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Invariant Subspaces and Condition Numbers

The Schur form depends on the order of the eigenvalues on the diagonal of T and this may optionally be chosen by the user. Suppose the user chooses that $\lambda_1 , \ldots , \lambda_j$, $1 \leq j \leq n$, appear in the upper left corner of T. Then the first j columns of Z span the right invariant subspace of A corresponding to $\lambda_1 , \ldots , \lambda_j$.

The following routines perform this re-ordering and also compute condition numbers for eigenvalues, eigenvectors, and invariant subspaces:

1.
xTREXC will move an eigenvalue (or 2-by-2 block) on the diagonal of the Schur form from its original position to any other position. It may be used to choose the order in which eigenvalues appear in the Schur form.
2.
xTRSYL solves the Sylvester matrix equation $AX \pm XB=C$ for X, given matrices A, B and C, with A and B (quasi) triangular. It is used in the routines xTRSNA and xTRSEN, but it is also of independent interest.
3.
xTRSNA computes the condition numbers of the eigenvalues and/or right eigenvectors of a matrix T in Schur form. These are the same as the condition numbers of the eigenvalues and right eigenvectors of the original matrix A from which T is derived. The user may compute these condition numbers for all eigenvalue/eigenvector pairs, or for any selected subset. For more details, see section 4.8 and [12].

4.
xTRSEN moves a selected subset of the eigenvalues of a matrix T in Schur form to the upper left corner of T, and optionally computes the condition numbers of their average value and of their right invariant subspace. These are the same as the condition numbers of the average eigenvalue and right invariant subspace of the original matrix A from which T is derived. For more details, see section 4.8 and [12]

See Table 2.11 for a complete list of the routines.


Table 2.11: Computational routines for the nonsymmetric eigenproblem
Type of matrix Operation Single precision Double precision
and storage scheme   real complex real complex
general Hessenberg reduction SGEHRD CGEHRD DGEHRD ZGEHRD
  balancing SGEBAL CGEBAL DGEBAL ZGEBAL
  backtransforming SGEBAK CGEBAK DGEBAK ZGEBAK
orthogonal/unitary generate matrix after SORGHR CUNGHR DORGHR ZUNGHR
  Hessenberg reduction        
  multiply matrix after SORMHR CUNMHR DORMHR ZUNMHR
  Hessenberg reduction        
Hessenberg Schur factorization SHSEQR CHSEQR DHSEQR ZHSEQR
  eigenvectors by SHSEIN CHSEIN DHSEIN ZHSEIN
  inverse iteration        
(quasi)triangular eigenvectors STREVC CTREVC DTREVC ZTREVC
  reordering Schur STREXC CTREXC DTREXC ZTREXC
  factorization        
  Sylvester equation STRSYL CTRSYL DTRSYL ZTRSYL
  condition numbers of STRSNA CTRSNA DTRSNA ZTRSNA
  eigenvalues/vectors        
  condition numbers of STRSEN CTRSEN DTRSEN ZTRSEN
  eigenvalue cluster/        
  invariant subspace        


next up previous contents index
Next: Singular Value Decomposition Up: Nonsymmetric Eigenproblems Previous: Balancing   Contents   Index
Susan Blackford
1999-10-01