Error bounds for low-rank approximations of the first exponential integral kernel

A hierarchical matrix is an efficient data-sparse representation of a matrix, especially useful for large dimensional problems. It consists of low-rank subblocks leading to low memory requirements as well as inexpensive computational costs. In this work, we discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and, hence, it is weakly singular. We develop analytical expressions for the approximate degenerate kernels and deduce error upper bounds for these approximations. Some computational results illustrating the efficiency and robustness of the approach are presented.

The estimation of the return on firms’ investments – as to ISO 9001

The aim of this paper was to estimate the return on investment in QMS (quality management systems) certification undertaken in Portuguese firms, according to the ISO 9000 series. A total of 426 certified Portuguese firms were surveyed. The response rate was 61.03 percent. The different payback periods were validated through statistical analysis and the relationship between expected and perceived payback periods was discussed. This study suggests that a firm’s sector of activity, size and degree of internationalization are related to the length of the investment in QMS certification recovery period. Furthermore, our findings suggest, that the time taken to obtain the certification is not directly related to the economic component of the certification. The majority of Portuguese firms (58.9%) took up to three years to recoup their investment and 35.5% of companies said they had not yet recovered the initial investment made. The recoup of investment was measured by the increase in the number of customers and consequent volume of deliveries, improved profitability and productivity of the company, improvement of competitive position and performance (cost savings), reduction in the number of external complaints and internal defects/scrap, achievement of some important clientele, among others. We compared our work to similar studies undertaken in other countries. This paper provides a contribution to the research related to the return on investment for costs related to the certification QMS according to ISO 9000. This paper provides a valuable contribution to the field and is one of the first studies to undertake this type of analysis in Portugal.