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 ,
1 < = j < = 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 .   

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  for A, 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 [11].

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 [11]

See Table 2.11 for a complete list of the routines.

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Type of matrix                             Single precision  Double precision
and storage scheme  Operation              real     complex  real     complex
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general             Hessenberg reduction   SGEHRD   CGEHRD   DGEHRD   ZGEHRD
                    balancing              SGEBAL   CGEBAL   DGEBAL   ZGEBAL
                    backtransforming       SGEBAK   CGEBAK   DGEBAK   ZGEBAK
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orthogonal/unitary  generate matrix after  SORGHR   CUNGHR   DORGHR   ZUNGHR
                    Hessenberg reduction
                    multiply matrix after  SORMHR   CUNMHR   DORMHR   ZUNMHR
                    Hessenberg reduction
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Hessenberg          Schur factorization    SHSEQR   CHSEQR   DHSEQR   ZHSEQR
                    eigenvectors by        SHSEIN   CHSEIN   DHSEIN   ZHSEIN
                    inverse iteration
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(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
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