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 1713 articles , 4 standard articles )

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  1. Anzt, Hartwig; Cojean, Terry; Flegar, Goran; Göbel, Fritz; Grützmacher, Thomas; Nayak, Pratik; Ribizel, Tobias; Tsai, Yuhsiang Mike; Quintana-Ortí, Enrique S.: \textscGinkgo: a modern linear operator algebra framework for high performance computing (2022)
  2. Assing da Silva, Ivan; Dalledone Machado, Roberto; Arndt, Marcos; de Oliveira Weinhardt, Paulo: Assessment of the \textitflat-top stable GFEM for free vibration analysis (2022)
  3. Barlow, Jesse L.; Eisenstat, Stanley C.; Stor, Nevena Jakovčević; Slapnicar, Ivan: Deflation for the symmetric arrowhead and diagonal-plus-rank-one eigenvalue problems (2022)
  4. Endeve, Eirik; Hauck, Cory D.: Conservative DG method for the micro-macro decomposition of the Vlasov-Poisson-Lenard-Bernstein model (2022)
  5. Gopinath, Venkatesh; Fournier, Alexandre; Gastine, Thomas: An assessment of implicit-explicit time integrators for the pseudo-spectral approximation of Boussinesq thermal convection in an annulus (2022)
  6. Inghelbrecht, Gilles; Barbé, Kurt: Parallelization of Hermitian positive definite systems of equations: a hierarchical Jacobi approach (2022)
  7. Jiao, Xiangmin; Chen, Qiao: Approximate generalized inverses with iterative refinement for (\epsilon)-accurate preconditioning of singular systems (2022)
  8. King, J. R. C.; Lind, S. J.: High-order simulations of isothermal flows using the local anisotropic basis function method (LABFM) (2022)
  9. Krah, Philipp; Engels, Thomas; Schneider, Kai; Reiss, Julius: Wavelet adaptive proper orthogonal decomposition for large-scale flow data (2022)
  10. Mai, Tina; Mortari, Daniele: Theory of functional connections applied to quadratic and nonlinear programming under equality constraints (2022)
  11. Marchand, P.; Galkowski, J.; Spence, E. A.; Spence, A.: Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency? (2022)
  12. Mohan Ananth, Mario F. Trujillo: 2PJIT: Two-phase 3D jet instability tool in cylindrical coordinates (2022) not zbMATH
  13. Myllykoski, Mirko: Algorithm 1019: a task-based multi-shift QR/QZ algorithm with aggressive early deflation (2022)
  14. Novaković, Vedran; Singer, Sanja: A Kogbetliantz-type algorithm for the hyperbolic SVD (2022)
  15. Petkov, Petko H.; Konstantinov, Mihail M.: The numerical Jordan form (2022)
  16. Schneider, Florian; Leibner, Tobias: First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: realizability-preserving splitting scheme and numerical analysis (2022)
  17. Shi, Jia; Li, Ruipeng; Xi, Yuanzhe; Saad, Yousef; de Hoop, Maarten V.: A non-perturbative approach to computing seismic normal modes in rotating planets (2022)
  18. Souto, Mario; Garcia, Joaquim D.; Veiga, Álvaro: Exploiting low-rank structure in semidefinite programming by approximate operator splitting (2022)
  19. Abdelfattah, Ahmad; Costa, Timothy; Dongarra, Jack; Gates, Mark; Haidar, Azzam; Hammarling, Sven; Higham, Nicholas J.; Kurzak, Jakub; Luszczek, Piotr; Tomov, Stanimire; Zounon, Mawussi: A set of batched basic linear algebra subprograms and LAPACK routines (2021)
  20. Aguirre-Mesa, Andres M.; Garcia, Manuel J.; Aristizabal, Mauricio; Wagner, David; Ramirez-Tamayo, Daniel; Montoya, Arturo; Millwater, Harry: A block forward substitution method for solving the hypercomplex finite element system of equations (2021)

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