Symmetric Eigenproblems

Let A be a real symmetric or complex Hermitian n-by-n matrix. A scalar $\lambda$ is called an eigenvalue and a nonzero column vector z the corresponding eigenvector if $Az = \lambda z$. $\lambda$ is always real when A is real symmetric or complex Hermitian.

The basic task of the symmetric eigenproblem routines is to compute values of $\lambda$ and, optionally, corresponding vectors z for a given matrix A.

This computation proceeds in the following stages:

1.

The real symmetric or complex Hermitian matrix A is reduced to real tridiagonal form T. If A is real symmetric this decomposition is A=QTQ^T with Q orthogonal and T symmetric tridiagonal. If A is complex Hermitian, the decomposition is A=QTQ^H with Q unitary and T, as before, real symmetric tridiagonal.

2.

Eigenvalues and eigenvectors of the real symmetric tridiagonal matrix T are computed. If all eigenvalues and eigenvectors are computed, this is equivalent to factorizing T as $T = S \Lambda S^T$, where S is orthogonal and $\Lambda$ is diagonal. The diagonal entries of $\Lambda$ are the eigenvalues of T, which are also the eigenvalues of A, and the columns of S are the eigenvectors of T; the eigenvectors of A are the columns of Z=QS, so that $A=Z \Lambda Z^T$ ($Z \Lambda Z^H$ when A is complex Hermitian).

In the real case, the decomposition A = Q T Q^T 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  [77].

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 section 2.3.4.1 for a discussion.

xSTEQR

This routine uses the implicitly shifted QR algorithm. It switches between the QR and QL variants in order to handle graded matrices more effectively than the simple QL variant that is provided by the EISPACK routines IMTQL1 and IMTQL2. See [56] for details. This routine is used by drivers with names ending in -EV and -EVX to compute all the eigenvalues and eigenvectors (see section 2.3.4.1).

xSTERF

This routine uses a square-root free version of the QR algorithm, also switching between QR and QL variants, and can only compute all the eigenvalues. See [56] for details. This routine is used by drivers with names ending in -EV and -EVX to compute all the eigenvalues and no eigenvectors (see section 2.3.4.1).

xSTEDC

This routine uses Cuppen's divide and conquer algorithm to find the eigenvalues and the eigenvectors (if only eigenvalues are desired, xSTEDC calls xSTERF). xSTEDC can be many times faster than xSTEQR for large matrices but needs more work space (2n² or 3n²). See [20,57,89] and section 3.4.3 for details. This routine is used by drivers with names ending in -EVD to compute all the eigenvalues and eigenvectors (see section 2.3.4.1).

xSTEGR

This routine uses the relatively robust representation (RRR) algorithm to find eigenvalues and eigenvectors. This routine uses an LDL^T factorization of a number of translates T - sI of T, for one shift s near each cluster of eigenvalues. For each translate the algorithm computes very accurate eigenpairs for the tiny eigenvalues. xSTEGR is faster than all the other routines except in a few cases, and uses the least workspace. See [35] and section 3.4.3 for details.

xPTEQR

This routine applies to symmetric positive definite tridiagonal matrices only. It uses a combination of Cholesky factorization and bidiagonal QR iteration (see xBDSQR) and may be significantly more accurate than the other routines except xSTEGR. See [14,32,23,51] for details.

xSTEBZ

This routine uses bisection to compute some or all of the eigenvalues. Options provide for computing all the eigenvalues in a real interval or all the eigenvalues from the i^th to the j^th largest. It can be highly accurate, but may be adjusted to run faster if lower accuracy is acceptable. This routine is used by drivers with names ending in -EVX.

xSTEIN

Given accurate eigenvalues, this routine uses inverse iteration to compute some or all of the eigenvectors. This routine is used by drivers with names ending in -EVX.

See Table 2.10.

Table 2.10: Computational routines for the symmetric eigenproblem

Type of matrix	Operation	Single precision		Double precision
and storage scheme		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/	SSTEDC	CSTEDC	DSTEDC	ZSTEDC
	eigenvectors via
	divide and conquer
	eigenvalues/	SSTEGR	CSTEGR	DSTEGR	ZSTEGR
	eigenvectors via
	RRR
	eigenvalues only	SSTEBZ		DSTEBZ
	via bisection
	eigenvectors by	SSTEIN	CSTEIN	DSTEIN	ZSTEIN
	inverse iteration
symmetric	eigenvalues/	SPTEQR	CPTEQR	DPTEQR	ZPTEQR
tridiagonal	eigenvectors
positive definite