Book a Demo!
CoCalc Logo Icon
StoreFeaturesDocsShareSupportNewsAboutPoliciesSign UpSign In
Download
5 views
ubuntu2404
\relax 
\citation{saad2003iterative}
\citation{hestenes1952methods}
\citation{shewchuk1994introduction}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{}\protected@file@percent }
\newlabel{sec:intro}{{1}{1}{}{section.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Mathematical Framework}{2}{}\protected@file@percent }
\newlabel{sec:framework}{{2}{2}{}{section.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Classical Iterative Methods}{2}{}\protected@file@percent }
\newlabel{thm:cg_convergence}{{2.1}{2}{}{theorem.2.1}{}}
\newlabel{eq:cg_bound}{{1}{2}{}{equation.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Accelerated Variants}{2}{}\protected@file@percent }
\newlabel{def:adaptive_cg}{{2.2}{2}{}{theorem.2.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Computational Experiments}{2}{}\protected@file@percent }
\newlabel{sec:experiments}{{3}{2}{}{section.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Test Problem Generation}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Convergence Analysis}{2}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Convergence and accuracy comparison of CG and Adaptive Accelerated CG across varying condition numbers $ \kappa $.}}{3}{}\protected@file@percent }
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:conv_summary}{{1}{3}{}{figure.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Detailed Convergence Behavior}{3}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4}Spectral Analysis}{3}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4}Theoretical Analysis}{3}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Convergence Rate Bounds}{3}{}\protected@file@percent }
\newlabel{thm:acc_bound}{{4.1}{3}{}{theorem.4.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Comprehensive convergence analysis of classical and accelerated conjugate gradient methods: (a) Residual norm convergence for different condition numbers, (b) Iteration count scaling with condition number, (c) Performance improvement quantification, (d) Comparison of theoretical and empirical convergence rates.}}{4}{}\protected@file@percent }
\newlabel{fig:convergence_analysis}{{2}{4}{}{figure.2}{}}
\newlabel{eq:acc_bound}{{5}{4}{}{equation.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Computational Complexity}{4}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Eigenvalue distributions for test matrices with different condition numbers. The acceleration technique shows greatest improvement for matrices with clustered eigenvalue distributions.}}{5}{}\protected@file@percent }
\newlabel{fig:eigenvalue_distributions}{{3}{5}{}{figure.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Practical Considerations}{5}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Implementation Guidelines}{5}{}\protected@file@percent }
\newlabel{alg:adaptive_cg}{{5.1}{5}{}{theorem.5.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Convergence Monitoring}{5}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {6}Conclusion}{5}{}\protected@file@percent }
\bibcite{saad2003iterative}{1}
\bibcite{hestenes1952methods}{2}
\bibcite{shewchuk1994introduction}{3}
\bibcite{golub2013matrix}{4}
\bibcite{nocedal2006numerical}{5}
\bibcite{kelley1995iterative}{6}
\bibcite{barrett1994templates}{7}
\gdef \@abspage@last{6}