibility of gains in the results of simulation models of crops and diseases that use as predictive variables the variables generated by weather forecast models.