BIT Numerical Mathematics

, Volume 56, Issue 1, pp 33–50 | Cite as

A parallel algorithm for calculation of determinants and minors using arbitrary precision arithmetic

  • Gleb Beliakov
  • Yuri Matiyasevich


We present a parallel algorithm for calculating determinants of matrices in arbitrary precision arithmetic on computer clusters. This algorithm limits data movements between the nodes and computes not only the determinant but also all the minors corresponding to a particular row or column at a little extra cost, and also the determinants and minors of all the leading principal submatrices at no extra cost. We implemented the algorithm in arbitrary precision arithmetic, suitable for very ill conditioned matrices, and empirically estimated the loss of precision. In our scenario the cost of computation is bigger than that of data movement. The algorithm was applied to studies of Riemann’s zeta function.


Determinant Linear algebra Parallel algorithms  Message passing interface GPU Riemann’s zeta function 

Mathematics Subject Classification

65F40 68W10 



The authors wish to acknowledge support by the Victorian Partnership in Advanced Computing (VPAC) and Monash e-research centre for providing computing resources at their clusters, and specifically Mr. S. Michnowicz for his help in developing MPI parallelization code. Part of the calculations was performed on the “Chebyshev” supercomputer of Moscow State University Supercomputing Center. The work of the second author was partly supported by the programme of fundamental research “Modern problems of theoretical mathematics” of the Mathematics Branch of the Russian Academy of Sciences.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of Information TechnologyDeakin UniversityBurwoodAustralia
  2. 2.St. Petersburg DepartmentSteklov Institute of Mathematics of Russian Academy of SciencesSt. PetersburgRussia

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