LAPACK

LAPACK is written in Fortran 90 and provides routines for solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems. The associated matrix factorizations (LU, Cholesky, QR, SVD, Schur, generalized Schur) are also provided, as are related computations such as reordering of the Schur factorizations and estimating condition numbers. Dense and banded matrices are handled, but not general sparse matrices. In all areas, similar functionality is provided for real and complex matrices, in both single and double precision. The original goal of the LAPACK project was to make the widely used EISPACK and LINPACK libraries run efficiently on shared-memory vector and parallel processors. On these machines, LINPACK and EISPACK are inefficient because their memory access patterns disregard the multi-layered memory hierarchies of the machines, thereby spending too much time moving data instead of doing useful floating-point operations. LAPACK addresses this problem by reorganizing the algorithms to use block matrix operations, such as matrix multiplication, in the innermost loops. These block operations can be optimized for each architecture to account for the memory hierarchy, and so provide a transportable way to achieve high efficiency on diverse modern machines. We use the term ”transportable” instead of ”portable” because, for fastest possible performance, LAPACK requires that highly optimized block matrix operations be already implemented on each machine. LAPACK routines are written so that as much as possible of the computation is performed by calls to the Basic Linear Algebra Subprograms (BLAS). LAPACK is designed at the outset to exploit the Level 3 BLAS — a set of specifications for Fortran subprograms that do various types of matrix multiplication and the solution of triangular systems with multiple right-hand sides. Because of the coarse granularity of the Level 3 BLAS operations, their use promotes high efficiency on many high-performance computers, particularly if specially coded implementations are provided by the manufacturer. Highly efficient machine-specific implementations of the BLAS are available for many modern high-performance computers. For details of known vendor- or ISV-provided BLAS, consult the BLAS FAQ. Alternatively, the user can download ATLAS to automatically generate an optimized BLAS library for the architecture. A Fortran 77 reference implementation of the BLAS is available from netlib; however, its use is discouraged as it will not perform as well as a specifically tuned implementation.

This software is also referenced in ORMS.


References in zbMATH (referenced in 1671 articles , 4 standard articles )

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  3. Bocharov, G. A.; Nechepurenko, Yu. M.; Khristichenko, M. Yu.; Grebennikov, D. S.: Optimal perturbations of systems with delayed independent variables for control of dynamics of infectious diseases based on multicomponent actions (2021)
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  7. Hirshikesh; Pramod, A. L. N.; Ooi, Ean Tat; Song, Chongmin; Natarajan, Sundararajan: An adaptive scaled boundary finite element method for contact analysis (2021)
  8. Hrga, Timotej; Povh, Janez: \textttMADAM: a parallel exact solver for max-cut based on semidefinite programming and ADMM (2021)
  9. Jason Rumengan, Terry Yue Zhuo, Conrad Sanderson: PyArmadillo: a streamlined linear algebra library for Python (2021) arXiv
  10. Kahl, Karsten; Lang, Bruno: Hypergraph edge elimination -- a symbolic phase for Hermitian eigensolvers based on rank-1 modifications (2021)
  11. Kuhlman, Kristopher L.; Malama, Bwalya: Uncoupling electrokinetic flow solutions (2021)
  12. Kużelewski, Andrzej; Zieniuk, Eugeniusz: The FMM accelerated PIES with the modified binary tree in solving potential problems for the domains with curvilinear boundaries (2021)
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  14. Marshall, Joshua P.; Richardson, J. D.: A three-dimensional, (p)-version BEM: high-order refinement leveraged through regularization (2021)
  15. Mitjana, F.; Cafieri, S.; Bugarin, F.; Segonds, S.; Castanie, F.; Duysinx, P.: Topological gradient in structural optimization under stress and buckling constraints (2021)
  16. Nohra, Carlos J.; Raghunathan, Arvind U.; Sahinidis, Nikolaos: Spectral relaxations and branching strategies for global optimization of mixed-integer quadratic programs (2021)
  17. Petkov, Petko H.: Componentwise perturbation analysis of the Schur decomposition of a matrix (2021)
  18. Rek, Zlatko; Šarler, Božidar: The method of fundamental solutions for the Stokes flow with the subdomain technique (2021)
  19. Saurabh, Kumar; Gao, Boshun; Fernando, Milinda; Xu, Songzhe; Khanwale, Makrand A.; Khara, Biswajit; Hsu, Ming-Chen; Krishnamurthy, Adarsh; Sundar, Hari; Ganapathysubramanian, Baskar: Industrial scale large eddy simulations with adaptive octree meshes using immersogeometric analysis (2021)
  20. Staub, Ruben; Steinmann, Stephan N.: Efficient recursive least squares solver for rank-deficient matrices (2021)

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