From: hackbard Date: Tue, 28 Jun 2005 15:56:35 +0000 (+0000) Subject: more X-Git-Url: https://hackdaworld.org/gitweb/?a=commitdiff_plain;h=a5ba283663603f254cff004e2c7842f4a63286de;p=lectures%2Flatex.git more --- diff --git a/computational_physics/cp.tex b/computational_physics/cp.tex index 7b52a7a..a320c48 100644 --- a/computational_physics/cp.tex +++ b/computational_physics/cp.tex @@ -116,26 +116,47 @@ $\Rightarrow$ study and implementation of numerical algorithms \end{itemstep} \end{slide}} -\overlays{3}{ -\begin{slide}{warning} +\overlays{4}{ +\begin{slide}{warning - machine accuracy $\epsilon_m$} \begin{itemstep} \item numerical precision of 64-bit floating point \\ - {\small ieee floating point format:} $v = -1^s 2^{-e} m$ - \[ + ieee floating point format: $v = -1^s 2^{-e} m$ + \[ \begin{array}{lll} s: & \textrm{signe} & \textrm{1 bit} \\ m: & \textrm{mantissa} & \textrm{52 bit} \\ e: & \textrm{exponent} & \textrm{11 bit} \\ \end{array} \] - \item foo - \item bar + \item $\epsilon_m$: smallest floating point with $1 + \epsilon_m \neq 1$ \\ + $\epsilon_m \approx 2 \times 10^{-18}$ \hspace{2pt} (roundoff error) + \item $N$ arithmetic operations $\Rightarrow$ error of order $N \epsilon_m$ + \item subtraction of very nearly equal numbers\\ + (difference in few significant low-order bits) \end{itemstep} \end{slide}} -\begin{slide}{} +\overlays{6}{ +\begin{slide}{warning - truncation error} + \begin{itemstep} + \item discrete approximation of continuous quantity + \item truncation error $\equiv$ discrepancy between true answer and practical calculation + \item persists even on hypothetical perfect computer ($\epsilon_m = 0$) + \item machine independent, characteristic of used algorithm + \item numerical analysis: minimizing truncation error + \item unstable method: roundoff error interacting at early stage + \end{itemstep} +\end{slide}} -\end{slide} +\overlays{4}{ +\begin{slide}{warning - recursive functions} + \begin{itemstep} + \item avoid recursive functions! + \verbatiminput{fak1.c} + \item better: + \verbatiminput{fak2.c} + \end{itemstep} +\end{slide}} \begin{slide}{computational techniques} techniques discussed in the talk: @@ -174,4 +195,115 @@ techniques \textcolor{red}{not yet} discussed in the talk:\footnote{if time is a \hspace{6cm} \end{slide} +\overlays{2}{ +\begin{slide}{rough discretization} + \begin{itemstep} + \item example: homogenous field of force $\vec{F} = (0,-mg)$ \\ + \begin{tabular}{ll} + equation of motion: & $\vec{F} = m \vec{a} = m \frac{d^2 \vec{r}}{dt^2}$ \\ + initial condition: & $\vec{r}(t=0) = \vec{r_0} = (x_0,y_0)$ \\ + & $\frac{d \vec{r}}{dt}|_{t=0} = (v_{x_0},v_{y_0})$ \\ + \end{tabular} + \item algorithm using discretized time ($T_{total} = N \tau$): + \begin{tabular}{lll} + $x^1 = x_0;$ & $y^1 = y_0;$ & \\ + $v^1_x = v_{x_0};$ & $v^1_y = v_{y_0};$ & \\ + loop: & $x^2 = x^1 + \tau v^1_x;$ & $y^2 = y^1 + \tau v^1_y;$ \\ + & $v^2_x = v^1_x;$ & $v^2_y = v^1_y + (-mg) \tau;$ \\ + & $x^1 = x^2;$ & $y^1 = y^2$ \\ + & $v^1_x = v^2_x;$ & $v^1_y = v^2_y;$ \\ + \end{tabular} + \end{itemstep} +\end{slide}} + +%\overlays{3}{ +%\begin{slide}{} +% \begin{itemstep} +% \item +% \item +% \item +% \end{itemstep} +%\end{slide}} + +%\overlays{3}{ +%\begin{slide}{} +% \begin{itemstep} +% \item +% \item +% \item +% \end{itemstep} +%\end{slide}} + +%\overlays{3}{ +%\begin{slide}{} +% \begin{itemstep} +% \item +% \item +% \item +% \end{itemstep} +%\end{slide}} + +%\overlays{3}{ +%\begin{slide}{} +% \begin{itemstep} +% \item +% \item +% \item +% \end{itemstep} +%\end{slide}} + +%\overlays{3}{ +%\begin{slide}{} +% \begin{itemstep} +% \item +% \item +% \item +% \end{itemstep} +%\end{slide}} + +%\overlays{3}{ +%\begin{slide}{} +% \begin{itemstep} +% \item +% \item +% \item +% \end{itemstep} +%\end{slide}} + +%\overlays{3}{ +%\begin{slide}{} +% \begin{itemstep} +% \item +% \item +% \item +% \end{itemstep} +%\end{slide}} + +%\overlays{3}{ +%\begin{slide}{} +% \begin{itemstep} +% \item +% \item +% \item +% \end{itemstep} +%\end{slide}} + +%\overlays{3}{ +%\begin{slide}{} +% \begin{itemstep} +% \item +% \item +% \item +% \end{itemstep} +%\end{slide}} + +%\overlays{3}{ +%\begin{slide}{} +% \begin{itemstep} +% \item +% \item +% \item +% \end{itemstep} +%\end{slide}} + \end{document} diff --git a/computational_physics/fak1.c b/computational_physics/fak1.c new file mode 100644 index 0000000..6031845 --- /dev/null +++ b/computational_physics/fak1.c @@ -0,0 +1,4 @@ +int fak(n) { + if(n==1) return 1; + else return n*fak(n-1); +} diff --git a/computational_physics/fak2.c b/computational_physics/fak2.c new file mode 100644 index 0000000..b749ccf --- /dev/null +++ b/computational_physics/fak2.c @@ -0,0 +1,8 @@ +int fak(n) { + int help=1; + while(n>1) { + help=help*n; + n=n-1; + } +return help; +}