\relax \citation{La} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.1}{Explicit Runge-Kutta Methods}}{1}} \newlabel{ode}{{1.1}{1}} \citation{CdB} \citation{La} \citation{HW} \citation{Bu} \citation{HJ} \newlabel{RKformula}{{2}{2}} \citation{La} \citation{RR} \citation{Ku} \citation{HJ} \citation{Fe} \citation{DP} \citation{PP} \@writefile{toc}{\contentsline {subsubsection}{\tocsubsubsection {}{1.1.1}{Previous work on optimization of four-stage fourth-order Runge-Kutta formulas}}{3}} \newlabel{tableau}{{5}{3}} \@writefile{toc}{\contentsline {subsubsection}{\tocsubsubsection {}{1.1.2}{Previous work on optimization of (4,5) Runge-Kutta formula pairs}}{3}} \citation{SR} \citation{Sh1} \citation{Fe} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.2}{ODE Software and the Control of Local Error}}{4}} \citation{Sh2} \citation{Sh1} \citation{Bu} \citation{Ro} \citation{AL} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.3}{Gr\"obner Basis Techniques for Deriving and Optimizing Runge-Kutta Formulas}}{5}} \citation{Bo} \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Design and Optimization of a Four-Stage Fourth-Order Runge-Kutta Formula}}{6}} \newlabel{fourfour}{{2}{6}} \newlabel{minfive}{{2.1}{7}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.1}{Minimizing the Sum of Squares of the Fifth-Order Truncation Error Terms}}{7}} \citation{Sh1} \citation{Wa} \citation{Sh2} \citation{CdB} \citation{SG} \newlabel{matlab}{{2.2}{8}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.2}{Testing of Runge-Kutta Formulas via \textsc {MATLAB}}}{8}} \citation{Sh2} \citation{Bo} \citation{HW} \newlabel{elevenfive}{{2.3}{9}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.3}{Considering Step-Doubling of a Fourth-Order Formula as an Eleven-Stage Fifth-Order Formula}}{9}} \newlabel{ratio}{{8}{10}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.4}{A Relationship Among the Double-Step Order Conditions}}{11}} \newlabel{trees}{{2.4.1}{11}} \newlabel{final}{{9}{11}} \newlabel{adjusted}{{10}{11}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Future Plans}}{12}} \newlabel{fourfive}{{3.1}{12}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{Design and Optimization of a Seven-Stage (4,5) Formula Pair}}{12}} \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Acknowledgements}}{12}} \@writefile{toc}{\contentsline {section}{\tocsection {Appendix}{A}{Fifth-Order Conditions Reduced by Gr\"obner Basis for a Fourth-Order Runge-Kutta Formula}}{13}} \newlabel{trunc}{{A}{13}} \@writefile{toc}{\contentsline {section}{\tocsection {Appendix}{B}{Error vs. Effort graph for Fourth-Order Runge-Kutta Formulas and ode45}}{14}} \newlabel{err_vs_eff}{{B}{14}} \@writefile{toc}{\contentsline {section}{\tocsection {Appendix}{C}{Matlab Code for Fourth-Order Runge-Kutta Test Program}}{14}} \newlabel{rk44auto}{{C}{14}} \bibcite{AL}{1} \bibcite{Bo}{2} \bibcite{Bu}{3} \bibcite{CdB}{4} \bibcite{DP}{5} \bibcite{Fe}{6} \bibcite{HW}{7} \bibcite{HJ}{8} \bibcite{Ku}{9} \bibcite{La}{10} \bibcite{LS}{11} \bibcite{PP}{12} \bibcite{RR}{13} \bibcite{Ro}{14} \bibcite{Sh1}{15} \bibcite{Sh2}{16} \bibcite{SG}{17} \bibcite{SR}{18} \bibcite{Wa}{19} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{68.86317pt} \newlabel{tocindent1}{77.67142pt} \newlabel{tocindent2}{34.5pt} \newlabel{tocindent3}{0pt} \@writefile{toc}{\contentsline {section}{\tocsection {Appendix}{}{References}}{21}}