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<DL> <DT><A NAME="779">...output</A><DD>This example program computes the relative machine precision 
which causes, on some systems, the IEEE floating-point exception 
flags to be raised. This may result in the printing of a warning
message. This is normal.
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</PRE><DT><A NAME="782">...output</A><DD>This example program computes the relative machine precision
which causes, on some systems, the IEEE floating-point exception
flags to be raised. This may result in the printing of a warning
message. This is normal.
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</PRE><DT><A NAME="2653">...output),</A><DD>(Local input or local
       output) means that the argument may be either a local input 
       argument or a local output argument, depending on the values
       of other arguments; for example, in the PxyySVX driver routines,
       some arguments are used either as local output arguments to return
       details of a factorization, or as local input arguments to supply
       details of a previously computed factorization.
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</PRE><DT><A NAME="2654">...input),</A><DD>(local or global input) is used
       to describe the length of the workspace arguments, e.g., LWORK,
       where the value can be local input specifying the size of the
       local WORK array, or global input LWORK=-1 specifying a global
       query for the amount of workspace required.
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</PRE><DT><A NAME="3346">...input)</A><DD>(local or global input) is used
       to describe the length of the workspace arguments, e.g., LWORK,
       where the value can be local input specifying the size of the
       local WORK array, or global input LWORK=-1 specifying a global
       query for the amount of workspace required.
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</PRE><DT><A NAME="3563">...size</A><DD>The block size must be large enough 
that the local matrix multiply is efficient.  A block size of 64
suffices for most
computers that have only one processor per node.  Computers that
have multiple shared-memory processors on each node may require a larger
block size.
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</PRE><DT><A NAME="3796">...(DSSL)</A><DD> Dakota Scientific Software, Inc., 501 East Saint Joseph
Street, Rapid City, SD  57701-3995  USA, (605) 394-2471
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</PRE><DT><A NAME="3846">...(DSSL)</A><DD> Dakota Scientific Software, Inc., 501 East Saint Joseph
Street, Rapid City, SD  57701-3995  USA, (605) 394-2471
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</PRE><DT><A NAME="3903">...PSLAHQR/PDLAHQR.</A><DD>Strictly
       speaking, PSLAHQR/PDLAHQR is an auxiliary routine
       for computing the eigenvalues and optionally the
       corresponding eigenvectors of the more general case
       of nonsymmetric matrices.
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</PRE><DT><A NAME="7993">...computer.</A><DD>The ScaLAPACK sample
timer page (contained within the ScaLAPACK examples
directory on <EM>netlib</EM> and on the CD-ROM) has a BLACS port
of the message-passing program and instructions for
building the BLAS timing program.
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</PRE><DT><A NAME="8006">...PxSYEVX.</A><DD>See section&nbsp;<A HREF="node42.html#subsecnaming">3.1.3</A> for explanation 
of the naming convention used for ScaLAPACK routines.
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</PRE><DT><A NAME="8007">...'E'))</A><DD>ICTXT refers to the BLACS CONTEXT parameter.
Refer to section&nbsp;<A HREF="node71.html#secblacscontext">4.1.2</A> for further details.
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</PRE><DT><A NAME="4433">...digit,</A><DD>This is the
case on the Cray C90 and its predecessors and emulators.
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</PRE><DT><A NAME="4440">...PCs,</A><DD>Important machines that do not implement the IEEE standard
include the CRAY X-MP, CRAY Y-MP, CRAY 2, CRAY C90, IBM 370, DEC Vax,
and their emulators.
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</PRE><DT><A NAME="8008">...[#lawn112##1#].</A><DD>Running 
either machine in non-default mode to avoid this problem, either so that
the IBM RS/6000 flushes denormalized numbers to zero or so that the DEC Alpha 
handles denormalized numbers correctly by doing gradual underflow, slows
down the machine significantly [<A HREF="node189.html#demmelli93">42</A>].
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</PRE><DT><A NAME="5012">....</A><DD>
Sometimes our algorithms satisfy only 535#535 where both
532#532 and 536#536 are small. This does not significantly change the following
analysis.
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</PRE><DT><A NAME="5013">...small.</A><DD>More generally,
we need only Lipschitz continuity of <I>f</I> and may use the Lipschitz
constant in place of <I>f</I>' in deriving error bounds.
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</PRE><DT><A NAME="5039">...exist).</A><DD>This is
a different use of the term ill-posed from that used in other contexts. For
example, to be well-posed (not ill-posed) in the sense of Hadamard,
it is sufficient for <I>f</I> to be continuous, 
whereas we require Lipschitz continuity.
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</PRE><DT><A NAME="8055">...described</A><DD>There are some
caveats to this statement. When computing the inverse of a matrix,
the backward error <I>E</I> is small, taking the
columns of the computed inverse one at
a time, with a different <I>E</I> for each column [<A HREF="node189.html#lapwn27">62</A>].
The same is true when computing
the eigenvectors of a nonsymmetric matrix.
When computing the eigenvalues and eigenvectors
of 550#550, 551#551 or 552#552, 
with <I>A</I> symmetric and <I>B</I> symmetric and positive definite
(using PxSYGVX or PxHEGVX), the method may not be backward normwise
stable if
<A NAME="5047">&#160;</A><A NAME="5048">&#160;</A><A NAME="5049">&#160;</A><A NAME="5050">&#160;</A>
<I>B</I> has a large condition number 553#553,
although it has useful error bounds in this case too
(see section <A HREF="node144.html#secgendef">6.9</A>). Solving the Sylvester equation<A NAME="5052">&#160;</A>
<I>AX</I>+<I>XB</I>=<I>C</I> for the matrix <I>X</I> may not be backward stable, although
there are again useful error bounds for <I>X</I> [<A HREF="node189.html#higham93">83</A>].
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</PRE><DT><A NAME="8056">...error.</A><DD>For other algorithms, the answers (and computed error bounds) 
are as accurate as though the algorithms were componentwise relatively backward 
stable, even though they are not. These algorithms are called 
<EM>componentwise relatively forward stable</EM>.
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</PRE><DT><A NAME="8057">...bound</A><DD>As discussed in 
section&nbsp;<A HREF="node135.html#secnormnotation">6.3</A>, this approximate error bound
may underestimate the true error by a factor <I>p</I>(<I>n</I>),
which is a modestly growing function of the problem dimension <I>n</I>.
Often 569#569.
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</PRE><DT><A NAME="8103">...vectors</A><DD>These bounds are special
cases of those in section&nbsp;<A HREF="node141.html#secsym">6.7</A> 
since the singular values
and vectors of <I>A</I> are simply related to the eigenvalues and eigenvectors of
the Hermitian matrix 700#700 [<A HREF="node189.html#GVL2">71</A>, p. 427,].
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</PRE><DT><A NAME="6125">...magnitude:</A><DD>This bound is guaranteed
only if the Level 3 BLAS are implemented in a conventional way,
not in a fast way.
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</PRE><DT><A NAME="8108">...large.</A><DD>Another interpretation of chordal distance is as half the usual 
Euclidean distance between the projections of 714#714 and
637#637 on the Riemann sphere, i.e., half the length of the chord
connecting the projections.
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</PRE> </DL>
<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
</ADDRESS>
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