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.

PODEM AS DÍVIDAS DO BES SER DO NOVO BANCO?

No ponto de vista jurídico existe um velho instituto jurídico que se chama “Levantamento ou desconsideração da personalidade colectiva” que pode permitir – por palavras breves – imputar as dívidas do BES ao Novo Banco. Também é possível nos socorrermos do instituto do Abuso do Direito (art. 334º do Código Civil): “É ilegítimo o exercício de um direito, quando o titular exceda manifestamente os limites impostos pela boa fé, pelos bons costumes ou pelo fim social ou económico desse direito”. Aplica-se o art. 11º/8 do Código Penal: «8 - A cisão e a fusão não determinam a extinção da responsabilidade criminal da pessoa colectiva ou entidade equiparada, respondendo pela prática do crime: § a) A pessoa colectiva ou entidade equiparada em que a fusão se tiver efectivado; § e § b) As pessoas colectivas ou entidades equiparadas que resultaram da cisão». § In the legal point of view there is an old legal principle called "Lifting or disregard of legal personality" which can allow - for brief words - charge the debts of the BES to the New Bank. It is also possible in socorrermos Law Abuse Institute (Article 334 of the Civil Code.): "It is illegitimate exercise of a right, where the proprietor clearly exceed the bounds of good faith, morality or the social or economic purpose this right ". Applies the art. 11/8 of the Penal Code: "8 - The split and the merger does not determine the extinction of criminal liability of the legal or related entity person, accounting for the crime: § a) The legal person or related entity in the merger if paid up; § and § b) A legal entity or similar entities resulting from the split. "