Let A be a real symmetric or complex Hermitian n-by-n matrix. A scalar is called an eigenvalue and a nonzero column vector z the corresponding eigenvector if . is always real when A is real symmetric or complex Hermitian.
The basic task of the symmetric eigenproblem routines is to compute values of and, optionally, corresponding vectors z for a given matrix A.
This computation proceeds in the following stages:
In the real case, the decomposition is computed by one of the routines xSYTRD , xSPTRD, or xSBTRD, depending on how the matrix is stored (see Table 2.10). The complex analogues of these routines are called xHETRD, xHPTRD, and xHBTRD. The routine xSYTRD (or xHETRD) represents the matrix Q as a product of elementary reflectors, as described in section 5.4. The routine xORGTR (or in the complex case xUNMTR) is provided to form Q explicitly; this is needed in particular before calling xSTEQR to compute all the eigenvectors of A by the QR algorithm. The routine xORMTR (or in the complex case xUNMTR) is provided to multiply another matrix by Q without forming Q explicitly; this can be used to transform eigenvectors of T computed by xSTEIN, back to eigenvectors of A.
When packed storage is used, the corresponding routines for forming Q or multiplying another matrix by Q are xOPGTR and xOPMTR (in the complex case, xUPGTR and xUPMTR).
When A is banded and xSBTRD (or xHBTRD) is used to reduce it to tridiagonal form , Q is determined as a product of Givens rotations , not as a product of elementary reflectors; if Q is required, it must be formed explicitly by the reduction routine. xSBTRD is based on the vectorizable algorithm due to Kaufman [57].
There are several routines for computing eigenvalues and eigenvectors of T, to cover the cases of computing some or all of the eigenvalues, and some or all of the eigenvectors. In addition, some routines run faster in some computing environments or for some matrices than for others. Also, some routines are more accurate than other routines.
See Table 2.10.
------------------------------------------------------------------------------ Type of matrix Single precision Double precision and storage scheme Operation real complex real complex ------------------------------------------------------------------------------ dense symmetric tridiagonal reduction SSYTRD CHETRD DSYTRD ZHETRD (or Hermitian) ------------------------------------------------------------------------------ packed symmetric tridiagonal reduction SSPTRD CHPTRD DSPTRD ZHPTRD (or Hermitian) ------------------------------------------------------------------------------ band symmetric tridiagonal reduction SSBTRD CHBTRD DSBTRD ZHBTRD (or Hermitian) orthogonal/unitary generate matrix after SORGTR CUNGTR DORGTR ZUNGTR reduction by xSYTRD multiply matrix after SORMTR CUNMTR DORMTR ZUNMTR reduction by xSYTRD ------------------------------------------------------------------------------ orthogonal/unitary generate matrix after SOPGTR CUPGTR DOPGTR ZUPGTR (packed storage) reduction by xSPTRD multiply matrix after SOPMTR CUPMTR DOPMTR ZUPMTR reduction by xSPTRD ------------------------------------------------------------------------------ symmetric eigenvalues/ SSTEQR CSTEQR DSTEQR ZSTEQR tridiagonal eigenvectors via QR eigenvalues only SSTERF DSTERF via root-free QR eigenvalues only SSTEBZ DSTEBZ via bisection eigenvectors by SSTEIN CSTEIN DSTEIN ZSTEIN inverse iteration ------------------------------------------------------------------------------ symmetric eigenvalues/ SPTEQR CPTEQR DPTEQR ZPTEQR tridiagonal eigenvectors positive definite ------------------------------------------------------------------------------Table 2.10: Computational routines for the symmetric eigenproblem