\relax 
\providecommand\hyper@newdestlabel[2]{}
\@writefile{toc}{\contentsline {chapter}{\numberline {4}Experiments}{12}{chapter.4}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{loa}{\addvspace {10\p@ }}
\newlabel{ch: experiments}{{4}{12}{Experiments}{chapter.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.1}QR}{13}{section.4.1}\protected@file@percent }
\newlabel{fig:QR-error-lambda}{{4.1a}{13}{\textit {QR decomposition errors and backward stability for different} $\lambda $}{figure.caption.3}{}}
\newlabel{sub@fig:QR-error-lambda}{{a}{13}{\textit {QR decomposition errors and backward stability for different} $\lambda $}{figure.caption.3}{}}
\newlabel{fig:QR-forward}{{4.1b}{13}{\textit {QR factorization forward stability on Q and R for different} $\lambda $}{figure.caption.3}{}}
\newlabel{sub@fig:QR-forward}{{b}{13}{\textit {QR factorization forward stability on Q and R for different} $\lambda $}{figure.caption.3}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.1}{\ignorespaces \textit  {Errors and scalability of the QR decomposition for different values of} $\lambda $}}{13}{figure.caption.3}\protected@file@percent }
\newlabel{fig:qrtests}{{4.1}{13}{\textit {Errors and scalability of the QR decomposition for different values of} $\lambda $}{figure.caption.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.2}L-BFGS}{14}{section.4.2}\protected@file@percent }
\newlabel{fig:gradnorm-res-rel-ill}{{4.2a}{14}{\textit {Ill-conditioned matrix}}{figure.caption.4}{}}
\newlabel{sub@fig:gradnorm-res-rel-ill}{{a}{14}{\textit {Ill-conditioned matrix}}{figure.caption.4}{}}
\newlabel{fig:gradnorm-res-rel-wellll}{{4.2b}{14}{\textit {Well-conditioned matrix}}{figure.caption.4}{}}
\newlabel{sub@fig:gradnorm-res-rel-wellll}{{b}{14}{\textit {Well-conditioned matrix}}{figure.caption.4}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.2}{\ignorespaces $\left \lVert \nabla f\right \rVert $\textit  {, Residual, Relative Error of L-BFGS execution on ill and well-conditioned matrices}}}{14}{figure.caption.4}\protected@file@percent }
\newlabel{fig:gradnorm-res-rel}{{4.2}{14}{$\norm {\nabla f}$\textit {, Residual, Relative Error of L-BFGS execution on ill and well-conditioned matrices}}{figure.caption.4}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.3}{\ignorespaces \textit  {Line Search algorithms comparison}}}{15}{figure.caption.5}\protected@file@percent }
\newlabel{fig:LS-comparison}{{4.3}{15}{\textit {Line Search algorithms comparison}}{figure.caption.5}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.3}Comparison between QR and L-BFGS}{15}{section.4.3}\protected@file@percent }
\newlabel{fig:time-comparison-illcond-n}{{4.4a}{16}{\textit {Ill-conditioned matrix}}{figure.caption.6}{}}
\newlabel{sub@fig:time-comparison-illcond-n}{{a}{16}{\textit {Ill-conditioned matrix}}{figure.caption.6}{}}
\newlabel{fig:time-comparison-wellcond-n}{{4.4b}{16}{\textit {Well-conditioned matrix}}{figure.caption.6}{}}
\newlabel{sub@fig:time-comparison-wellcond-n}{{b}{16}{\textit {Well-conditioned matrix}}{figure.caption.6}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.4}{\ignorespaces \textit  {Time and Memory scalability comparison of QR and L-BFGS on ill and well-conditioned matrices, varying \textbf  {n}}}}{16}{figure.caption.6}\protected@file@percent }
\newlabel{fig:QRvsLBFGS-time-comparison-n}{{4.4}{16}{\textit {Time and Memory scalability comparison of QR and L-BFGS on ill and well-conditioned matrices, varying \textbf {n}}}{figure.caption.6}{}}
\newlabel{fig:time-comparison-illcond-m}{{4.5a}{16}{\textit {Ill-conditioned matrix}}{figure.caption.7}{}}
\newlabel{sub@fig:time-comparison-illcond-m}{{a}{16}{\textit {Ill-conditioned matrix}}{figure.caption.7}{}}
\newlabel{fig:time-comparison-wellcond-m}{{4.5b}{16}{\textit {Well-conditioned matrix}}{figure.caption.7}{}}
\newlabel{sub@fig:time-comparison-wellcond-m}{{b}{16}{\textit {Well-conditioned matrix}}{figure.caption.7}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.5}{\ignorespaces \textit  {Time and Memory scalability comparison of QR and L-BFGS on ill and well-conditioned matrices, varying \textbf  {m}}}}{16}{figure.caption.7}\protected@file@percent }
\newlabel{fig:QRvsLBFGS-time-comparison-m}{{4.5}{16}{\textit {Time and Memory scalability comparison of QR and L-BFGS on ill and well-conditioned matrices, varying \textbf {m}}}{figure.caption.7}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.4}Other Experiments}{17}{section.4.4}\protected@file@percent }
\newlabel{sec:other_experiments}{{4.4}{17}{Other Experiments}{section.4.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.1}The Effect of the Memory Size}{17}{subsection.4.4.1}\protected@file@percent }
\newlabel{fig:error-memory-size-illcond}{{4.6a}{17}{\textit {Ill-conditioned matrix}}{figure.caption.8}{}}
\newlabel{sub@fig:error-memory-size-illcond}{{a}{17}{\textit {Ill-conditioned matrix}}{figure.caption.8}{}}
\newlabel{fig:error-memory-size-wellcond}{{4.6b}{17}{\textit {Well-conditioned matrix}}{figure.caption.8}{}}
\newlabel{sub@fig:error-memory-size-wellcond}{{b}{17}{\textit {Well-conditioned matrix}}{figure.caption.8}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.6}{\ignorespaces \textit  {The effect of the memory size}}}{17}{figure.caption.8}\protected@file@percent }
\newlabel{fig:error-memory-size}{{4.6}{17}{\textit {The effect of the memory size}}{figure.caption.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.2}A Comparison of Quasi-Newton Methods}{18}{subsection.4.4.2}\protected@file@percent }
\newlabel{fig:BFGSvsLBFGS-rate}{{4.7a}{19}{\textit {Convergence rate}}{figure.caption.9}{}}
\newlabel{sub@fig:BFGSvsLBFGS-rate}{{a}{19}{\textit {Convergence rate}}{figure.caption.9}{}}
\newlabel{fig:BFGSvsLBFGS-scalability}{{4.7b}{19}{\textit {Time and memory scalability}}{figure.caption.9}{}}
\newlabel{sub@fig:BFGSvsLBFGS-scalability}{{b}{19}{\textit {Time and memory scalability}}{figure.caption.9}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.7}{\ignorespaces \textit  {BFGS vs L-BFGS}}}{19}{figure.caption.9}\protected@file@percent }
\newlabel{fig:BFGSvsLBFGS-comparison}{{4.7}{19}{\textit {BFGS vs L-BFGS}}{figure.caption.9}{}}
\newlabel{fig:Quasi-newton-dogleg-time}{{4.8a}{20}{\textit {With Dogleg}}{figure.caption.10}{}}
\newlabel{sub@fig:Quasi-newton-dogleg-time}{{a}{20}{\textit {With Dogleg}}{figure.caption.10}{}}
\newlabel{fig:Quasi-newton-no-dogleg-time}{{4.8b}{20}{\textit {Without Dogleg}}{figure.caption.10}{}}
\newlabel{sub@fig:Quasi-newton-no-dogleg-time}{{b}{20}{\textit {Without Dogleg}}{figure.caption.10}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.8}{\ignorespaces \textit  {Quasi-Newton methods running time comparison}}}{20}{figure.caption.10}\protected@file@percent }
\newlabel{fig:Quasi-newton-time}{{4.8}{20}{\textit {Quasi-Newton methods running time comparison}}{figure.caption.10}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.9}{\ignorespaces \textit  {Memory allocation of the different Quasi-Newton methods}}}{20}{figure.caption.11}\protected@file@percent }
\newlabel{fig:Quasi newton comparison memory}{{4.9}{20}{\textit {Memory allocation of the different Quasi-Newton methods}}{figure.caption.11}{}}
\@setckpt{(4) - experiments/experiments}{
\setcounter{page}{21}
\setcounter{equation}{0}
\setcounter{enumi}{3}
\setcounter{enumii}{0}
\setcounter{enumiii}{0}
\setcounter{enumiv}{0}
\setcounter{footnote}{0}
\setcounter{mpfootnote}{0}
\setcounter{part}{0}
\setcounter{chapter}{4}
\setcounter{section}{4}
\setcounter{subsection}{2}
\setcounter{subsubsection}{0}
\setcounter{paragraph}{0}
\setcounter{subparagraph}{0}
\setcounter{figure}{9}
\setcounter{table}{0}
\setcounter{tabx@nest}{0}
\setcounter{listtotal}{0}
\setcounter{listcount}{0}
\setcounter{liststart}{0}
\setcounter{liststop}{0}
\setcounter{citecount}{0}
\setcounter{citetotal}{0}
\setcounter{multicitecount}{0}
\setcounter{multicitetotal}{0}
\setcounter{instcount}{5}
\setcounter{maxnames}{3}
\setcounter{minnames}{3}
\setcounter{maxitems}{3}
\setcounter{minitems}{1}
\setcounter{citecounter}{0}
\setcounter{maxcitecounter}{0}
\setcounter{savedcitecounter}{0}
\setcounter{uniquelist}{0}
\setcounter{uniquename}{0}
\setcounter{refsection}{0}
\setcounter{refsegment}{0}
\setcounter{maxextratitle}{0}
\setcounter{maxextratitleyear}{0}
\setcounter{maxextraname}{0}
\setcounter{maxextradate}{0}
\setcounter{maxextraalpha}{0}
\setcounter{abbrvpenalty}{50}
\setcounter{highnamepenalty}{50}
\setcounter{lownamepenalty}{25}
\setcounter{maxparens}{3}
\setcounter{parenlevel}{0}
\setcounter{blx@maxsection}{0}
\setcounter{mincomprange}{10}
\setcounter{maxcomprange}{100000}
\setcounter{mincompwidth}{1}
\setcounter{afterword}{0}
\setcounter{savedafterword}{0}
\setcounter{annotator}{0}
\setcounter{savedannotator}{0}
\setcounter{author}{0}
\setcounter{savedauthor}{0}
\setcounter{bookauthor}{0}
\setcounter{savedbookauthor}{0}
\setcounter{commentator}{0}
\setcounter{savedcommentator}{0}
\setcounter{editor}{0}
\setcounter{savededitor}{0}
\setcounter{editora}{0}
\setcounter{savededitora}{0}
\setcounter{editorb}{0}
\setcounter{savededitorb}{0}
\setcounter{editorc}{0}
\setcounter{savededitorc}{0}
\setcounter{foreword}{0}
\setcounter{savedforeword}{0}
\setcounter{holder}{0}
\setcounter{savedholder}{0}
\setcounter{introduction}{0}
\setcounter{savedintroduction}{0}
\setcounter{namea}{0}
\setcounter{savednamea}{0}
\setcounter{nameb}{0}
\setcounter{savednameb}{0}
\setcounter{namec}{0}
\setcounter{savednamec}{0}
\setcounter{translator}{0}
\setcounter{savedtranslator}{0}
\setcounter{shortauthor}{0}
\setcounter{savedshortauthor}{0}
\setcounter{shorteditor}{0}
\setcounter{savedshorteditor}{0}
\setcounter{labelname}{0}
\setcounter{savedlabelname}{0}
\setcounter{institution}{0}
\setcounter{savedinstitution}{0}
\setcounter{lista}{0}
\setcounter{savedlista}{0}
\setcounter{listb}{0}
\setcounter{savedlistb}{0}
\setcounter{listc}{0}
\setcounter{savedlistc}{0}
\setcounter{listd}{0}
\setcounter{savedlistd}{0}
\setcounter{liste}{0}
\setcounter{savedliste}{0}
\setcounter{listf}{0}
\setcounter{savedlistf}{0}
\setcounter{location}{0}
\setcounter{savedlocation}{0}
\setcounter{organization}{0}
\setcounter{savedorganization}{0}
\setcounter{origlocation}{0}
\setcounter{savedoriglocation}{0}
\setcounter{origpublisher}{0}
\setcounter{savedorigpublisher}{0}
\setcounter{publisher}{0}
\setcounter{savedpublisher}{0}
\setcounter{language}{0}
\setcounter{savedlanguage}{0}
\setcounter{origlanguage}{0}
\setcounter{savedoriglanguage}{0}
\setcounter{pageref}{0}
\setcounter{savedpageref}{0}
\setcounter{textcitecount}{0}
\setcounter{textcitetotal}{0}
\setcounter{textcitemaxnames}{0}
\setcounter{biburlbigbreakpenalty}{100}
\setcounter{biburlbreakpenalty}{200}
\setcounter{biburlnumpenalty}{0}
\setcounter{biburlucpenalty}{0}
\setcounter{biburllcpenalty}{0}
\setcounter{smartand}{1}
\setcounter{bbx:relatedcount}{0}
\setcounter{bbx:relatedtotal}{0}
\setcounter{cbx@tempcnta}{0}
\setcounter{cbx@tempcntb}{2}
\setcounter{cbx@tempcntc}{0}
\setcounter{cbx@tempcntd}{-1}
\setcounter{float@type}{4}
\setcounter{parentequation}{0}
\setcounter{ALG@line}{0}
\setcounter{ALG@rem}{0}
\setcounter{ALG@nested}{0}
\setcounter{ALG@Lnr}{2}
\setcounter{ALG@blocknr}{10}
\setcounter{ALG@storecount}{0}
\setcounter{ALG@tmpcounter}{0}
\setcounter{thmt@dummyctr}{2}
\setcounter{nlinenum}{0}
\setcounter{caption@flags}{2}
\setcounter{continuedfloat}{0}
\setcounter{subfigure}{0}
\setcounter{subtable}{0}
\setcounter{section@level}{0}
\setcounter{Item}{3}
\setcounter{Hfootnote}{0}
\setcounter{bookmark@seq@number}{0}
\setcounter{AlgoLine}{11}
\setcounter{algocfline}{4}
\setcounter{algocfproc}{4}
\setcounter{algocf}{4}
\setcounter{mlemma}{0}
}