Solving a kind of BVP for second-order ODEs using novel data formats for dense matrices

The aim of this paper is to show that a kind of boundary value problem for second-order ordinary differential equations which reduces to the problem of solving tridiagonal systems of linear equations can be efficiently solved on modern multicore computer architectures. A new method for solving such tridiagonal systems of linear equations, based on recently developed algorithms for solving linear recurrence systems with constant coefficients, can be easily vectorized and parallelized. Further improvements can be achieved when novel data formats for dense matrices are used.

[1]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[2]  Julien Langou,et al.  The Impact of Multicore on Math Software , 2006, PARA.

[3]  Przemyslaw Stpiczynski,et al.  Solving Linear Recurrence Systems Using Level 2 and 3 BLAS Routines , 2003, PPAM.

[4]  Fred G. Gustavson,et al.  New Generalized Data Structures for Matrices Lead to a Variety of High Performance Algorithms , 2001, PPAM.

[5]  Fred G. Gustavson.  The Relevance of New Data Structure Approaches for Dense Linear Algebra in the New Multi-Core / Many Core Environments , 2007, PPAM.

[6]  H. H. Wang,et al.  A Parallel Method for Tridiagonal Equations , 1981, TOMS.

[7]  Jack Dongarra,et al.  Numerical Linear Algebra for High-Performance Computers , 1998 .

[8]  Markus Kowarschik,et al.  An Overview of Cache Optimization Techniques and Cache-Aware Numerical Algorithms , 2002, Algorithms for Memory Hierarchies.

[9]  F. G. Gustavson,et al.  High-performance linear algebra algorithms using new generalized data structures for matrices , 2003, IBM J. Res. Dev..

[10]  Rohit Chandra,et al.  Parallel programming in openMP , 2000 .

[11]  Gene H. Golub,et al.  Scientific computing: an introduction with parallel computing , 1993 .

[12]  L. Ridgway Scott,et al.  Scientific Parallel Computing , 2005 .

[13]  Erik Elmroth,et al.  SIAM REVIEW c ○ 2004 Society for Industrial and Applied Mathematics Vol. 46, No. 1, pp. 3–45 Recursive Blocked Algorithms and Hybrid Data Structures for Dense Matrix Library Software ∗ , 2022 .

[14]  Przemyslaw Stpiczynski,et al.  Evaluating Linear Recursive Filters Using Novel Data Formats for Dense Matrices , 2007, PPAM.