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 1386 articles , 4 standard articles )

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  1. Bylina, Beata: The block WZ factorization (2018)
  2. Carson, Erin; Higham, Nicholas J.: Accelerating the solution of linear systems by iterative refinement in three precisions (2018)
  3. González-Calderón, Alfredo; Vivas-Cruz, Luis X.; Herrera-Hernández, Erik César: Application of the $\varTheta$-method to a telegraphic model of fluid flow in a dual-porosity medium (2018)
  4. Grigori, Laura; Cayrols, Sebastien; Demmel, James W.: Low rank approximation of a sparse matrix based on LU factorization with column and row tournament pivoting (2018)
  5. Li, Hanyu: Structured condition numbers for some matrix factorizations of structured matrices (2018)
  6. Oh, Duk-Soon; Widlund, Olof B.; Zampini, Stefano; Dohrmann, Clark R.: BDDC algorithms with deluxe scaling and adaptive selection of primal constraints for Raviart-Thomas vector fields (2018)
  7. O’Neil, Michael; Cerfon, Antoine J.: An integral equation-based numerical solver for Taylor states in toroidal geometries (2018)
  8. Sastre, J.: Efficient evaluation of matrix polynomials (2018)
  9. Steger, Carsten: Algorithms for the orthographic-$n$-point problem (2018)
  10. Wang, Xuezhong; Che, Maolin; Wei, Yimin: Partial orthogonal rank-one decomposition of complex symmetric tensors based on the Takagi factorization (2018)
  11. Amestoy, Patrick; Buttari, Alfredo; L’Excellent, Jean-Yves; Mary, Theo: On the complexity of the block low-rank multifrontal factorization (2017)
  12. Balasubramanian Narasimhan and Daniel Rubin and Samuel Gross and Marina Bendersky and Philip Lavori: Software for Distributed Computation on Medical Databases: A Demonstration Project (2017)
  13. Beilina, Larisa; Karchevskii, Evgenii; Karchevskii, Mikhail: Numerical linear algebra: theory and applications (2017)
  14. Bezanson, Jeff; Edelman, Alan; Karpinski, Stefan; Shah, Viral B.: Julia: a fresh approach to numerical computing (2017)
  15. Birgin, E.G.; Martínez, J.M.: The use of quadratic regularization with a cubic descent condition for unconstrained optimization (2017)
  16. Boedec, Gwenn; Leonetti, Marc; Jaeger, Marc: Isogeometric FEM-BEM simulations of drop, capsule and vesicle dynamics in Stokes flow (2017)
  17. Boiko, Andrey V.; Demyanko, Kirill V.; Nechepurenko, Yuri M.: On computing the location of laminar-turbulent transition in compressible boundary layers (2017)
  18. Bosner, Nela; Karlsson, Lars: Parallel and heterogeneous $m$-Hessenberg-triangular-triangular reduction (2017)
  19. Crivellaro, Alberto; Perotto, Simona; Zonca, Stefano: Reconstruction of 3D scattered data via radial basis functions by efficient and robust techniques (2017)
  20. Cui, Yiran; del Baño Rollin, Sebastian; Germano, Guido: Full and fast calibration of the Heston stochastic volatility model (2017)

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