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List: scilab-users
Subject: Re: [Scilab-users] numderivative Heart function
From: Rafael Guerra <jrafaelbguerra () hotmail ! com>
Date: 2017-11-30 18:23:32
Message-ID: VI1PR04MB322973019E73CBF921D385E2CC380 () VI1PR04MB3229 ! eurprd04 ! prod ! outlook ! com
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Hi Hermes,
I do not recall your problem and it would help some explanation.
To plot the contours of a gradient with contour2d, you have to make it a scalar \
(otherwise look at champ, etc.).
Check out the edited code below.
Note that your gradient does not reach 0 and so a contour |g|=5 is plotted.
// START OF CODE
clear; clf;
funcprot(0);
function val=H(x1, x2)
val=(1.25*x2-sqrt(abs(x1))).^2+x1.^2-1;
endfunction;
function [gx, gy]=g(x1, x2)
gx= 2*x1-(x1.*(1.25*x2-sqrt(abs(x1))))./abs(x1).^(3/2);
gy = 2.5*(1.25*x2-sqrt(abs(x1)));
endfunction
function val=HH(z)
x1=z(1);
x2=z(2);
val=(1.25*x2-sqrt(abs(x1))).^2+x1.^2-1;
endfunction;
function z=dH(x)
z=numderivative(HH,x)
endfunction
//test
x0=[0.2 -0.5];
disp(H(x0(1),x0(2)));
[gx,gy]=g(x0(1),x0(2))
disp(gx,gy);
disp(dH(x0));
x = linspace(-1,1,1002);
y = 2*x';
xx = x.*.ones(x');
yy = y.*.ones(y');
[gx,gy] = g(xx,yy);
g1= abs(gx + %i*gy);
contour2d(x, y, g1, [5 5]);
replot([-1.415,-1,1.415,1.415]);
// END OF CODE
[cid:image003.jpg@01D36A10.B2E54FF0]
Regards,
Rafael
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<div class="WordSection1">
<p class="MsoPlainText"><font size="2" face="Calibri"><span \
style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Hi \
Hermes,<o:p></o:p></span></span></font></p> <p class="MsoPlainText"><font size="2" \
face="Calibri"><o:p> </o:p></font></p> <p class="MsoPlainText"><font size="2" \
face="Calibri"><span style="font-size:11.0pt"><span \
style="mso-bidi-font-size:10.5pt">I do not recall your problem and it would help some \
explanation.<o:p></o:p></span></span></font></p> <p class="MsoPlainText"><font \
size="2" face="Calibri"><span style="font-size:11.0pt"><span \
style="mso-bidi-font-size:10.5pt">To plot the contours of a gradient with contour2d, \
you have to make it a scalar (otherwise look at champ, \
etc.).<o:p></o:p></span></span></font></p> <p class="MsoPlainText"><font size="2" \
face="Calibri"><span style="font-size:11.0pt"><span \
style="mso-bidi-font-size:10.5pt">Check out the edited code \
below.<o:p></o:p></span></span></font></p> <p class="MsoPlainText"><font size="2" \
face="Calibri"><span style="font-size:11.0pt"><span \
style="mso-bidi-font-size:10.5pt">Note that your gradient does not reach 0 and so a \
contour |g|=5 is plotted.<o:p></o:p></span></span></font></p> <p \
class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p> \
<pre><i><font size="2" color="#64ae64" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#64AE64;font-style:italic">// \
START OF CODE</span></font></i><font face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p></o:p></span></font></pre>
<pre><font size="2" color="#32b9b9" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#32B9B9">clear</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">;</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> <span \
class="SpellE"><u><font color="#ae5cb0"><span \
style="color:#AE5CB0">clf</span></font></u></span><font color="black"><span \
style="color:black">;</span></font><o:p></o:p></span></font></pre> <pre><span \
class="SpellE"><font size="2" color="#32b9b9" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#32B9B9">funcprot</span></font></span><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">(</span></font><font \
color="rosybrown" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:rosybrown">0</span></font><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">)</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">;</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p></o:p></span></font></pre>
<pre><font size="2" color="#b01813" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#B01813">function</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> <span \
class="SpellE"><b><font color="#834310"><span \
style="color:#834310;font-weight:bold">val</span></font></b></span><font \
color="#5c5c5c"><span style="color:#5C5C5C">=</span></font><u><font \
color="black"><span style="color:black">H</span></font></u><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x1</span></font></b><font color="black"><span \
style="color:black">, </span></font><b><font color="#834310"><span \
style="color:#834310;font-weight:bold">x2</span></font></b><font \
color="#4a55db"><span style="color:#4A55DB">)</span></font> \
<o:p></o:p></span></font></pre> <pre><font size="2" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><span \
style="mso-spacerun:yes"> </span><span class="SpellE"><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">val</span></font></b></span><font \
color="#5c5c5c"><span style="color:#5C5C5C">=</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><font \
color="rosybrown"><span style="color:rosybrown">1.25</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">*</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x2</span></font></b><font \
color="#5c5c5c"><span style="color:#5C5C5C">-</span></font><font \
color="#32b9b9"><span style="color:#32B9B9">sqrt</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><font \
color="#32b9b9"><span style="color:#32B9B9">abs</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x1</span></font></b><font \
color="#4a55db"><span style="color:#4A55DB">)))</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">.^</span></font><font \
color="rosybrown"><span style="color:rosybrown">2</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">+</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x1</span></font></b><font \
color="#5c5c5c"><span style="color:#5C5C5C">.^</span></font><font \
color="rosybrown"><span style="color:rosybrown">2</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">-</span></font><font \
color="rosybrown"><span style="color:rosybrown">1</span></font><font \
color="black"><span style="color:black">;</span></font><o:p></o:p></span></font></pre>
<pre><span class="SpellE"><font size="2" color="#b01813" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#B01813">endfunction</span></font></span><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">;</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p></o:p></span></font></pre>
<pre><font size="2" color="#b01813" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#B01813">function</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> <font \
color="#4a55db"><span style="color:#4A55DB">[</span></font><span \
class="SpellE"><b><font color="#834310"><span \
style="color:#834310;font-weight:bold">gx</span></font></b></span><font \
color="black"><span style="color:black">, </span></font><span class="SpellE"><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">gy</span></font></b></span><font \
color="#4a55db"><span style="color:#4A55DB">]</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">=</span></font><u><font \
color="black"><span style="color:black">g</span></font></u><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x1</span></font></b><font color="black"><span \
style="color:black">, </span></font><b><font color="#834310"><span \
style="color:#834310;font-weight:bold">x2</span></font></b><font \
color="#4a55db"><span \
style="color:#4A55DB">)</span></font><o:p></o:p></span></font></pre> <pre><font \
size="2" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><span \
style="mso-spacerun:yes"> </span><span class="SpellE"><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">gx</span></font></b></span><font \
color="#5c5c5c"><span style="color:#5C5C5C">=</span></font> <font \
color="rosybrown"><span style="color:rosybrown">2</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">*</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x1</span></font></b><font \
color="#5c5c5c"><span style="color:#5C5C5C">-</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x1</span></font></b><font \
color="#5c5c5c"><span style="color:#5C5C5C">.*</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><font \
color="rosybrown"><span style="color:rosybrown">1.25</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">*</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x2</span></font></b><font \
color="#5c5c5c"><span style="color:#5C5C5C">-</span></font><font \
color="#32b9b9"><span style="color:#32B9B9">sqrt</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><font \
color="#32b9b9"><span style="color:#32B9B9">abs</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x1</span></font></b><font \
color="#4a55db"><span style="color:#4A55DB">))))</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">./</span></font><font \
color="#32b9b9"><span style="color:#32B9B9">abs</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x1</span></font></b><font \
color="#4a55db"><span style="color:#4A55DB">)</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">.^</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><font \
color="rosybrown"><span style="color:rosybrown">3</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">/</span></font><font \
color="rosybrown"><span style="color:rosybrown">2</span></font><font \
color="#4a55db"><span style="color:#4A55DB">)</span></font><font color="black"><span \
style="color:black">;</span></font><o:p></o:p></span></font></pre> <pre><font \
size="2" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><span \
style="mso-spacerun:yes"> </span><span class="SpellE"><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">gy</span></font></b></span> <font \
color="#5c5c5c"><span style="color:#5C5C5C">=</span></font> <font \
color="rosybrown"><span style="color:rosybrown">2.5</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">*</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><font \
color="rosybrown"><span style="color:rosybrown">1.25</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">*</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x2</span></font></b><font \
color="#5c5c5c"><span style="color:#5C5C5C">-</span></font><font \
color="#32b9b9"><span style="color:#32B9B9">sqrt</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><font \
color="#32b9b9"><span style="color:#32B9B9">abs</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">x1</span></font></b><font \
color="#4a55db"><span style="color:#4A55DB">)))</span></font><font \
color="black"><span style="color:black">;</span></font><span \
style="mso-spacerun:yes"> </span><o:p></o:p></span></font></pre> \
<pre><span class="SpellE"><font size="2" color="#b01813" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#B01813">endfunction</span></font></span><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p></o:p></span></font></pre>
<pre><font size="2" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p> </o:p></span></font></pre>
<pre><font size="2" color="#b01813" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#B01813">function</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> <span \
class="SpellE"><b><font color="#834310"><span \
style="color:#834310;font-weight:bold">val</span></font></b></span><font \
color="#5c5c5c"><span style="color:#5C5C5C">=</span></font><u><font \
color="black"><span style="color:black">HH</span></font></u><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><b><font \
color="#834310"><span style="color:#834310;font-weight:bold">z</span></font></b><font \
color="#4a55db"><span style="color:#4A55DB">)</span></font> \
<o:p></o:p></span></font></pre> <pre><font size="2" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><span \
style="mso-spacerun:yes"> </span><font color="black"><span \
style="color:black">x1</span></font><font color="#5c5c5c"><span \
style="color:#5C5C5C">=</span></font><b><font color="#834310"><span \
style="color:#834310;font-weight:bold">z</span></font></b><font color="#4a55db"><span \
style="color:#4A55DB">(</span></font><font color="rosybrown"><span \
style="color:rosybrown">1</span></font><font color="#4a55db"><span \
style="color:#4A55DB">)</span></font><font color="black"><span \
<pre><font size="2" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><span \
style="mso-spacerun:yes"> </span><span class="SpellE"><b><font \
color="#834310"><span \
style="color:#834310;font-weight:bold">val</span></font></b></span><font \
color="#5c5c5c"><span style="color:#5C5C5C">=</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><font \
color="rosybrown"><span style="color:rosybrown">1.25</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">*</span></font><font color="black"><span \
style="color:black">x2</span></font><font color="#5c5c5c"><span \
style="color:#5C5C5C">-</span></font><font color="#32b9b9"><span \
style="color:#32B9B9">sqrt</span></font><font color="#4a55db"><span \
style="color:#4A55DB">(</span></font><font color="#32b9b9"><span \
style="color:#32B9B9">abs</span></font><font color="#4a55db"><span \
style="color:#4A55DB">(</span></font><font color="black"><span \
style="color:black">x1</span></font><font color="#4a55db"><span \
style="color:#4A55DB">)))</span></font><font color="#5c5c5c"><span \
style="color:#5C5C5C">.^</span></font><font color="rosybrown"><span \
style="color:rosybrown">2</span></font><font color="#5c5c5c"><span \
style="color:#5C5C5C">+</span></font><font color="black"><span \
style="color:black">x1</span></font><font color="#5c5c5c"><span \
style="color:#5C5C5C">.^</span></font><font color="rosybrown"><span \
style="color:rosybrown">2</span></font><font color="#5c5c5c"><span \
style="color:#5C5C5C">-</span></font><font color="rosybrown"><span \
style="color:rosybrown">1</span></font><font color="black"><span \
style="color:black">;</span></font><o:p></o:p></span></font></pre> <pre><span \
class="SpellE"><font size="2" color="#b01813" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#B01813">endfunction</span></font></span><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">;</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> \
<o:p></o:p></span></font></pre> <pre><font size="2" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p> </o:p></span></font></pre>
<pre><font size="2" color="#b01813" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#B01813">function</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> <b><font \
color="#834310"><span style="color:#834310;font-weight:bold">z</span></font></b><font \
color="#5c5c5c"><span style="color:#5C5C5C">=</span></font><span \
class="SpellE"><u><font color="black"><span \
style="color:black">dH</span></font></u></span><font color="#4a55db"><span \
style="color:#4A55DB">(</span></font><b><font color="#834310"><span \
style="color:#834310;font-weight:bold">x</span></font></b><font color="#4a55db"><span \
style="color:#4A55DB">)</span></font><span style="mso-spacerun:yes"> \
</span><o:p></o:p></span></font></pre> <pre><font size="2" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><span \
style="mso-spacerun:yes"> </span><b><font \
color="#834310"><span style="color:#834310;font-weight:bold">z</span></font></b><font \
color="#5c5c5c"><span style="color:#5C5C5C">=</span></font><span \
class="SpellE"><u><font color="#ae5cb0"><span \
style="color:#AE5CB0">numderivative</span></font></u></span><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><span \
class="SpellE"><u><font color="black"><span \
style="color:black">HH</span></font></u><font color="black"><span \
style="color:black">,</span></font><b><font color="#834310"><span \
style="color:#834310;font-weight:bold">x</span></font></b></span><font \
color="#4a55db"><span style="color:#4A55DB">)</span></font> \
<o:p></o:p></span></font></pre> <pre><span class="SpellE"><font size="2" \
color="#b01813" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#B01813">endfunction</span></font></span><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> \
<o:p></o:p></span></font></pre> <pre><font size="2" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p> </o:p></span></font></pre>
<pre><i><font size="2" color="#64ae64" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#64AE64;font-style:italic">//test \
</span></font></i><font face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p></o:p></span></font></pre>
<pre><font size="2" color="black" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">x0</span></font><font \
color="#5c5c5c" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#5C5C5C">=</span></font><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">[</span></font><font \
color="rosybrown" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:rosybrown">0.2</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> <font \
color="#5c5c5c"><span style="color:#5C5C5C">-</span></font><font \
color="rosybrown"><span style="color:rosybrown">0.5</span></font><font \
color="#4a55db"><span style="color:#4A55DB">]</span></font><font color="black"><span \
style="color:black">;</span></font><o:p></o:p></span></font></pre> <pre><span \
class="SpellE"><font size="2" color="#32b9b9" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#32B9B9">disp</span></font></span><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">(</span></font><u><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">H</span></font></u><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">(</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">x0</span></font><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">(</span></font><font \
color="rosybrown" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:rosybrown">1</span></font><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">)</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">,x0</span></font><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">(</span></font><font \
color="rosybrown" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:rosybrown">2</span></font><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">)))</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">;</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p></o:p></span></font></pre>
<pre><font size="2" color="#4a55db" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">[</span></font><span \
class="SpellE"><font color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">gx,gy</span></font></span><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">]</span></font><font \
color="#5c5c5c" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#5C5C5C">=</span></font><u><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">g</span></font></u><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">(</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">x0</span></font><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">(</span></font><font \
color="rosybrown" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:rosybrown">1</span></font><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">)</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">,x0</span></font><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">(</span></font><font \
color="rosybrown" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:rosybrown">2</span></font><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">))</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p></o:p></span></font></pre>
<pre><span class="SpellE"><font size="2" color="#32b9b9" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#32B9B9">disp</span></font></span><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">(</span></font><span \
class="SpellE"><font color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">gx,gy</span></font></span><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">)</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">;</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p></o:p></span></font></pre>
<pre><span class="SpellE"><font size="2" color="#32b9b9" face="Monospaced"><span \
lang="ES" style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospac \
ed",serif;color:#32B9B9;mso-ansi-language:ES">disp</span></font></span><font \
color="#4a55db" face="Monospaced"><span lang="ES" \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB;mso-ansi-language:ES">(</span></font><span \
class="SpellE"><u><font color="black" face="Monospaced"><span lang="ES" \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black;mso-ansi-language:ES">dH</span></font></u></span><font \
color="#4a55db" face="Monospaced"><span lang="ES" \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB;mso-ansi-language:ES">(</span></font><font \
color="black" face="Monospaced"><span lang="ES" \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black;mso-ansi-language:ES">x0</span></font><font \
color="#4a55db" face="Monospaced"><span lang="ES" \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB;mso-ansi-language:ES">))</span></font><font \
color="black" face="Monospaced"><span lang="ES" \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black;mso-ansi-language:ES">;</span></font><font \
face="Monospaced"><span lang="ES" \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;mso-ansi-language:ES"><o:p></o:p></span></font></pre>
<pre><font size="2" face="Monospaced"><span lang="ES" \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;mso-ansi-language:ES"><o:p> </o:p></span></font></pre>
<pre><font size="2" color="black" face="Monospaced"><span lang="ES" \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black;mso-ansi-language:ES">x</span></font><font \
face="Monospaced"><span lang="ES" \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;mso-ansi-language:ES"> \
<font color="#5c5c5c"><span style="color:#5C5C5C">=</span></font> <span \
class="SpellE"><u><font color="#ae5cb0"><span \
style="color:#AE5CB0">linspace</span></font></u></span><font color="#4a55db"><span \
style="color:#4A55DB">(</span></font><font color="#5c5c5c"><span \
style="color:#5C5C5C">-</span></font><font color="rosybrown"><span \
style="color:rosybrown">1</span></font><font color="black"><span \
style="color:black">,</span></font><font color="rosybrown"><span \
style="color:rosybrown">1</span></font><font color="black"><span \
style="color:black">,</span></font><font color="rosybrown"><span \
style="color:rosybrown">1002</span></font><font color="#4a55db"><span \
style="color:#4A55DB">)</span></font><font color="black"><span \
style="color:black">;</span></font><o:p></o:p></span></font></pre> <pre><font \
size="2" color="black" face="Monospaced"><span lang="ES" \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black;mso-ansi-language:ES">y</span></font><font \
face="Monospaced"><span lang="ES" \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;mso-ansi-language:ES"> \
<font color="#5c5c5c"><span style="color:#5C5C5C">=</span></font> <font \
color="rosybrown"><span style="color:rosybrown">2</span></font><font \
color="#5c5c5c"><span style="color:#5C5C5C">*</span></font><font color="black"><span \
style="color:black">x</span></font><font color="#5c5c5c"><span \
style="color:#5C5C5C">'</span></font><font color="black"><span \
style="color:black">;</span></font><o:p></o:p></span></font></pre> <pre><font \
size="2" color="black" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">xx</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> <font \
<pre><span class="SpellE"><font size="2" color="black" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">yy</span></font></span><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> <font \
color="#5c5c5c"><span style="color:#5C5C5C">=</span></font> <font color="black"><span \
style="color:black">y</span></font><font color="#5c5c5c"><span \
style="color:#5C5C5C">.*.</span></font><font color="#32b9b9"><span \
style="color:#32B9B9">ones</span></font><font color="#4a55db"><span \
style="color:#4A55DB">(</span></font><font color="black"><span \
style="color:black">y</span></font><font color="#5c5c5c"><span \
style="color:#5C5C5C">'</span></font><font color="#4a55db"><span \
style="color:#4A55DB">)</span></font><font color="black"><span \
style="color:black">;</span></font><o:p></o:p></span></font></pre> <pre><font \
size="2" color="#4a55db" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">[</span></font><span \
class="SpellE"><font color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">gx,gy</span></font></span><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">]</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> <font \
color="#5c5c5c"><span style="color:#5C5C5C">=</span></font> <u><font \
color="black"><span style="color:black">g</span></font></u><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><span class="SpellE"><font \
color="black"><span style="color:black">xx,yy</span></font></span><font \
color="#4a55db"><span style="color:#4A55DB">)</span></font><font color="black"><span \
style="color:black">;</span></font><o:p></o:p></span></font></pre> <pre><font \
size="2" color="black" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">g1</span></font><font \
color="#5c5c5c" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#5C5C5C">=</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> <font \
color="#32b9b9"><span style="color:#32B9B9">abs</span></font><font \
color="#4a55db"><span style="color:#4A55DB">(</span></font><span class="SpellE"><font \
color="black"><span style="color:black">gx</span></font></span> <font \
color="#5c5c5c"><span style="color:#5C5C5C">+</span></font> <font \
color="orchid"><span style="color:orchid">%<span \
class="SpellE">i</span></span></font><font color="#5c5c5c"><span \
style="color:#5C5C5C">*</span></font><span class="SpellE"><font color="black"><span \
style="color:black">gy</span></font></span><font color="#4a55db"><span \
style="color:#4A55DB">)</span></font><font color="black"><span \
style="color:black">;</span></font><o:p></o:p></span></font></pre> <pre><u><font \
size="2" color="#ae5cb0" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#AE5CB0">contour2d</span></font></u><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">(</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">x,</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"> <font \
color="black"><span style="color:black">y,</span></font> <font color="black"><span \
style="color:black">g1,</span></font> <font color="#4a55db"><span \
style="color:#4A55DB">[</span></font><font color="rosybrown"><span \
style="color:rosybrown">5</span></font> <font color="rosybrown"><span \
style="color:rosybrown">5</span></font><font color="#4a55db"><span \
style="color:#4A55DB">])</span></font><font color="black"><span \
style="color:black">;</span></font><o:p></o:p></span></font></pre> <pre><font \
size="2" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p> </o:p></span></font></pre>
<pre><u><font size="2" color="#ae5cb0" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#AE5CB0">replot</span></font></u><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">([</span></font><font \
color="#5c5c5c" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#5C5C5C">-</span></font><font \
color="rosybrown" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:rosybrown">1.415</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">,</span></font><font \
color="#5c5c5c" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#5C5C5C">-</span></font><font \
color="rosybrown" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:rosybrown">1</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">,</span></font><font \
color="rosybrown" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:rosybrown">1.415</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">,</span></font><font \
color="rosybrown" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:rosybrown">1.415</span></font><font \
color="#4a55db" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#4A55DB">])</span></font><font \
color="black" face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:black">;</span></font><font \
face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p></o:p></span></font></pre>
<pre><i><font size="2" color="#64ae64" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif;color:#64AE64;font-style:italic">// \
END OF CODE</span></font></i><font face="Monospaced"><span \
style="mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p></o:p></span></font></pre>
<pre><font size="2" face="Monospaced"><span \
style="font-size:10.0pt;mso-bidi-font-size:13.0pt;font-family:"Monospaced",serif"><o:p> </o:p></span></font></pre>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span \
style="font-size:11.0pt;mso-no-proof:yes"><img width="336" height="254" \
id="Picture_x0020_1" \
src="cid:image003.jpg@01D36A10.B2E54FF0"></span><o:p></o:p></font></p> <p \
class="MsoPlainText"><font size="2" color="black" \
face="Calibri"><o:p> </o:p></font></p> <p class="MsoPlainText"><font size="2" \
color="black" face="Calibri"><span style="font-size:11.0pt;color:black"><span \
style="mso-bidi-font-size:10.5pt">Regards,<o:p></o:p></span></span></font></p> <p \
class="MsoPlainText"><font size="2" color="black" face="Calibri"><span \
style="font-size:11.0pt;color:black"><span \
style="mso-bidi-font-size:10.5pt">Rafael<o:p></o:p></span></span></font></p> </div>
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