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<BR> <P>
<H1><A NAME="SECTION04600000000000000000">Accuracy and Stability</A></H1>
<A NAME="chapaccstab">&#160;</A>
<P>
In this chapter we explain our overall approach to obtaining error bounds
and provide enough information to use the software.
The comments at the beginning of the individual routines should be consulted 
for more details.
It is beyond the scope of this chapter to justify all the bounds
we present. Instead, we give references to the literature.
For example, standard material on error analysis can be found 
in [<A HREF="node189.html#GVL2">71</A>, <A HREF="node189.html#wilkinson1">114</A>, <A HREF="node189.html#higham96">84</A>, <A HREF="node189.html#demmelMA221">38</A>].
<P>
To make this chapter easy to read, we have labeled parts
not essential for a first reading as <B>Further Details</B>. The sections
not labeled as <B>Further Details</B> should provide all the information
needed to understand and use the main error bounds computed by
ScaLAPACK.  The <B>Further Details</B> sections provide mathematical background,
references, and tighter but more expensive error bounds, and may be read later.
<P>
Since ScaLAPACK uses the same overall algorithmic approach as
LAPACK, its error<A NAME="4355">&#160;</A> bounds<A NAME="4356">&#160;</A> are 
essentially the same as those for LAPACK.  
Therefore, this chapter is largely analogous to Chapter&nbsp;4 
of the LAPACK Users' Guide [<A HREF="node189.html#laug">3</A>].
Significant differences between LAPACK and ScaLAPACK include the
following:
<UL>
<LI> Section&nbsp;<A HREF="node133.html#secroundoff">6.1</A> discusses how
machine constants in a heterogeneous
network of machines with differing floating-point arithmetics must be
redefined. ScaLAPACK can also exploit arithmetic with <IMG WIDTH=29 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline17191" SRC="img428.gif">,
which is available in IEEE standard floating-point arithmetic.
<A NAME="4360">&#160;</A>
<LI> Section&nbsp;<A HREF="node134.html#sec_Hetero">6.2</A> discusses reliability problems that can
arise on heterogeneous networks of machines and how to guarantee
reliability on a homogeneous network.
<LI> Section&nbsp;<A HREF="node139.html#secAxb">6.5</A> discusses
some routines that do Gaussian elimination on band matrices with 
the pivot order chosen for parallelism rather than numerical stability.
These routines are numerically stable only when applied to matrices that 
are do not require partial pivoting for stability
(such as diagonally dominant and symmetric positive definite  matrices).
<LI> Section&nbsp;<A HREF="node141.html#secsym">6.7</A> discusses
PxSYEVX.<A NAME="tex2html1062" HREF="footnode.html#8006"><IMG  ALIGN=BOTTOM ALT="gif" SRC="http://www.netlib.org/utk/icons/foot_motif.gif"></A>
In contrast to its LAPACK analogue, xSYEVX, PxSYEVX allows
the user to trade off orthogonality of computed eigenvectors and
runtime.
</UL>
<P>
In section&nbsp;<A HREF="node133.html#secroundoff">6.1</A>
we discuss the sources of numerical error, in particular roundoff 
error. We also briefly discuss IEEE arithmetic.
Section&nbsp;<A HREF="node134.html#sec_Hetero">6.2</A> discusses the new sources of numerical
error specific to parallel libraries, and the restrictions they
impose on the reliable use of ScaLAPACK.
Section&nbsp;<A HREF="node135.html#secnormnotation">6.3</A> discusses how to measure errors,
as well as some standard notation.
Section&nbsp;<A HREF="node136.html#secwhereerrorboundscomefrom">6.4</A> discusses further details
of how error bounds are derived.
Sections&nbsp;<A HREF="node139.html#secAxb">6.5</A> through <A HREF="node144.html#secgendef">6.9</A> present error bounds
for linear equations, linear least squares problems, 
the symmetric eigenproblem, 
the singular value decomposition, and
the generalized symmetric definite eigenproblem, respectively.
<P>
<BR> <HR>
<UL><A NAME="CHILD_LINKS">&#160;</A>
<LI> <A NAME="tex2html3858" HREF="node133.html#SECTION04610000000000000000">Sources of Error in Numerical Calculations</A>
<LI> <A NAME="tex2html3859" HREF="node134.html#SECTION04620000000000000000">New Sources of Error in Parallel Numerical Computations</A>
<LI> <A NAME="tex2html3860" HREF="node135.html#SECTION04630000000000000000">How to Measure Errors</A>
<LI> <A NAME="tex2html3861" HREF="node136.html#SECTION04640000000000000000">Further Details:  How Error Bounds Are Derived</A>
<UL>
<LI> <A NAME="tex2html3862" HREF="node137.html#SECTION04641000000000000000">Standard Error Analysis</A>
<LI> <A NAME="tex2html3863" HREF="node138.html#SECTION04642000000000000000">Improved Error Bounds</A>
</UL> 
<LI> <A NAME="tex2html3864" HREF="node139.html#SECTION04650000000000000000">Error Bounds for Linear Equation Solving</A>
<LI> <A NAME="tex2html3865" HREF="node140.html#SECTION04660000000000000000">Error Bounds for Linear Least Squares Problems</A>
<LI> <A NAME="tex2html3866" HREF="node141.html#SECTION04670000000000000000">Error Bounds for the Symmetric Eigenproblem </A>
<LI> <A NAME="tex2html3867" HREF="node142.html#SECTION04680000000000000000">Error Bounds for the Singular Value Decomposition</A>
<UL>
<LI> <A NAME="tex2html3868" HREF="node143.html#SECTION04681000000000000000">Further Details:  Error Bounds for the Singular Value Decomposition</A>
</UL> 
<LI> <A NAME="tex2html3869" HREF="node144.html#SECTION04690000000000000000">Error Bounds for the Generalized Symmetric Definite
Eigenproblem</A>
</UL>
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<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
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