A set of level 3 basic linear algebra subprograms
ACM Transactions on Mathematical Software (TOMS)
Accurate Symmetric Indefinite Linear Equation Solvers
SIAM Journal on Matrix Analysis and Applications
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
A Comparison of Algorithms for Solving Symmetric Indefinite Systems of Linear Equations
ACM Transactions on Mathematical Software (TOMS)
Numerical Linear Algebra for High Performance Computers
Numerical Linear Algebra for High Performance Computers
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Evaluating Block Algorithm Variants in LAPACK
Proceedings of the Fourth SIAM Conference on Parallel Processing for Scientific Computing
The science of deriving dense linear algebra algorithms
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
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We present a partitioned algorithm for reducing a symmetric matrix to a tridiagonal form, with partial pivoting. That is, the algorithm computes a factorization PAPT = LTLT, where, P is a permutation matrix, L is lower triangular with a unit diagonal and entries’ magnitudes bounded by 1, and T is symmetric and tridiagonal. The algorithm is based on the basic (nonpartitioned) methods of Parlett and Reid and of Aasen. We show that our factorization algorithm is componentwise backward stable (provided that the growth factor is not too large), with a similar behavior to that of Aasen’s basic algorithm. Our implementation also computes the QR factorization of T and solves linear systems of equations using the computed factorization. The partitioning allows our algorithm to exploit modern computer architectures (in particular, cache memories and high-performance blas libraries). Experimental results demonstrate that our algorithms achieve approximately the same level of performance as the partitioned Bunch-Kaufman factor and solve routines in lapack.