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The design and implementation of the MRRR algorithm

In the 1990's, Dhillon and Parlett devised the algorithm of multiple relatively robust representations (MRRR) for computing numerically orthogonal eigenvectors of a symmetric tridiagonal matrix T with O(n2) cost. While previous publications related to MRRR focused on theoretical aspects of the algorithm, a documentation of software issues has been missing. In this article, we discuss the design and implementation of the new MRRR version STEGR that will be included in the next LAPACK release. By giving an algorithmic description of MRRR and identifying governing parameters, we hope to make STEGR more easily accessible and suitable for future performance tuning. Furthermore, this should help users understand design choices and tradeoffs when using the code.