NumericalAnalysisSolution2

HW2_Sol.tex

\documentclass[12pt]{article}
\usepackage{amsmath}
\title{Numerical Analysis Homework 2 Solution}
\begin{document}

\begin{enumerate}
    \item 
        \begin{enumerate}
            \item Find $f(0.43)$
            \begin{enumerate}
                \item Pick the two nearest points, $x_0=0.5, x_1=0.25$ \\
                    \begin{align*}
                        P_1(0.43) &= 2.71828 \times \frac{0.43-0.25}{0.5-0.25} + 1.64872\times \frac{0.43-0.5}{0.25-0.5}  \\
                                  &= 2.4188 
                    \end{align*}
                \item Pick the three nearest points, $x_0=0.5, x_1=0.25, x_2=0.75$ \\
                    \begin{align*}
                        P_2(0.43) &= 2.71828 \times \frac{(0.43-0.25)(0.43-0.75)}{(0.5-0.25)(0.5-0.75)} \\
                                  &\qquad + 1.64872\times \frac{(0.43-0.5)(0.43-0.75)}{(0.25-0.5)(0.25-0.75)} \\
                                  &\qquad + 4.48169 \times \frac{(0.43-0.5)(0.43-0.25)}{(0.75-0.5)(0.75-0.25)} \\
                                  &= 2.34886
                    \end{align*}
                \item Pick the all points, $X_0=0.5, x_1=0.25, x_2=0.75, x_3=0$\\
                    \begin{align*}
                        P_3(0.43) &= 2.71828 \times \frac{(0.43-0.25)(0.43-0.75)(0.43-0)}{(0.5-0.25)(0.5-0.75)(0.5-0)} \\
                                  &\qquad + 1.64872\times \frac{(0.43-0.5)(0.43-0.75)(0.43-0)}{(0.25-0.5)(0.25-0.75)(0.25-0)} \\
                                  &\qquad + 4.48169 \times \frac{(0.43-0.5)(0.43-0.25)(0.43-0)}{(0.75-0.5)(0.75-0.25)(0.75-0)} \\
                                  &\qquad + 1 \times \frac{(0.43-0.5)(0.43-0.25)(0.43-0.75)}{(0-0.5)(0-0.25)(0-0.75)} \\
                                  &= 2.36060
                    \end{align*}
            \end{enumerate}

            \newpage

            \item find $f(0.18)$ 
                \begin{enumerate}
                    \item Pick the two nearest points, $x_0=0.1, x_1=0.2$ \\
                        \begin{align*}
                            P_1(0.18) &= -0.29004986 \times \frac{0.18-0.2}{0.1-0.2} + (-0.56079734)\times \frac{0.18-0.1}{0.2-0.1}  \\
                                      &= -0.50665
                        \end{align*}
                    \item Pick the three nearest points, $x_0=0.1, x_1=0.2, x_2=0.3$ \\
                        \begin{align*}
                            P_2(0.18) &= -0.29004986 \times \frac{(0.18-0.2)(0.18-0.3)}{(0.1-0.2)(0.1-0.3)} \\
                                      &\qquad + (-0.56079734) \times \frac{(0.18-0.1)(0.18-0.3)}{(0.2-0.1)(0.2-0.3)} \\
                                      &\qquad + (-0.81401972) \times \frac{(0.18-0.1)(0.18-0.2)}{(0.3-0.1)(0.3-0.2)} \\
                                      &= -0.50805
                        \end{align*}
                    \item Pick the all points, $X_0=0.1, x_1=0.2, x_2=0.3, x_3=0.4$\\
                        \begin{align*}
                            P_3(0.18) &= -0.29004986 \times \frac{(0.18-0.2)(0.18-0.3)(0.18-0.4)}{(0.1-0.2)(0.1-0.3)(0.1-0.4)} \\
                                      &\qquad + (-0.56079734)\times \frac{(0.18-0.1)(0.18-0.3)(0.18-0.4)}{(0.2-0.1)(0.2-0.3)(0.2-0.4)} \\
                                      &\qquad + (-0.81401972) \times \frac{(0.18-0.1)(0.18-0.2)(0.18-0.4)}{(0.3-0.1)(0.3-0.2)(0.3-0.4)} \\
                                      &\qquad + (-1.0526302) \times \frac{(0.18-0.1)(0.18-0.2)(0.18-0.3)}{(0.4-0.1)(0.4-0.2)(0.4-0.3)} \\
                                      &= -0.50814
                        \end{align*}
                \end{enumerate}


        \end{enumerate}
\end{enumerate}

\end{document}

makefile

compile:
	latex ${input}.tex
	xdvi ${input}.dvi

export:
	latex ${input}.tex
	dvips ${input}.dvi
	ps2pdf ${input}.ps


remove: 
	rm ${input}.aux ${input}.log ${input}.dvi