stemr Interface

public interface stemr

STEMR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T. Any such unreduced matrix has a well defined set of pairwise different real eigenvalues, the corresponding real eigenvectors are pairwise orthogonal. The spectrum may be computed either completely or partially by specifying either an interval (VL,VU] or a range of indices IL:IU for the desired eigenvalues. Depending on the number of desired eigenvalues, these are computed either by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are computed by the use of various suitable L D L^T factorizations near clusters of close eigenvalues (referred to as RRRs, Relatively Robust Representations). An informal sketch of the algorithm follows. For each unreduced block (submatrix) of T, (a) Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general. (b) Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d). (c) For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy. (d) For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain. For more details, see: - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices," Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154. - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997. Further Details 1.STEMR works only on machines which follow IEEE-754 floating-point standard in their handling of infinities and NaNs. This permits the use of efficient inner loops avoiding a check for zero divisors. 2. LAPACK routines can be used to reduce a complex Hermitean matrix to real symmetric tridiagonal form. (Any complex Hermitean tridiagonal matrix has real values on its diagonal and potentially complex numbers on its off-diagonals. By applying a similarity transform with an appropriate diagonal matrix diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean matrix can be transformed into a real symmetric matrix and complex arithmetic can be entirely avoided.) While the eigenvectors of the real symmetric tridiagonal matrix are real, the eigenvectors of original complex Hermitean matrix have complex entries in general. Since LAPACK drivers overwrite the matrix data with the eigenvectors, STEMR accepts complex workspace to facilitate interoperability with CUNMTR or CUPMTR.

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