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<H3><A NAME="SECTION03245300000000000000">
Invariant Subspaces and Condition Numbers</A>
</H3>
<P>
The Schur form<A NAME="3309"></A> depends on the order of the eigenvalues on the diagonal
of <B><I>T</I></B> and this may optionally be chosen by the user. Suppose the user chooses
that
<!-- MATH
$\lambda_1 , \ldots , \lambda_j$
-->
<IMG
WIDTH="78" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img154.png"
ALT="$\lambda_1 , \ldots , \lambda_j$">,
<!-- MATH
$1 \leq j \leq n$
-->
<IMG
WIDTH="78" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img155.png"
ALT="$1 \leq j \leq n$">,
appear in the upper left
corner of <B><I>T</I></B>. Then the first <B><I>j</I></B> columns of <B><I>Z</I></B> span the <B>right invariant
subspace</B> of <B><I>A</I></B> corresponding to
<!-- MATH
$\lambda_1 , \ldots , \lambda_j$
-->
<IMG
WIDTH="78" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img154.png"
ALT="$\lambda_1 , \ldots , \lambda_j$">.
<A NAME="3311"></A><A NAME="3312"></A>
<P>
The following routines perform this re-ordering and also
<A NAME="3313"></A>
compute condition numbers for eigenvalues, eigenvectors,
and invariant subspaces:
<P>
<DL COMPACT>
<DT>1.
<DD>xTREXC<A NAME="3315"></A><A NAME="3316"></A><A NAME="3317"></A><A NAME="3318"></A> will move an eigenvalue (or <B>2</B>-by-<B>2</B> block) on the diagonal of
the Schur form<A NAME="3319"></A> from its original position to any other position. It may be used to
choose the order in which eigenvalues appear in the Schur form.
<DT>2.
<DD>xTRSYL<A NAME="3320"></A><A NAME="3321"></A><A NAME="3322"></A><A NAME="3323"></A> solves
the Sylvester matrix equation<A NAME="3324"></A> <IMG
WIDTH="122" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img156.png"
ALT="$AX \pm XB=C$">
for <B><I>X</I></B>, given
matrices <B><I>A</I></B>, <B><I>B</I></B> and <B><I>C</I></B>, with <B><I>A</I></B> and <B><I>B</I></B> (quasi) triangular.
It is used in the routines xTRSNA and xTRSEN, but it is also of independent
interest.
<DT>3.
<DD>xTRSNA<A NAME="3325"></A><A NAME="3326"></A><A NAME="3327"></A><A NAME="3328"></A> computes the condition numbers of the eigenvalues and/or
right eigenvectors of a matrix <B><I>T</I></B> in Schur form.
<A NAME="3329"></A>
These are the same as the condition<A NAME="3330"></A>
numbers of the eigenvalues and right eigenvectors of the original matrix
<B><I>A</I></B> from which <B><I>T</I></B> is derived. The user may compute these condition numbers
for all eigenvalue/eigenvector pairs, or for any selected subset.
For more details, see section <A HREF="node91.html#secnonsym">4.8</A> and [<A
HREF="node151.html#baidemmelmckenney">12</A>].
<P>
<DT>4.
<DD>xTRSEN<A NAME="3333"></A><A NAME="3334"></A><A NAME="3335"></A><A NAME="3336"></A> moves
<A NAME="3337"></A>
a selected subset of the eigenvalues of a matrix <B><I>T</I></B>
in Schur form to the upper left corner of <B><I>T</I></B>, and optionally computes
the condition numbers<A NAME="3338"></A> 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 <B><I>A</I></B>
from which <B><I>T</I></B> is derived.
For more details, see section <A HREF="node91.html#secnonsym">4.8</A> and
[<A
HREF="node151.html#baidemmelmckenney">12</A>]
</DL>
<P>
See Table <A HREF="node52.html#tabcompeig2">2.11</A> for a complete list of the routines.
<P>
<BR>
<DIV ALIGN="CENTER">
<A NAME="tabcompeig2"></A>
<DIV ALIGN="CENTER">
<A NAME="3344"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 2.11:</STRONG>
Computational routines for the nonsymmetric eigenproblem</CAPTION>
<TR><TD ALIGN="LEFT">Type of matrix</TD>
<TD ALIGN="LEFT">Operation</TD>
<TD ALIGN="CENTER" COLSPAN=2>Single precision</TD>
<TD ALIGN="CENTER" COLSPAN=2>Double precision</TD>
</TR>
<TR><TD ALIGN="LEFT">and storage scheme</TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
<TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
</TR>
<TR><TD ALIGN="LEFT">general</TD>
<TD ALIGN="LEFT">Hessenberg reduction</TD>
<TD ALIGN="LEFT">SGEHRD<A NAME="3356"></A></TD>
<TD ALIGN="LEFT">CGEHRD<A NAME="3357"></A></TD>
<TD ALIGN="LEFT">DGEHRD<A NAME="3358"></A></TD>
<TD ALIGN="LEFT">ZGEHRD<A NAME="3359"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">balancing</TD>
<TD ALIGN="LEFT">SGEBAL<A NAME="3360"></A></TD>
<TD ALIGN="LEFT">CGEBAL<A NAME="3361"></A></TD>
<TD ALIGN="LEFT">DGEBAL<A NAME="3362"></A></TD>
<TD ALIGN="LEFT">ZGEBAL<A NAME="3363"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">backtransforming</TD>
<TD ALIGN="LEFT">SGEBAK<A NAME="3364"></A></TD>
<TD ALIGN="LEFT">CGEBAK<A NAME="3365"></A></TD>
<TD ALIGN="LEFT">DGEBAK<A NAME="3366"></A></TD>
<TD ALIGN="LEFT">ZGEBAK<A NAME="3367"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">orthogonal/unitary</TD>
<TD ALIGN="LEFT">generate matrix after</TD>
<TD ALIGN="LEFT">SORGHR<A NAME="3368"></A></TD>
<TD ALIGN="LEFT">CUNGHR<A NAME="3369"></A></TD>
<TD ALIGN="LEFT">DORGHR<A NAME="3370"></A></TD>
<TD ALIGN="LEFT">ZUNGHR<A NAME="3371"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">Hessenberg reduction</TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">multiply matrix after</TD>
<TD ALIGN="LEFT">SORMHR<A NAME="3372"></A></TD>
<TD ALIGN="LEFT">CUNMHR<A NAME="3373"></A></TD>
<TD ALIGN="LEFT">DORMHR<A NAME="3374"></A></TD>
<TD ALIGN="LEFT">ZUNMHR<A NAME="3375"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">Hessenberg reduction</TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
</TR>
<TR><TD ALIGN="LEFT">Hessenberg</TD>
<TD ALIGN="LEFT">Schur factorization</TD>
<TD ALIGN="LEFT">SHSEQR<A NAME="3376"></A></TD>
<TD ALIGN="LEFT">CHSEQR<A NAME="3377"></A></TD>
<TD ALIGN="LEFT">DHSEQR<A NAME="3378"></A></TD>
<TD ALIGN="LEFT">ZHSEQR<A NAME="3379"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">eigenvectors by</TD>
<TD ALIGN="LEFT">SHSEIN<A NAME="3380"></A></TD>
<TD ALIGN="LEFT">CHSEIN<A NAME="3381"></A></TD>
<TD ALIGN="LEFT">DHSEIN<A NAME="3382"></A></TD>
<TD ALIGN="LEFT">ZHSEIN<A NAME="3383"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">inverse iteration</TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
</TR>
<TR><TD ALIGN="LEFT">(quasi)triangular</TD>
<TD ALIGN="LEFT">eigenvectors</TD>
<TD ALIGN="LEFT">STREVC<A NAME="3384"></A></TD>
<TD ALIGN="LEFT">CTREVC<A NAME="3385"></A></TD>
<TD ALIGN="LEFT">DTREVC<A NAME="3386"></A></TD>
<TD ALIGN="LEFT">ZTREVC<A NAME="3387"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">reordering Schur</TD>
<TD ALIGN="LEFT">STREXC<A NAME="3388"></A></TD>
<TD ALIGN="LEFT">CTREXC<A NAME="3389"></A></TD>
<TD ALIGN="LEFT">DTREXC<A NAME="3390"></A></TD>
<TD ALIGN="LEFT">ZTREXC<A NAME="3391"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">factorization</TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">Sylvester equation</TD>
<TD ALIGN="LEFT">STRSYL<A NAME="3392"></A></TD>
<TD ALIGN="LEFT">CTRSYL<A NAME="3393"></A></TD>
<TD ALIGN="LEFT">DTRSYL<A NAME="3394"></A></TD>
<TD ALIGN="LEFT">ZTRSYL<A NAME="3395"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">condition numbers of</TD>
<TD ALIGN="LEFT">STRSNA<A NAME="3396"></A></TD>
<TD ALIGN="LEFT">CTRSNA<A NAME="3397"></A></TD>
<TD ALIGN="LEFT">DTRSNA<A NAME="3398"></A></TD>
<TD ALIGN="LEFT">ZTRSNA<A NAME="3399"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">eigenvalues/vectors</TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">condition numbers of</TD>
<TD ALIGN="LEFT">STRSEN<A NAME="3400"></A></TD>
<TD ALIGN="LEFT">CTRSEN<A NAME="3401"></A></TD>
<TD ALIGN="LEFT">DTRSEN<A NAME="3402"></A></TD>
<TD ALIGN="LEFT">ZTRSEN<A NAME="3403"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">eigenvalue cluster/</TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">invariant subspace</TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
</TR>
</TABLE>
</DIV>
</DIV>
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<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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