From e3e798ccc5c0d0dad3955c4d24bd361871ece335 Mon Sep 17 00:00:00 2001 From: =?utf8?q?Bj=C3=B8rn=20Rustad?= Date: Tue, 4 Oct 2011 00:05:54 +0200 Subject: [PATCH] Finishing up, report handed in --- main.py | 6 +- rapport.tex | 42 +- snapshot.eps | Bin 0 -> 98094 bytes snapshot.png | Bin 0 -> 56810 bytes steepest.eps | 1280 ++++++++++++++++++++++++++++++-------------------- 5 files changed, 803 insertions(+), 525 deletions(-) create mode 100644 snapshot.eps create mode 100644 snapshot.png diff --git a/main.py b/main.py index 78c24a4..5e05335 100644 --- a/main.py +++ b/main.py @@ -13,11 +13,11 @@ def main(): b = np.array([-0.3119187, 0.14508516]) c = np.array([ 0.47736331, 1.48834978]) - x0 = np.array([-50, 50]) + x0 = np.array([-1.28, -1.2]) function_values = [] - alpha = 0.001 + alpha = 0.1 tolerance = 0.000000001 start = time.time() @@ -47,7 +47,7 @@ def main(): mlab.figure(figure="Surface", bgcolor=(1,1,1), fgcolor=(0,0,0)) - plen = 15 + plen = 200 for point in optimal_points[:plen]: z = function_g(point, H, b, c) f_optimal.append(z) diff --git a/rapport.tex b/rapport.tex index 6634a4f..5c30469 100644 --- a/rapport.tex +++ b/rapport.tex @@ -1,6 +1,5 @@ % Fag: Numerisk matematikk % Project 1 -% Utført av: Bjørn, Mats og Edvard \documentclass[12pt, a4paper]{article} \usepackage[latin1]{inputenc} @@ -61,7 +60,7 @@ The function analyzed in this project was \begin{equation} \label{eq:func_g} - g(\mathbf{x}) := -\mathbf{b}^{T}\mathbf{x} + + g(\mathbf{x}) = -\mathbf{b}^{T}\mathbf{x} + \frac{1}{2}\mathbf{x}^{T}H\mathbf{x} + \frac{1}{12}\mathbf{x}^{T}C(\mathbf{x})\mathbf{x}, \end{equation} @@ -73,13 +72,13 @@ this purpose we used the following randomly generated data: \begin{bmatrix} 1.8580682 & -1.01197101 \\ -1.01197101 & 0.63456543 - \end{bmatrix} - , \mathbf{b} = + \end{bmatrix}, + \mathbf{b} = \begin{bmatrix} -0.3119187 \\ 0.14508516 - \end{bmatrix} - , \mathbf{c} = + \end{bmatrix}, + \mathbf{c} = \begin{bmatrix} 0.47736331 \\ 1.48834978 @@ -105,6 +104,7 @@ From equation \eqref{eq:func_g} we get \end{eqnarray} The three functions, \eqref{eq:func_g}, \eqref{eq:grad_g} and \eqref{eq:hess_g}, are implemented in the Python code below. +\newpage \lstinputlisting[language=Python]{bj_function.py} @@ -143,9 +143,9 @@ Consider the equation \mathbf{r}^{(0)}. \end{equation} One can view the function $\mathbf{r}(t)$ as a parameterisation of the curve -going through the starting point to the minima, always going in the direction of -the steepest descent. By applying the forward Euler method to this differential -vector equation, we get the following iteration: +going through the starting point and to the minima, always going in the +direction of the steepest descent. By applying the forward Euler method to this +differential vector equation, we get the following iteration: \begin{equation} \label{eq:forward_iter} \mathbf{r}^{(k+1)} = \mathbf{r}^{(k)} - \alpha \nabla g\left(\mathbf{r}^{(k)}\right) @@ -154,8 +154,9 @@ where $t_{n+1} = t_n + \alpha$. Looking at this result, we see that it is equivalent to the steepest descent iteration with constant step size $\alpha$. To find the optimal step length, $\alpha^*$, we walk along the line of steepest -descent until we get to the lowest point. This point can be found analytically -by solving the following equation for $\alpha$: +descent until we get to the lowest point and it starts going upwards again. +This point can be found analytically by solving the following equation for +$\alpha$: \begin{eqnarray} \label{eq:poly} \frac{\partial}{\partial @@ -171,7 +172,8 @@ Where $A$, $B$, C &=& \mathbf{u}^TH\mathbf{u}+\mathbf{u}^TC(\mathbf{x})\mathbf{u} \nonumber \\ D &=& \frac{1}{3}\mathbf{u}^TC(\mathbf{x})\mathbf{x}+\mathbf{u}^TH\mathbf{x}-\mathbf{b}^T\mathbf{u}. \end{eqnarray} -Differentiating the polynomial results in: +Differentiating the polynomial and using the fact that the Hessian is positive +definite we get the following: \begin{eqnarray} \frac{\partial^2}{\partial \alpha^2}\left(g\left(\mathbf{x} + \alpha\mathbf{u}\right)\right) &=& \mathbf{u}^T H \mathbf{u} + \mathbf{u}^T @@ -180,7 +182,7 @@ Differentiating the polynomial results in: C\left(\mathbf{u}\right) \mathbf{u}\alpha \nonumber \\ &=& \mathbf{u}^T H \mathbf{u} + \mathbf{u}^T C\left(\mathbf{x} + \alpha\mathbf{u}\right)\mathbf{u} \nonumber \\ - &=& \mathbf{u}^T\nabla^2 g\left(\mathbf{x} + \alpha\mathbf{u}\right)\mathbf{u}. + &=& \mathbf{u}^T\nabla^2 g\left(\mathbf{x} + \alpha\mathbf{u}\right)\mathbf{u} > 0. \end{eqnarray} From this we see that the polynomial \eqref{eq:poly} is monotonically increasing and therefore only has one real zero. This is the zero we want to use as our @@ -207,15 +209,17 @@ The Newton method was also implemented to solve $\nabla g\left(\mathbf{x}^*\righ \end{center} \label{tab:time} \end{table} -We see from table \ref{tab:time} that newtons method uses the least time to -solve the problem, even though figure \ref{fig:fn_resid} shows that the -combination method results in fewer iterations. This comes from the fact that -computing the optimal step size requires solving a cubic equation and is -computationally expensive. +We see from table \ref{tab:time} that Newtons method uses the least time to +solve an instance of the problem, even though we adjusted the combination method +to use as few iterations as possible as shown in figure \ref{fig:fn_resid}. +This comes from the fact that computing the optimal step size requires solving a +cubic equation and is computationally expensive. A faster and more approximated +solution for the optimal step size could make the steepest descent method even +faster. \begin{figure}[hb] \centering - \includegraphics[width=0.9\textwidth]{surface} + \includegraphics[width=0.9\textwidth]{snapshot} \caption{\sf Surface plot of function with points showing iteration of steepest descent method, both with optimal and constant $\alpha$} \label{fig:surface} diff --git a/snapshot.eps b/snapshot.eps new file mode 100644 index 0000000000000000000000000000000000000000..e4a98e321f2d4e7d2babf221deaf932ac25b1cc6 GIT binary patch literal 98094 zcmeFY2UL?=w>BC?L;(RI$i@NzK?uD|@1Y}v-jNcD^n~6mC>Z-|yUU{&B~+7l)d>YrSjDHRm&*S>6{eb^{{;WhZY3 zlz^C^FbrW}q)z%k#l@wHM%nv$qh(-Pp7t&%J$n~7M|YT{proKE7&LbC^FYbKFkbdP z7*}sUL6i^X_bqPTUTXG!U{F+8L_|PDSU_0JSX4|#L{vshluuYxMwqm?inqU)lbe@| zig%CQCs*WSy;A4sH&3P5?tz{J2U2H-131||YNBD|bb zy*)iqUVbEjR8TH%UIu7y4{sMLA(dc1l#w4em8TlY(c1~7?&jf#LQ{oO!C*omFjSDE zE6fSyO!|iY+c(F*ehUGzqI_YYFd=0lRV^(kiJxo1oDUX*g~QxPD}=yAz%1~Av^WeV 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+0.000000 0.171875 m /five glyphshow +7.634766 0.171875 m /zero glyphshow grestore gsave /o { @@ -1859,7 +1899,7 @@ translate stroke grestore } bind def -347.657 59.76 o +299.812 44.1 o grestore gsave /o { @@ -1871,12 +1911,14 @@ translate stroke grestore } bind def -347.657 258.96 o +299.812 191.1 o grestore gsave -344.641518 46.681875 translate +289.406250 31.021875 translate 0.000000 rotate -0.000000 0.171875 m /six glyphshow +0.000000 0.171875 m /one glyphshow +7.634766 0.171875 m /zero glyphshow +15.269531 0.171875 m /zero glyphshow grestore gsave /o { @@ -1888,7 +1930,7 @@ translate stroke grestore } bind def -431.855 59.76 o +413.156 44.1 o grestore gsave /o { @@ -1900,12 +1942,14 @@ translate stroke grestore } bind def -431.855 258.96 o +413.156 191.1 o grestore gsave -428.855357 46.681875 translate +402.750000 31.021875 translate 0.000000 rotate -0.000000 0.171875 m /eight glyphshow +0.000000 0.171875 m /one glyphshow +7.634766 0.171875 m /five glyphshow +15.269531 0.171875 m /zero glyphshow grestore gsave /o { @@ -1917,7 +1961,7 @@ translate stroke grestore } bind def -516.054 59.76 o +526.5 44.1 o grestore gsave /o { @@ -1929,25 +1973,29 @@ translate stroke grestore } bind def -516.054 258.96 o +526.5 191.1 o grestore gsave -509.467634 46.681875 translate +515.875000 31.021875 translate 0.000000 rotate -0.000000 0.171875 m /one glyphshow +0.000000 0.171875 m /two glyphshow 7.634766 0.171875 m /zero glyphshow +15.269531 0.171875 m /zero glyphshow grestore +275.445 16.725 m +0 0.172 rmoveto +(Iteration) show gsave /o { gsave newpath translate 0 0 m -0 4 l +4 0 l stroke grestore } bind def -600.252 59.76 o +73.125 44.1 o grestore gsave /o { @@ -1955,17 +2003,37 @@ gsave newpath translate 0 0 m -0 -4 l +-4 0 l stroke grestore } bind def -600.252 258.96 o +526.5 44.1 o grestore gsave -593.868973 46.853750 translate +38.125000 36.600000 translate 0.000000 rotate -0.000000 0.000000 m /one glyphshow -7.634766 0.000000 m /two glyphshow +/DejaVuSans findfont +12.0 scalefont +setfont +0.000000 1.828125 moveto +/one glyphshow + +7.634766 1.828125 moveto +/zero glyphshow + +/DejaVuSans findfont +8.4 scalefont +setfont +15.269531 8.765625 moveto +/hyphen glyphshow + +18.300586 8.765625 moveto +/one glyphshow + +23.644922 8.765625 moveto +/zero glyphshow + + grestore gsave /o { @@ -1973,11 +2041,11 @@ gsave newpath translate 0 0 m -0 4 l +4 0 l stroke grestore } bind def -684.45 59.76 o +73.125 58.8 o grestore gsave /o { @@ -1985,21 +2053,35 @@ gsave newpath translate 0 0 m -0 -4 l +-4 0 l stroke grestore } bind def -684.45 258.96 o +526.5 58.8 o grestore gsave -677.801563 47.010000 translate +44.125000 51.300000 translate 0.000000 rotate -0.000000 0.000000 m /one glyphshow -7.634766 0.000000 m /four glyphshow +/DejaVuSans findfont +12.0 scalefont +setfont +0.000000 1.828125 moveto +/one glyphshow + +7.634766 1.828125 moveto +/zero glyphshow + +/DejaVuSans findfont +8.4 scalefont +setfont +15.269531 8.765625 moveto +/hyphen glyphshow + +18.300586 8.765625 moveto +/nine glyphshow + + grestore -365.389 32.385 m -0 0.172 rmoveto -(Iteration) show gsave /o { gsave @@ -2010,7 +2092,7 @@ translate stroke grestore } bind def -95.0625 59.76 o +73.125 73.5 o grestore gsave /o { @@ -2022,10 +2104,10 @@ translate stroke grestore } bind def -684.45 59.76 o +526.5 73.5 o grestore gsave -66.062500 52.260000 translate +44.125000 66.000000 translate 0.000000 rotate /DejaVuSans findfont 12.0 scalefont @@ -2057,7 +2139,7 @@ translate stroke grestore } bind def -95.0625 84.66 o +73.125 88.2 o grestore gsave /o { @@ -2069,10 +2151,10 @@ translate stroke grestore } bind def -684.45 84.66 o +526.5 88.2 o grestore gsave -66.062500 77.160000 translate +44.125000 80.700000 translate 0.000000 rotate /DejaVuSans findfont 12.0 scalefont @@ -2104,7 +2186,7 @@ translate stroke grestore } bind def -95.0625 109.56 o +73.125 102.9 o grestore gsave /o { @@ -2116,10 +2198,10 @@ translate stroke grestore } bind def -684.45 109.56 o +526.5 102.9 o grestore gsave -66.062500 102.060000 translate +44.125000 95.400000 translate 0.000000 rotate /DejaVuSans findfont 12.0 scalefont @@ -2151,7 +2233,7 @@ translate stroke grestore } bind def -95.0625 134.46 o +73.125 117.6 o grestore gsave /o { @@ -2163,10 +2245,10 @@ translate stroke grestore } bind def -684.45 134.46 o +526.5 117.6 o grestore gsave -66.062500 126.960000 translate +44.125000 110.100000 translate 0.000000 rotate /DejaVuSans findfont 12.0 scalefont @@ -2198,7 +2280,7 @@ translate stroke grestore } bind def -95.0625 159.36 o +73.125 132.3 o grestore gsave /o { @@ -2210,10 +2292,10 @@ translate stroke grestore } bind def -684.45 159.36 o +526.5 132.3 o grestore gsave -66.062500 151.860000 translate +44.125000 124.800000 translate 0.000000 rotate /DejaVuSans findfont 12.0 scalefont @@ -2245,7 +2327,7 @@ translate stroke grestore } bind def -95.0625 184.26 o +73.125 147 o grestore gsave /o { @@ -2257,10 +2339,10 @@ translate stroke grestore } bind def -684.45 184.26 o +526.5 147 o grestore gsave -66.062500 176.760000 translate +44.125000 139.500000 translate 0.000000 rotate /DejaVuSans findfont 12.0 scalefont @@ -2292,7 +2374,7 @@ translate stroke grestore } bind def -95.0625 209.16 o +73.125 161.7 o grestore gsave /o { @@ -2304,10 +2386,10 @@ translate stroke grestore } bind def -684.45 209.16 o +526.5 161.7 o grestore gsave -66.062500 201.660000 translate +44.125000 154.200000 translate 0.000000 rotate /DejaVuSans findfont 12.0 scalefont @@ -2339,7 +2421,7 @@ translate stroke grestore } bind def -95.0625 234.06 o +73.125 176.4 o grestore gsave /o { @@ -2351,10 +2433,10 @@ translate stroke grestore } bind def -684.45 234.06 o +526.5 176.4 o grestore gsave -66.062500 226.560000 translate +44.125000 168.900000 translate 0.000000 rotate /DejaVuSans findfont 12.0 scalefont @@ -2386,7 +2468,7 @@ translate stroke grestore } bind def -95.0625 258.96 o +73.125 191.1 o grestore gsave /o { @@ -2398,10 +2480,10 @@ translate stroke grestore } bind def -684.45 258.96 o +526.5 191.1 o grestore gsave -69.062500 251.460000 translate +47.125000 183.600000 translate 0.000000 rotate /DejaVuSans findfont 12.0 scalefont @@ -2430,7 +2512,199 @@ translate stroke grestore } bind def -95.0625 67.2556 o +73.125 48.5251 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +-2 0 l +stroke +grestore +} bind def +526.5 48.5251 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +2 0 l +stroke +grestore +} bind def +73.125 52.9503 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +-2 0 l +stroke +grestore +} bind def +526.5 52.9503 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +2 0 l +stroke +grestore +} bind def +73.125 55.5388 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +-2 0 l +stroke +grestore +} bind def +526.5 55.5388 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +2 0 l +stroke +grestore +} bind def +73.125 57.3754 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +-2 0 l +stroke +grestore +} bind def +526.5 57.3754 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +2 0 l +stroke +grestore +} bind def +73.125 63.2251 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +-2 0 l +stroke +grestore +} bind def +526.5 63.2251 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +2 0 l +stroke +grestore +} bind def +73.125 67.6503 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +-2 0 l +stroke +grestore +} bind def +526.5 67.6503 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +2 0 l +stroke +grestore +} bind def +73.125 70.2388 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +-2 0 l +stroke +grestore +} bind def +526.5 70.2388 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +2 0 l +stroke +grestore +} bind def +73.125 72.0754 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +-2 0 l +stroke +grestore +} bind def +526.5 72.0754 o +grestore +gsave +/o { +gsave +newpath +translate +0 0 m +2 0 l +stroke +grestore +} bind def +73.125 77.9251 o grestore gsave /o { @@ -2442,7 +2716,7 @@ translate stroke grestore } bind def -684.45 67.2556 o +526.5 77.9251 o grestore gsave /o { @@ -2454,7 +2728,7 @@ translate stroke grestore } bind def -95.0625 74.7513 o +73.125 82.3503 o grestore gsave /o { @@ -2466,7 +2740,7 @@ translate stroke grestore } bind def -684.45 74.7513 o +526.5 82.3503 o grestore gsave /o { @@ -2478,7 +2752,7 @@ translate stroke grestore } bind def -95.0625 79.136 o +73.125 84.9388 o grestore gsave /o { @@ -2490,7 +2764,7 @@ translate stroke grestore } bind def -684.45 79.136 o +526.5 84.9388 o grestore gsave /o { @@ -2502,7 +2776,7 @@ translate stroke grestore } bind def -95.0625 82.2469 o +73.125 86.7754 o grestore gsave /o { @@ -2514,7 +2788,7 @@ translate stroke grestore } bind def -684.45 82.2469 o +526.5 86.7754 o grestore gsave /o { @@ -2526,7 +2800,7 @@ translate stroke grestore } bind def -95.0625 92.1556 o +73.125 92.6251 o grestore gsave /o { @@ -2538,7 +2812,7 @@ translate stroke grestore } bind def -684.45 92.1556 o +526.5 92.6251 o grestore gsave /o { @@ -2550,7 +2824,7 @@ translate stroke grestore } bind def -95.0625 99.6513 o +73.125 97.0503 o grestore gsave /o { @@ -2562,7 +2836,7 @@ translate stroke grestore } bind def -684.45 99.6513 o +526.5 97.0503 o grestore gsave /o { @@ -2574,7 +2848,7 @@ translate stroke grestore } bind def -95.0625 104.036 o +73.125 99.6388 o grestore gsave /o { @@ -2586,7 +2860,7 @@ translate stroke grestore } bind def -684.45 104.036 o +526.5 99.6388 o grestore gsave /o { @@ -2598,7 +2872,7 @@ translate stroke grestore } bind def -95.0625 107.147 o +73.125 101.475 o grestore gsave /o { @@ -2610,7 +2884,7 @@ translate stroke grestore } bind def -684.45 107.147 o +526.5 101.475 o grestore gsave /o { @@ -2622,7 +2896,7 @@ translate stroke grestore } bind def -95.0625 117.056 o +73.125 107.325 o grestore gsave /o { @@ -2634,7 +2908,7 @@ translate stroke grestore } bind def -684.45 117.056 o +526.5 107.325 o grestore gsave /o { @@ -2646,7 +2920,7 @@ translate stroke grestore } bind def -95.0625 124.551 o +73.125 111.75 o grestore gsave /o { @@ -2658,7 +2932,7 @@ translate stroke grestore } bind def -684.45 124.551 o +526.5 111.75 o grestore gsave /o { @@ -2670,7 +2944,7 @@ translate stroke grestore } bind def -95.0625 128.936 o +73.125 114.339 o grestore gsave /o { @@ -2682,7 +2956,7 @@ translate stroke grestore } bind def -684.45 128.936 o +526.5 114.339 o grestore gsave /o { @@ -2694,7 +2968,7 @@ translate stroke grestore } bind def -95.0625 132.047 o +73.125 116.175 o grestore gsave /o { @@ -2706,7 +2980,7 @@ translate stroke grestore } bind def -684.45 132.047 o +526.5 116.175 o grestore gsave /o { @@ -2718,7 +2992,7 @@ translate stroke grestore } bind def -95.0625 141.956 o +73.125 122.025 o grestore gsave /o { @@ -2730,7 +3004,7 @@ translate stroke grestore } bind def -684.45 141.956 o +526.5 122.025 o grestore gsave /o { @@ -2742,7 +3016,7 @@ translate stroke grestore } bind def -95.0625 149.451 o +73.125 126.45 o grestore gsave /o { @@ -2754,7 +3028,7 @@ translate stroke grestore } bind def -684.45 149.451 o +526.5 126.45 o grestore gsave /o { @@ -2766,7 +3040,7 @@ translate stroke grestore } bind def -95.0625 153.836 o +73.125 129.039 o grestore gsave /o { @@ -2778,7 +3052,7 @@ translate stroke grestore } bind def -684.45 153.836 o +526.5 129.039 o grestore gsave /o { @@ -2790,7 +3064,7 @@ translate stroke grestore } bind def -95.0625 156.947 o +73.125 130.875 o grestore gsave /o { @@ -2802,7 +3076,7 @@ translate stroke grestore } bind def -684.45 156.947 o +526.5 130.875 o grestore gsave /o { @@ -2814,7 +3088,7 @@ translate stroke grestore } bind def -95.0625 166.856 o +73.125 136.725 o grestore gsave /o { @@ -2826,7 +3100,7 @@ translate stroke grestore } bind def -684.45 166.856 o +526.5 136.725 o grestore gsave /o { @@ -2838,7 +3112,7 @@ translate stroke grestore } bind def -95.0625 174.351 o +73.125 141.15 o grestore gsave /o { @@ -2850,7 +3124,7 @@ translate stroke grestore } bind def -684.45 174.351 o +526.5 141.15 o grestore gsave /o { @@ -2862,7 +3136,7 @@ translate stroke grestore } bind def -95.0625 178.736 o +73.125 143.739 o grestore gsave /o { @@ -2874,7 +3148,7 @@ translate stroke grestore } bind def -684.45 178.736 o +526.5 143.739 o grestore gsave /o { @@ -2886,7 +3160,7 @@ translate stroke grestore } bind def -95.0625 181.847 o +73.125 145.575 o grestore gsave /o { @@ -2898,7 +3172,7 @@ translate stroke grestore } bind def -684.45 181.847 o +526.5 145.575 o grestore gsave /o { @@ -2910,7 +3184,7 @@ translate stroke grestore } bind def -95.0625 191.756 o +73.125 151.425 o grestore gsave /o { @@ -2922,7 +3196,7 @@ translate stroke grestore } bind def -684.45 191.756 o +526.5 151.425 o grestore gsave /o { @@ -2934,7 +3208,7 @@ translate stroke grestore } bind def -95.0625 199.251 o +73.125 155.85 o grestore gsave /o { @@ -2946,7 +3220,7 @@ translate stroke grestore } bind def -684.45 199.251 o +526.5 155.85 o grestore gsave /o { @@ -2958,7 +3232,7 @@ translate stroke grestore } bind def -95.0625 203.636 o +73.125 158.439 o grestore gsave /o { @@ -2970,7 +3244,7 @@ translate stroke grestore } bind def -684.45 203.636 o +526.5 158.439 o grestore gsave /o { @@ -2982,7 +3256,7 @@ translate stroke grestore } bind def -95.0625 206.747 o +73.125 160.275 o grestore gsave /o { @@ -2994,7 +3268,7 @@ translate stroke grestore } bind def -684.45 206.747 o +526.5 160.275 o grestore gsave /o { @@ -3006,7 +3280,7 @@ translate stroke grestore } bind def -95.0625 216.656 o +73.125 166.125 o grestore gsave /o { @@ -3018,7 +3292,7 @@ translate stroke grestore } bind def -684.45 216.656 o +526.5 166.125 o grestore gsave /o { @@ -3030,7 +3304,7 @@ translate stroke grestore } bind def -95.0625 224.151 o +73.125 170.55 o grestore gsave /o { @@ -3042,7 +3316,7 @@ translate stroke grestore } bind def -684.45 224.151 o +526.5 170.55 o grestore gsave /o { @@ -3054,7 +3328,7 @@ translate stroke grestore } bind def -95.0625 228.536 o +73.125 173.139 o grestore gsave /o { @@ -3066,7 +3340,7 @@ translate stroke grestore } bind def -684.45 228.536 o +526.5 173.139 o grestore gsave /o { @@ -3078,7 +3352,7 @@ translate stroke grestore } bind def -95.0625 231.647 o +73.125 174.975 o grestore gsave /o { @@ -3090,7 +3364,7 @@ translate stroke grestore } bind def -684.45 231.647 o +526.5 174.975 o grestore gsave /o { @@ -3102,7 +3376,7 @@ translate stroke grestore } bind def -95.0625 241.556 o +73.125 180.825 o grestore gsave /o { @@ -3114,7 +3388,7 @@ translate stroke grestore } bind def -684.45 241.556 o +526.5 180.825 o grestore gsave /o { @@ -3126,7 +3400,7 @@ translate stroke grestore } bind def -95.0625 249.051 o +73.125 185.25 o grestore gsave /o { @@ -3138,7 +3412,7 @@ translate stroke grestore } bind def -684.45 249.051 o +526.5 185.25 o grestore gsave /o { @@ -3150,7 +3424,7 @@ translate stroke grestore } bind def -95.0625 253.436 o +73.125 187.839 o grestore gsave /o { @@ -3162,7 +3436,7 @@ translate stroke grestore } bind def -684.45 253.436 o +526.5 187.839 o grestore gsave /o { @@ -3174,7 +3448,7 @@ translate stroke grestore } bind def -95.0625 256.547 o +73.125 189.675 o grestore gsave /o { @@ -3186,9 +3460,9 @@ translate stroke grestore } bind def -684.45 256.547 o +526.5 189.675 o grestore -61.062 110.001 m +33.125 68.241 m gsave 90 rotate 0 0.172 rmoveto @@ -3197,29 +3471,29 @@ grestore 1.000 setlinewidth 2 setlinecap gsave -95.0625 258.96 m -684.45 258.96 l +73.125 191.1 m +526.5 191.1 l stroke grestore gsave -684.45 59.76 m -684.45 258.96 l +526.5 44.1 m +526.5 191.1 l stroke grestore gsave -95.0625 59.76 m -684.45 59.76 l +73.125 44.1 m +526.5 44.1 l stroke grestore gsave -95.0625 59.76 m -95.0625 258.96 l +73.125 44.1 m +73.125 191.1 l stroke grestore /DejaVuSans findfont 14.400 scalefont setfont -330.686 263.96 m +240.742 196.1 m 0 0.203 rmoveto (Relative residual) show -- 2.47.3