# 2009 Research Summary

## Improving LAPACK and ScaLAPACK (LAPACK)

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Yozo Hida, Jason Riedy, Christof Vömel^{1}, Xiaoye Li^{2}, Osni Marques^{3}, William M. Kahan, Beresford N. Parlett, James Demmel, Ming Gu^{4} and Vasily Volkov

National Science Foundation CCF-0444486, Department of Energy DOE-DE-FC02-06ER25786 and National Science Foundation CNS-0325873

Users and distributors of LAPACK and ScaLAPACK expect these libraries to represent the state-of-the-art in high-performance dense linear algebra. Numerical algorithms, implementation techniques, and computer architectures have progressed significantly since the last major update of these libraries. With collaborators around the world, we are incorporating the latest advances to address user demands as collected in an ongoing survey. We are also growing a community to support for long-term, continuous development.

The improvements address accuracy, performance, functionality, engineering, and ease of use. The prospectus [1] surveys the entire project. Selected Berkeley thrusts include:

- Iterative refinement for linear systems [2] and least-square problems [3] to provide excellent accuracy and dependable error estimates;
- Use of IEEE-754 arithmetic features for performance and reliability [4];
- Exploration of using high-performance coprocessors and graphics adapters;
- Implementing parallel ScaLAPACK routines to include more LAPACK functionality;
- Integration of higher arithmetic precisions throughout LAPACK and ScaLAPACK; and
- Performance optimizations for matrices with limited non-zero structures that are stored as full, dense matrices [5].

Other functionality being incorporated includes automatic tuning of parameters and parallel data distributions, factorization updating facilities, quadratic eigenvalue problems, matrix functions, high-accuracy Jacobi SVD routines, pivoting and scaling for symmetric linear systems, recursive data layouts, fast Hessenberg QR, generalized SVDs, and many more.

- [1]
- J. Demmel and J. Dongarra, "LAPACK 2005 Prospectus: Reliable and Scalable Software for Linear Algebra Computations on High End Computers,"
*LAPACK Working Note 164,*February 2005. http://www.netlib.org/lapack/lawnspdf/lawn164.pdf. - [2]
- J. Demmel, Y. Hida, W. Kahan, X. Li, S. Mukherjee, and J. Riedy, "Error Bounds from Extra Precise Iterative Refinement,"
*ACM Trans. Mathematical Software (TOMS),*Vol. 32, No. 2, June 2006, pp. 325-351. DOI: http://dx.doi.org/10.1145/1141885.1141894 - [3]
- J. Demmel, Y. Hida, X. Li, and J. Riedy, "Extra-Precise Iterative Refinement for Overdetermined Least Squares Problems,"
*ACM Trans. Mathematical Software (TOMS),*Vol. 35, No. 4 (to appear). Also, LAPACK Working Note 188, http://www.netlib.org/lapack/lawnspdf/lawn188.pdf. - [4]
- O. A. Marques, E. J. Riedy, and C. Vömel, "Benefits of IEEE-754 Features in Modern Symmetric Tridiagonal Eigensolvers,"
*SIAM Journal on Scientific Computing,*Vol. 28, No. 5, 2006, pp. 1613-1633. DOI: http://dx.doi.org/10.1137/050641624 - [5]
- J. Demmel, M. Hoemmen, Y. Hida, and J. Riedy, "Non-Negative Diagonals and High Performance on Low-Profile Matrices from Householder QR,"
*LAPACK Working Note 203,*May 2008. http://www.netlib.org/lapack/lawnspdf/lawn203.pdf.

^{1}NERSC / Lawrence Berkeley National Laboratory

^{2}NERSC / Lawrence Berkeley National Laboratory

^{3}NERSC / Lawrence Berkeley National Laboratory

^{4}Mathematics