Параллельные вычисления в ИММ УрО РАН
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Next: Nonsymmetric Eigenproblems Up: Computational Routines Previous: Generalized RQ factorization
Symmetric EigenproblemsLet 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 the routine PxSYTRD (see table 3.8). The complex analogue of this routine is called PxHETRD. The routine PxSYTRD (or PxHETRD) represents the matrix Q as a product of elementary reflectors, as described in section 3.4. The routine PxORMTR (or in the complex case PxUNMTR) is provided to multiply another matrix by Q without forming Q explicitly; this can be used to transform eigenvectors of T, computed by PxSTEIN, back to eigenvectors of A. The following routines compute eigenvalues and eigenvectors of T.
Without any reorthogonalization, inverse iteration may produce vectors that have large dot products. To cure this, most implementations of inverse iteration such as LAPACK's xSTEIN reorthogonalize when eigenvalues differ by less than . As a result, the eigenvectors computed by xSTEIN are almost always orthogonal, but the increase in cost can result in work. On some rare examples, xSTEIN may still fail to deliver accurate answers; see [43, 44]. The orthogonalization done by PxSTEIN is limited by the amount of workspace provided; whenever it performs less reorthogonalization than xSTEIN, there is a danger that the dot products may not be satisfactory.
Next: Nonsymmetric Eigenproblems Up: Computational Routines Previous: Generalized RQ factorization Susan Blackford Tue May 13 09:21:01 EDT 1997 |