Computes selected eigenvalues and eigenvectors of a real symmetric tridiagonal matrix.
The routine 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:iufor the desired eigenvalues.
Depending on the number of desired eigenvalues, these are computed either by bisection or the
dqdsalgorithm. Numerically orthogonal eigenvectors are computed by the use of various suitable
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
- Compute, so thatT- sigma*I=L*D*LTLandDdefine all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries ofLandDcause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrixTdoes not have this property in general.
- 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.
- 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.
- For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to step c for any clusters that remain.
Normal execution of
?stemrmay create NaNs and infinities and may abort due to a floating point exception in environments that do not handle NaNs and infinities in the IEEE standard default manner.