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<H2><A NAME="SECTIONREF">Bibliography</A>
</H2>
<DL COMPACT><DD><P></P><DT><A NAME="lawn20">1</A>
<DD>
E. A<SMALL>NDERSON, </SMALL>Z. B<SMALL>AI, </SMALL>C. B<SMALL>ISCHOF, </SMALL>J. D<SMALL>EMMEL, </SMALL>J. D<SMALL>ONGARRA, </SMALL>J. D<SMALL>U </SMALL>C<SMALL>ROZ,
  </SMALL>A. G<SMALL>REENBAUM, </SMALL>S. H<SMALL>AMMARLING, </SMALL>A. M<SMALL>C</SMALL>K<SMALL>ENNEY, AND </SMALL>D. S<SMALL>ORENSEN</SMALL>, <EM>LAPACK: A
  portable linear algebra library for high-performance computers</EM>, Computer
  Science Dept. Technical Report CS-90-105, University of Tennessee,
  Knoxville, TN, May 1990.
<BR>(Also LAPACK Working Note #20).

<P></P><DT><A NAME="lawn31">2</A>
<DD>
E. A<SMALL>NDERSON, </SMALL>Z. B<SMALL>AI, AND </SMALL>J. D<SMALL>ONGARRA</SMALL>, <EM>Generalized QR factorization
  and its applications</EM>, Linear Algebra and Its Applications, 162-164 (1992),
  pp. 243-271.
<BR>(Also LAPACK Working Note #31).

<P></P><DT><A NAME="lawn41">3</A>
<DD>
E. A<SMALL>NDERSON, </SMALL>J. D<SMALL>ONGARRA, AND </SMALL>S. O<SMALL>STROUCHOV</SMALL>, <EM>Installation guide for
  LAPACK</EM>, Computer Science Dept. Technical Report CS-92-151,
  University of Tennessee, Knoxville, TN, March 1992.
<BR>(Also LAPACK Working Note #41).

<P></P><DT><A NAME="ieee754">4</A>
<DD>
ANSI/IEEE, <EM>  IEEE Standard for Binary Floating Point Arithmetic</EM>, New York, Std
  754-1985 ed., 1985.

<P></P><DT><A NAME="ieee854">5</A>
<DD>
ANSI/IEEE, <EM>  IEEE Standard for Radix Independent Floating Point Arithmetic</EM>, New
  York, Std 854-1987 ed., 1987.

<P></P><DT><A NAME="ariolidemmelduff">6</A>
<DD>
M. A<SMALL>RIOLI, </SMALL>J. W. D<SMALL>EMMEL, AND </SMALL>I. S. D<SMALL>UFF</SMALL>, <EM>Solving sparse linear
  systems with sparse backward error</EM>, SIAM J. Matrix Anal. Appl., 10 (1989),
  pp. 165-190.

<P></P><DT><A NAME="arioliduffderijk">7</A>
<DD>
M. A<SMALL>RIOLI, </SMALL>I. S. D<SMALL>UFF, AND </SMALL>P. P. M. <SMALL>DE </SMALL>R<SMALL>IJK</SMALL>, <EM>On the augmented system
  approach to sparse least squares problems</EM>, Num. Math., 55 (1989),
  pp. 667-684.

<P></P><DT><A NAME="baizha93">8</A>
<DD>
Z. B<SMALL>AI, , AND </SMALL>H. Z<SMALL>HA</SMALL>, <EM>A new preprocessing algorithm for the
  computation of the generalized singular value decomposition</EM>, SIAM J. Sci.
  Comp., 14 (1993), pp. 1007-1012.

<P></P><DT><A NAME="baidemmel">9</A>
<DD>
Z. B<SMALL>AI AND </SMALL>J. W. D<SMALL>EMMEL</SMALL>, <EM>On a block implementation of Hessenberg
  multishift QR iteration</EM>, International Journal of High Speed Computing, 1
  (1989), pp. 97-112.
<BR>(Also LAPACK Working Note #8).

<P></P><DT><A NAME="baidemmel92b">10</A>
<DD>
Z. B<SMALL>AI AND </SMALL>J. W. D<SMALL>EMMEL</SMALL>, <EM>Computing the
  generalized singular value decomposition</EM>, SIAM J. Sci. Comp., 14 (1993),
  pp. 1464-1486.
<BR>(Also LAPACK Working Note #46).

<P></P><DT><A NAME="baidemmel92a">11</A>
<DD>
Z. B<SMALL>AI AND </SMALL>J. W. D<SMALL>EMMEL</SMALL>, <EM>Design of a parallel
  nonsymmetric eigenroutine toolbox, Part I</EM>, in Proceedings of the Sixth
  SIAM Conference on Parallel Processing for Scientific Computing, R. F. <EM>  et al</EM>. Sincovec, ed., Philadelphia, PA, 1993, Society for Industrial and
  Applied Mathematics, pp. 391-398.
<BR>Long version available as Computer Science Report CSD-92-718,
  University of California, Berkeley, 1992.

<P></P><DT><A NAME="baidemmelmckenney">12</A>
<DD>
Z. B<SMALL>AI, </SMALL>J. W. D<SMALL>EMMEL, AND </SMALL>A. M<SMALL>C</SMALL>K<SMALL>ENNEY</SMALL>, <EM>On computing condition
  numbers for the nonsymmetric eigenproblem</EM>, ACM Trans. Math. Softw., 19
  (1993), pp. 202-223.
<BR>(LAPACK Working Note #13).

<P></P><DT><A NAME="baifahey97">13</A>
<DD>
Z. B<SMALL>AI AND </SMALL>M. F<SMALL>AHEY</SMALL>, <EM>Computation of error bounds in linear least
  squares problems with equality constraints and generalized linear model
  problems</EM>.
<BR>to appear, 1997.

<P></P><DT><A NAME="barlowdemmel">14</A>
<DD>
J. B<SMALL>ARLOW AND </SMALL>J. D<SMALL>EMMEL</SMALL>, <EM>Computing accurate eigensystems of scaled
  diagonally dominant matrices</EM>, SIAM J. Num. Anal., 27 (1990), pp. 762-791.
<BR>(Also LAPACK Working Note #7).

<P></P><DT><A NAME="lawn111">15</A>
<DD>
J. B<SMALL>ILMES, </SMALL>K. A<SMALL>SANOVIC, </SMALL>J. D<SMALL>EMMEL, </SMALL>D. L<SMALL>AM, AND </SMALL>C. C<SMALL>HIN</SMALL>, <EM>Optimizing
  matrix multiply using PHiPAC: A portable, high-performance, ANSI C
  coding methodology</EM>, Computer Science Dept. Technical Report
  CS-96-326, University of Tennessee, Knoxville, TN, 1996.
<BR>(Also LAPACK Working Note #111).

<P></P><DT><A NAME="bjorck3">16</A>
<DD>
<SMALL>&#197;</SMALL>. B<SMALL>J&#168;ORCK</SMALL>, <EM>Numerical Methods for Least Squares Problem</EM>,
  SIAM, 1996.

<P></P><DT><A NAME="slug">17</A>
<DD>
L. S. B<SMALL>LACKFORD, </SMALL>J. C<SMALL>HOI, </SMALL>A. C<SMALL>LEARY, </SMALL>E. D'A<SMALL>ZEVEDO, </SMALL>J. D<SMALL>EMMEL, </SMALL>I. D<SMALL>HILLON,
  </SMALL>J. D<SMALL>ONGARRA, </SMALL>S. H<SMALL>AMMARLING, </SMALL>G. H<SMALL>ENRY, </SMALL>A. P<SMALL>ETITET, </SMALL>K. S<SMALL>TANLEY, </SMALL>D. W<SMALL>ALKER, AND
  </SMALL>R. C. W<SMALL>HALEY</SMALL>, <EM>ScaLAPACK Users' Guide</EM>, Society for Industrial and
  Applied Mathematics, Philadelphia, PA, 1997.

<P></P><DT><A NAME="coxhigham">18</A>
<DD>
A. J. C<SMALL>OX AND </SMALL>N. J. H<SMALL>IGHAM</SMALL>, <EM>Backward error bounds for constrained
  least squares problems</EM>, BIT, 39 (1999), pp. 210-227.

<P></P><DT><A NAME="crawford">19</A>
<DD>
C. R. C<SMALL>RAWFORD</SMALL>, <EM>Reduction of a band-symmetric generalized eigenvalue
  problem</EM>, Comm. ACM, 16 (1973), pp. 41-44.

<P></P><DT><A NAME="cuppen">20</A>
<DD>
J. J. M. C<SMALL>UPPEN</SMALL>, <EM>A divide and conquer method for the symmetric
  tridiagonal eigenproblem</EM>, Numerische Math., 36 (1981), pp. 177-195.

<P></P><DT><A NAME="dayde94a">21</A>
<DD>
M. D<SMALL>AYDE, </SMALL>I. D<SMALL>UFF, AND </SMALL>A. P<SMALL>ETITET</SMALL>, <EM>A Parallel Block Implementation
  of Level 3 BLAS for MIMD Vector Processors</EM>, ACM Trans. Math. Softw., 20
  (1994), pp. 178-193.

<P></P><DT><A NAME="demoorvandooren92">22</A>
<DD>
B. D<SMALL>E </SMALL>M<SMALL>OOR AND </SMALL>P. V<SMALL>AN </SMALL>D<SMALL>OOREN</SMALL>, <EM>Generalization of the singular value
  and QR decompositions</EM>, SIAM J. Matrix Anal. Appl., 13 (1992),
  pp. 993-1014.

<P></P><DT><A NAME="deiftdemmellitomei">23</A>
<DD>
P. D<SMALL>EIFT, </SMALL>J. W. D<SMALL>EMMEL, </SMALL>L.-C. L<SMALL>I, AND </SMALL>C. T<SMALL>OMEI</SMALL>, <EM>The bidiagonal
  singular values decomposition and Hamiltonian mechanics</EM>, SIAM J. Numer.
  Anal., 28 (1991), pp. 1463-1516.
<BR>(LAPACK Working Note #11).

<P></P><DT><A NAME="demmel84">24</A>
<DD>
J. D<SMALL>EMMEL</SMALL>, <EM>Underflow and the reliability of numerical software</EM>,
  SIAM J. Sci. Stat. Comput., 5 (1984), pp. 887-919.

<P></P><DT><A NAME="demmelMA221">25</A>
<DD>
J. D<SMALL>EMMEL</SMALL>, <EM>Applied Numerical
  Linear Algebra</EM>, SIAM, Philadelphia, PA, 1997.

<P></P><DT><A NAME="demmel83">26</A>
<DD>
J. W. D<SMALL>EMMEL</SMALL>, <EM>The condition number of equivalence transformations
  that block diagonalize matrix pencils</EM>, SIAM J. Numer. Anal., 20 (1983),
  pp. 599-610.

<P></P><DT><A NAME="Demmel-Higham-Wnote22">27</A>
<DD>
J. W. D<SMALL>EMMEL AND </SMALL>N. J. H<SMALL>IGHAM</SMALL>, <EM>Stability of block algorithms with
  fast level 3 BLAS</EM>, ACM Trans. Math. Softw., 18 (1992), pp. 274-291.
<BR>(Also LAPACK Working Note #22).

<P></P><DT><A NAME="demmelhigham1">28</A>
<DD>
J. W. D<SMALL>EMMEL AND </SMALL>N. J. H<SMALL>IGHAM</SMALL>, <EM>Improved error
  bounds for underdetermined systems solvers</EM>, SIAM J. Matrix Anal. Appl., 14
  (1993), pp. 1-14.
<BR>(Also LAPACK Working Note #23).

<P></P><DT><A NAME="demmelkagstrom87">29</A>
<DD>
J. W. D<SMALL>EMMEL AND </SMALL>B. K<SMALL>&#197;GSTR&#168;OM</SMALL>, <EM>Computing stable
  eigendecompositions of matrix pencils</EM>, Lin. Alg. Appl., 88/89 (1987),
  pp. 139-186.

<P></P><DT><A NAME="demmelkagstrom93a">30</A>
<DD>
J. W. D<SMALL>EMMEL AND </SMALL>B. K<SMALL>&#197;GSTR&#168;OM</SMALL>, <EM>The generalized Schur
  decomposition of an arbitrary pencil <IMG
 WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img56.png"
 ALT="$A - \lambda B$">:
robust software with
  error bounds and applications, part I: Theory and algorithms</EM>, ACM Trans.
  Math. Softw., 19 (1993), pp. 160-174.

<P></P><DT><A NAME="demmelkagstrom93b">31</A>
<DD>
J. W. D<SMALL>EMMEL AND </SMALL>B. K<SMALL>&#197;GSTR&#168;OM</SMALL>, <EM>The generalized
  Schur decomposition of an arbitrary pencil <IMG
 WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img56.png"
 ALT="$A - \lambda B$">:
robust
  software with error bounds and applications, part II: Software and
  applications</EM>, ACM Trans. Math. Softw., 19 (1993), pp. 175-201.

<P></P><DT><A NAME="demmelkahan">32</A>
<DD>
J. W. D<SMALL>EMMEL AND </SMALL>W. K<SMALL>AHAN</SMALL>, <EM>Accurate singular values of bidiagonal
  matrices</EM>, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 873-912.
<BR>(Also LAPACK Working Note #3).

<P></P><DT><A NAME="demmelli93">33</A>
<DD>
J. W. D<SMALL>EMMEL AND </SMALL>X. L<SMALL>I</SMALL>, <EM>Faster numerical algorithms via exception
  handling</EM>, IEEE Trans. Comp., 43 (1994), pp. 983-992.
<BR>(Also LAPACK Working Note #59).

<P></P><DT><A NAME="demmelveselic">34</A>
<DD>
J. W. D<SMALL>EMMEL AND </SMALL>K. V<SMALL>ESELI&#180;C</SMALL>, <EM>Jacobi's method is more accurate
  than QR</EM>, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 1204-1246.
<BR>(Also LAPACK Working Note #15).

<P></P><DT><A NAME="holygrail">35</A>
<DD>
I. D<SMALL>HILLON</SMALL>, <EM>A new <B><I>O</I>(<I>n</I><SUP>2</SUP>)</B> algorithm for the symmetric tridiagonal
  eigenvalue/eigenvector problem</EM>, Computer Science Division Technical
  Report no. UCB/CSD-97-971, University of California, Berkeley, CA, May
  1997.

<P></P><DT><A NAME="dhillonparlett99b">36</A>
<DD>
I. S. D<SMALL>HILLON AND </SMALL>B. N. P<SMALL>ARLETT</SMALL>, <EM>Orthogonal eigenvectors and
  relative gaps</EM>, June 1999.
<BR>to appear.

<P></P><DT><A NAME="lawn81">37</A>
<DD>
J. D<SMALL>ONGARRA AND </SMALL>S. O<SMALL>STROUCHOV</SMALL>, <EM>Quick installation guide for LAPACK
  on unix systems</EM>, Computer Science Dept. Technical Report
  CS-94-249, University of Tennessee, Knoxville, TN, September 1994.
<BR>(LAPACK Working Note #81).

<P></P><DT><A NAME="dongarra79">38</A>
<DD>
J. J. D<SMALL>ONGARRA, </SMALL>J. R. B<SMALL>UNCH, </SMALL>C. B. M<SMALL>OLER, AND </SMALL>G. W. S<SMALL>TEWART</SMALL>, <EM>LINPACK
  Users' Guide</EM>, Society for Industrial and Applied Mathematics, Philadelphia,
  PA, 1979.

<P></P><DT><A NAME="blas3alg">39</A>
<DD>
J. J. D<SMALL>ONGARRA, </SMALL>J. D<SMALL>U </SMALL>C<SMALL>ROZ, </SMALL>I. S. D<SMALL>UFF, AND </SMALL>S. H<SMALL>AMMARLING</SMALL>, <EM>Algorithm
  679: A set of Level 3 Basic Linear Algebra Subprograms</EM>, ACM
  Trans. Math. Soft., 16 (1990), pp. 18-28.

<P></P><DT><A NAME="blas3">40</A>
<DD>
J. J. D<SMALL>ONGARRA, </SMALL>J. D<SMALL>U </SMALL>C<SMALL>ROZ, </SMALL>I. S. D
<SMALL>UFF, AND </SMALL>S. H<SMALL>AMMARLING</SMALL>, <EM>A set of Level 3
  Basic Linear Algebra Subprograms</EM>, ACM Trans. Math. Soft., 16
  (1990), pp. 1-17.

<P></P><DT><A NAME="blas2alg">41</A>
<DD>
J. J. D<SMALL>ONGARRA, </SMALL>J. D<SMALL>U </SMALL>C<SMALL>ROZ, </SMALL>S. H<SMALL>AMMARLING, AND </SMALL>R. J. H<SMALL>ANSON</SMALL>, <EM>  Algorithm 656: An extended set of FORTRAN Basic Linear Algebra
  Subroutines</EM>, ACM Trans. Math. Soft., 14 (1988), pp. 18-32.

<P></P><DT><A NAME="blas2">42</A>
<DD>
J. J. D<SMALL>ONGARRA, </SMALL>J. D<SMALL>U </SMALL>C<SMALL>ROZ, </SMALL>S. H<SM
ALL>AMMARLING, AND </SMALL>R. J. H<SMALL>ANSON</SMALL>, <EM>An extended set of
  FORTRAN basic linear algebra subroutines</EM>, ACM Trans. Math. Soft., 14
  (1988), pp. 1-17.

<P></P><DT><A NAME="dongarraetal2">43</A>
<DD>
J. J. D<SMALL>ONGARRA, </SMALL>I. S. D<SMALL>UFF, </SMALL>D. C. S<SMALL>ORENSEN, AND </SMALL>H. A. V<SMALL>AN DER </SMALL>V<SMALL>ORST</SMALL>, <EM>  Numerical Linear Algebra for High-Performance Computers</EM>, Society for
  Industrial and Applied Mathematics, Philadelphia, PA, 1998.

<P></P><DT><A NAME="Dongarra87e">44</A>
<DD>
J. J. D<SMALL>ONGARRA AND </SMALL>E. G<SMALL>ROSSE</SMALL>, <EM>Distribution of mathematical software
  via electronic mail</EM>, Communications of the ACM, 30 (1987), pp. 403-407.

<P></P><DT><A NAME="Dongarra84a">45</A>
<DD>
J. J. D<SMALL>ONGARRA, </SMALL>F. G. G<SMALL>USTAFSON, AND </SMALL>A. K<SMALL>ARP</SMALL>, <EM>Implementing linear
  algebra algorithms for dense matrices on a vector pipeline machine</EM>, SIAM
  Review, 26 (1984), pp. 91-112.

<P></P><DT><A NAME="lapwn2">46</A>
<DD>
J. J. D<SMALL>ONGARRA, </SMALL>S. H<SMALL>AMMARLING, AND </SMALL>D. C. S<SMALL>ORENSEN</SMALL>, <EM>Block reduction
  of matrices to condensed forms for eigenvalue computations</EM>, JCAM, 27
  (1989), pp. 215-227.
<BR>(LAPACK Working Note #2).

<P></P><DT><A NAME="lapwn27">47</A>
<DD>
J. D<SMALL>U </SMALL>C<SMALL>ROZ AND </SMALL>N. J. H<SMALL>IGHAM</SMALL>, <EM>Stability of methods for matrix
  inversion</EM>, IMA J. Numer. Anal., 12 (1992), pp. 1-19.
<BR>(Also LAPACK Working Note #27).

<P></P><DT><A NAME="lapwn21">48</A>
<DD>
J. D<SMALL>U </SMALL>C<SMALL>ROZ, </SMALL>P. J. D. M<SMALL>AYES, AND </SMALL>G. R<SMALL>ADICATI DI </SMALL>B<SMALL>ROZOLO</SMALL>, <EM>  Factorizations of band matrices using Level 3 BLAS</EM>, Computer Science
  Dept. Technical Report CS-90-109, University of Tennessee, Knoxville,
  TN, 1990.
<BR>(LAPACK Working Note #21).

<P></P><DT><A NAME="dubrulle">49</A>
<DD>
A. D<SMALL>UBRULLE</SMALL>, <EM>The multishift QR algorithm: is it worth the
  trouble?</EM>, Palo Alto Scientific Center Report G320-3558x, IBM Corp., 1530
  Page Mill Road, Palo Alto, CA 94304, 1991.

<P></P><DT><A NAME="elden">50</A>
<DD>
L. E<SMALL>LD&#180;EN</SMALL>, <EM>Perturbation theory for the least squares problem with
  linear equality constraints</EM>, SIAM J. Numer. Anal., 17 (1980),
  pp. 338-350.

<P></P><DT><A NAME="fernandoparlett">51</A>
<DD>
V. F<SMALL>ERNANDO AND </SMALL>B. P<SMALL>ARLETT</SMALL>, <EM>Accurate singular values and
  differential qd algorithms</EM>, Numerisch Math., 67 (1994), pp. 191-229.

<P></P><DT><A NAME="gallivanetal">52</A>
<DD>
K. A. G<SMALL>ALLIVAN, </SMALL>R. J. P<SMALL>LEMMONS, AND </SMALL>A. H. S<SMALL>AMEH</SMALL>, <EM>Parallel algorithms
  for dense linear algebra computations</EM>, SIAM Review, 32 (1990),
  pp. 54-135.

<P></P><DT><A NAME="gantmacher">53</A>
<DD>
F. G<SMALL>ANTMACHER</SMALL>, <EM>The Theory of Matrices, vol. II (transl.)</EM>,
  Chelsea, New York, 1959.

<P></P><DT><A NAME="Garbow77">54</A>
<DD>
B. S. G<SMALL>ARBOW, </SMALL>J. M. B<SMALL>OYLE, </SMALL>J. J. D<SMALL>ONGARRA, AND </SMALL>C. B. M<SMALL>OLER</SMALL>, <EM>Matrix
  Eigensystem Routines - EISPACK Guide Extension</EM>, vol. 51 of Lecture Notes
  in Computer Science, Springer-Verlag, Berlin, 1977.

<P></P><DT><A NAME="GVL2">55</A>
<DD>
G. G<SMALL>OLUB AND </SMALL>C. F. V<SMALL>AN </SMALL>L<SMALL>OAN</SMALL>, <EM>Matrix Computations</EM>, Johns Hopkins
  University Press, Baltimore, MD, third ed., 1996.

<P></P><DT><A NAME="greenbaumdongarra">56</A>
<DD>
A. G<SMALL>REENBAUM AND </SMALL>J. J. D<SMALL>ONGARRA</SMALL>, <EM>Experiments with QL/QR methods
  for the symmetric tridiagonal eigenproblem</EM>, Computer Science Dept.
  Technical Report CS-89-92, University of Tennessee, Knoxville,TN, 1989.
<BR>(LAPACK Working Note #17).

<P></P><DT><A NAME="gueisenstat">57</A>
<DD>
M. G<SMALL>U AND </SMALL>S. E<SMALL>ISENSTAT</SMALL>, <EM>A stable algorithm for the rank-1
  modification of the symmetric eigenproblem</EM>, Computer Science Department
  Report YALEU/DCS/RR-916, Yale University, New Haven, CT, 1992.

<P></P><DT><A NAME="gueisenstat3">58</A>
<DD>
M. G<SMALL>U AND </SMALL>S. E<SMALL>ISENSTAT</SMALL>, <EM>A divide-and-conquer
  algorithm for the bidiagonal SVD</EM>, SIAM J. Mat. Anal. Appl., 16 (1995),
  pp. 79-92.

<P></P><DT><A NAME="hager84">59</A>
<DD>
W. W. H<SMALL>AGER</SMALL>, <EM>Condition estimators</EM>, SIAM J. Sci. Stat. Comput., 5
  (1984), pp. 311-316.

<P></P><DT><A NAME="hammarling86">60</A>
<DD>
S. H<SMALL>AMMARLING</SMALL>, <EM>The numerical solution of the general
  Gauss-Markov linear model</EM>, in Mathematics in Signal Processing,
  T. S. <EM>et al.</EM>. Durani, ed., Clarendon Press, Oxford, UK, 1986.

<P></P><DT><A NAME="higham3">61</A>
<DD>
N. J. H<SMALL>IGHAM</SMALL>, <EM>Efficient algorithms for computing the condition
  number of a tridiagonal matrix</EM>, SIAM J. Sci. Stat. Comput., 7 (1986),
  pp. 150-165.

<P></P><DT><A NAME="higham1">62</A>
<DD>
N. J. H<SMALL>IGHAM</SMALL>, <EM>A survey of
  condition number estimation for triangular matrices</EM>, SIAM Review, 29
  (1987), pp. 575-596.

<P></P><DT><A NAME="nick2">63</A>
<DD>
N. J. H<SMALL>IGHAM</SMALL>, <EM>FORTRAN codes for
  estimating the one-norm of a real or complex matrix, with applications to
  condition estimation</EM>, ACM Trans. Math. Softw., 14 (1988), pp. 381-396.

<P></P><DT><A NAME="higham89">64</A>
<DD>
N. J. H<SMALL>IGHAM</SMALL>, <EM>Algorithm 674:
  FORTRAN codes for estimating the one-norm of a real or complex matrix, with
  applications to condition estimation</EM>, ACM Trans. Math. Softw., 15 (1989),
  p. 168.

<P></P><DT><A NAME="higham90">65</A>
<DD>
N. J. H<SMALL>IGHAM</SMALL>, <EM>Experience with a
  matrix norm estimator</EM>, SIAM J. Sci. Stat. Comput., 11 (1990),
  pp. 804-809.

<P></P><DT><A NAME="higham93">66</A>
<DD>
N. J. H<SMALL>IGHAM</SMALL>, <EM>Perturbation theory
  and backward error for <B><I>AX</I>-<I>XB</I>=<I>C</I></B></EM>, BIT, 33 (1993), pp. 124-136.

<P></P><DT><A NAME="higham96">67</A>
<DD>
N. J. H<SMALL>IGHAM</SMALL>, <EM>Accuracy and
  Stability of Numerical Algorithms</EM>, SIAM, Philadelphia, PA, 1996.

<P></P><DT><A NAME="hussledermantsaozhang93">68</A>
<DD>
S. H<SMALL>USS-</SMALL>L<SMALL>EDERMAN, </SMALL>A. T<SMALL>SAO, AND </SMALL>G. Z<SMALL>HANG</SMALL>, <EM>A parallel implementation
  of the invariant subspace decomposition algorithm for dense symmetric
  matrices</EM>, in Proceedings of the Sixth SIAM Conference on Parallel Processing
  for Scientific Computing, Society for Industrial and Applied Mathematics,
  1993, pp. 367-374.

<P></P><DT><A NAME="jessupsorensen">69</A>
<DD>
E. J<SMALL>ESSUP AND </SMALL>D. S<SMALL>ORENSEN</SMALL>, <EM>A parallel algorithm for computing the
  singular value decomposition of a matrix</EM>, Mathematics and Computer Science
  Division Report ANL/MCS-TM-102, Argonne National Laboratory, Argonne, IL,
  December 1987.

<P></P><DT><A NAME="kagstrom93">70</A>
<DD>
B. K<SMALL>&#197;GSTR&#214;M</SMALL>, <EM>A direct method for reordering eigenvalues in
  the generalized real Schur form of a regular matrix pair (a,b)</EM>, in Linear
  Algebra for Large Scale and Real-Time Applications, Kluwer Academic
  Publishers, 1993, pp. 195-218.

<P></P><DT><A NAME="kagstrom94">71</A>
<DD>
B. K<SMALL>&#197;GSTR&#214;M</SMALL>, <EM>A perturbation
  analysis of the generalized sylvester equation</EM>, SIAM J. Matrix Anal.
  Appl., 15 (1994), pp. 1045-1060.

<P></P><DT><A NAME="kagstrom95b">72</A>
<DD>
B. K<SMALL>&#197;GSTR&#168;OM, </SMALL>P. L<SMALL>ING, AND </SMALL>C. V. L<SMALL>OAN</SMALL>, <EM>GEMM-based level 3
  BLAS: High-performance model implementations and performance evaluation
  benchmark</EM>, Tech. Rep. UMINF 95-18, Department of Computing Science, Ume&#229;
  University, 1995.
<BR>Submitted to ACM Trans. Math. Softw.

<P></P><DT><A NAME="kagstromporomaa94a">73</A>
<DD>
B. K<SMALL>&#197;GSTR&#168;OM AND </SMALL>P. P<SMALL>OROMAA</SMALL>, <EM>Computing eigenspaces with
  specified eigenvalues of a regular matrix pair <B>(<I>A</I>,<I>B</I>)</B> and condition
  estimation: Theory, algorithms and software</EM>, Tech. Rep. UMINF 94.04,
  Department of Computing Science, Ume&#229; University, 1994.

<P></P><DT><A NAME="kagstromporomaa93a">74</A>
<DD>
B. K<SMALL>&#197;GSTR&#214;M AND </SMALL>P. P<SMALL>OROMAA</SMALL>, <EM>LAPACK-style algorithms and
  software for solving the generalized Sylvester equation and estimating the
  separation between regular matrix pairs</EM>, ACM Trans. Math. Softw., 22
  (1996), pp. 78-103.

<P></P><DT><A NAME="kagstromwestin89">75</A>
<DD>
B. K<SMALL>&#197;GSTR&#214;M AND </SMALL>L. W<SMALL>ESTIN</SMALL>, <EM>Generalized schur methods with
  condition estimators for solving the generalized Sylvester equation</EM>,
  IEEE Trans. Autom. Contr., 34 (1989), pp. 745-751.

<P></P><DT><A NAME="kato">76</A>
<DD>
T. K<SMALL>ATO</SMALL>, <EM>Perturbation Theory for Linear Operators</EM>, Springer-Verlag,
  Berlin, 2 ed., 1980.

<P></P><DT><A NAME="vbandr">77</A>
<DD>
L. K<SMALL>AUFMAN</SMALL>, <EM>Banded eigenvalue solvers on vector machines</EM>, ACM
  Trans. Math. Softw., 10 (1984), pp. 73-86.

<P></P><DT><A NAME="blas1">78</A>
<DD>
C. L. L<SMALL>AWSON, </SMALL>R. J. H<SMALL>ANSON, </SMALL>D. K<SMALL>INCAID, AND </SMALL>F. T. K<SMALL>ROGH</SMALL>, <EM>Basic
  linear algebra subprograms for Fortran usage</EM>, ACM Trans. Math. Soft., 5
  (1979), pp. 308-323.

<P></P><DT><A NAME="lawn72">79</A>
<DD>
R. L<SMALL>EHOUCQ</SMALL>, <EM>The computation of elementary unitary matrices</EM>,
  Computer Science Dept. Technical Report CS-94-233, University of
  Tennessee, Knoxville, TN, 1994.
<BR>(Also LAPACK Working Note 72).

<P></P><DT><A NAME="paige79b">80</A>
<DD>
C. P<SMALL>AIGE</SMALL>, <EM>Computer solution and perturbation analysis of generalized
  linear least squares problems</EM>, Math. of Comput., 33 (1979), pp. 171-183.

<P></P><DT><A NAME="paige79">81</A>
<DD>
C. P<SMALL>AIGE</SMALL>, <EM>Fast numerically
  stable computations for generalized linear least squares problems
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<P></P><DT><A NAME="paige84">82</A>
<DD>
C. P<SMALL>AIGE</SMALL>, <EM>A note on a result
  of sun ji-guang: sensitivity of the cs and gsv decomposition</EM>, SIAM J. Num.
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<P></P><DT><A NAME="paige86a">83</A>
<DD>
C. P<SMALL>AIGE</SMALL>, <EM>Computing the
  generalized singular value decomposition</EM>, SIAM J. Sci. Stat., 7 (1986),
  pp. 1126-1146.

<P></P><DT><A NAME="paige90">84</A>
<DD>
C. P<SMALL>AIGE</SMALL>, <EM>Some aspects of
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<P></P><DT><A NAME="parlett">85</A>
<DD>
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<DD>
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<BR>to appear.

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<DD>
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<BR>to appear.

<P></P><DT><A NAME="Toolpack">88</A>
<DD>
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<P></P><DT><A NAME="rutter">89</A>
<DD>
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<BR>(Also LAPACK Working Note 69).

<P></P><DT><A NAME="Schreiber87a">90</A>
<DD>
R. S<SMALL>CHREIBER AND </SMALL>C. F. V<SMALL>AN </SMALL>L<SMALL>OAN</SMALL>, <EM>A storage efficient WY
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<P></P><DT><A NAME="slapnicar1">91</A>
<DD>
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<P></P><DT><A NAME="Smith76">92</A>
<DD>
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  1976.

<P></P><DT><A NAME="stewart72">93</A>
<DD>
G. W. S<SMALL>TEWART</SMALL>, <EM>On the sensitivity of the eigenvalue problem 
<!-- MATH
 $Ax=
  \lambda Bx$
 -->
<IMG
 WIDTH="85" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img176.png"
 ALT="$Ax = \lambda Bx$"></EM>, SIAM J. Num. Anal., 9 (1972), pp. 669-686.

<P></P><DT><A NAME="stewart73">94</A>
<DD>
G. W. S<SMALL>TEWART</SMALL>, <EM>Error and
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<P></P><DT><A NAME="stewartsun90">95</A>
<DD>
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  Press, New York, 1990.

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<DD>
J. G. S<SMALL>UN</SMALL>, <EM>Perturbation analysis for the generalized singular value
  problem</EM>, SIAM J. Num. Anal., 20 (1983), pp. 611-625.

<P></P><DT><A NAME="vandooren79">97</A>
<DD>
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<DD>
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  Anal., 16 (1979), pp. 216-222.

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<DD>
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  Hermitian matrices</EM>, Linear Algebra and Appl., 195 (1993), pp. 81-116.

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<DD>
R. C. W<SMALL>ARD</SMALL>, <EM>Balancing the generalized eigenvalue problem</EM>, SIAM J.
  Sci. Stat. Comput., 2 (1981), pp. 141-152.

<P></P><DT><A NAME="watkinselsner">101</A>
<DD>
D. W<SMALL>ATKINS AND </SMALL>L. E<SMALL>LSNER</SMALL>, <EM>Convergence of algorithms of decomposition
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  Software</EM>.
<BR>
  http://www.supercomp.org/sc98/TechPapers/sc98_FullAbstracts/Whaley814/INDEX
.HTM, 1998.
<BR>Winner, best paper in the systems category, SC98: High Performance
  Networking and Computing.

<P></P><DT><A NAME="wilkinson1">103</A>
<DD>
J. H. W<SMALL>ILKINSON</SMALL>, <EM>The Algebraic Eigenvalue Problem</EM>, Oxford
  University Press, Oxford, UK, 1965.

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<DD>
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<DD>
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<P></P><DT><A NAME="wilkinson3">106</A>
<DD>
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</DL>
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<P>

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<BR><HR>
<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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