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<html>
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="Author" content="Dietmar Gerald Schrausser">
<title>
ConsoleApp_Integral/README
</title>
</head>
<body style="font-family:calibri;font-size:85%;background-color:#eceff1;color:#37474f">
<style>
a {color: #78909c; }
a:visited {color: #78909c;}
</style>
<h1 id="consoleapp_integral">ConsoleApp_Integral</h1>
<p>Console applications for <em>integral</em> and <em>interpolation</em> (German).</p>
<h2 id="romi">ROMI</h2>
<p>Approximiert (a) Flächen- oder (b) Kurven-Integrale</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munderover><mo data-mjx-texclass="OP">∫</mo><mi>a</mi><mi>b</mi></munderover><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></math></p>
<p>mittels Romberg Methode, dabei
ggf. Dateiausgabe nach <code>romi.txt</code> von Funkionsmatrix</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">F</mi><mo mathvariant="bold-italic">=</mo><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mrow data-mjx-texclass="ORD"><mo mathvariant="bold-italic" stretchy="false">|</mo></mrow><mi mathvariant="bold-italic">y</mi><mo mathvariant="bold-italic" stretchy="false">)</mo></mrow><mo>,</mo></math></p>
<p>bei</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mi>d</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mi>y</mi><mo>=</mo><munderover><mo data-mjx-texclass="OP">∫</mo><mi>a</mi><mi>b</mi></munderover><mrow data-mjx-texclass="ORD"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow><mo>.</mo></math></p>
<ul>
<li>Ausführung von <code>ROMI.bat</code>:</li>
<li>Definition von <math display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> in <code>ROMI.h</code>;</li>
<li>Compilieren von <code>ROMI.c</code>;</li>
<li>Ausführung von <code>ROMI.exe</code>.</li>
</ul>
<p>Handhabung</p>
<pre><code>ROMI [a] [b] [d] [m] [F]
[a] ..... Integrations Minimum a
[b] ......Integrations Maximum b
[d] ......Delta d
[m] ......Modus: (0)Flaechen.I (1)Kurven.I
[F] ......Funktionsmatrix: (0)keine (1)romi.txt
</code></pre>
<h2 id="rome">ROME</h2>
<p>Approximiert das Integral <math display="inline"><munderover><mo data-mjx-texclass="OP">∫</mo><mi>a</mi><mi>b</mi></munderover><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></math>
mittels <em>Romberg-Extrapolation</em>
(ROMBERG Integration nach Meyberg und Vachenauer, <a href="https://doi.org/10.1007/978-3-642-56654-7_4">2001</a>, S. 209).</p>
<ul>
<li>Ausführung von <code>ROME.bat</code>:</li>
<li>Definition von <math display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> in <code>ROME.h</code>;</li>
<li>Compilieren von <code>ROME.c</code>;</li>
<li>Ausführung von <code>ROME.exe</code>.</li>
</ul>
<p>Handhabung</p>
<pre><code>ROME [a] [b]
[a] ....... Integrations Minimum a
[b] ....... Integrations Maximum b
</code></pre>
<h2 id="kusi">KUSI</h2>
<p>Kubische Spline Interpolation: Berechnung der Koeffizientenmatrix <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">A</mi><mo mathvariant="bold-italic">=</mo><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">b</mi><mrow data-mjx-texclass="ORD"><mo mathvariant="bold-italic" stretchy="false">|</mo></mrow><mi mathvariant="bold-italic">c</mi><mrow data-mjx-texclass="ORD"><mo mathvariant="bold-italic" stretchy="false">|</mo></mrow><mi mathvariant="bold-italic">d</mi><mo mathvariant="bold-italic" stretchy="false">)</mo></mrow></math> sowie <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">s</mi><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo mathvariant="bold-italic" stretchy="false">)</mo></mrow></math> zu einer (empirischen) Funktionsmatrix <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">F</mi><mo mathvariant="bold-italic">=</mo><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mrow data-mjx-texclass="ORD"><mo mathvariant="bold-italic" stretchy="false">|</mo></mrow><mi mathvariant="bold-italic">y</mi><mo mathvariant="bold-italic" stretchy="false">)</mo></mrow></math>, wobei</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>s</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo>+</mo><msub><mi>b</mi><mi>i</mi></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>c</mi><mi>i</mi></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msub><mi>d</mi><mi>i</mi></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo>;</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1.</mn></math></p>
<ul>
<li>Übernahme einer ASCII Funktionsmatrix Datei <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">F</mi></mrow></math>;</li>
<li>Ausgabe der ASCII Koeffizientenmatrix Datei <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">A</mi></mrow></math> (<code>KUSI.txt</code>);</li>
<li>Berechnung von <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">s</mi><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo mathvariant="bold-italic" stretchy="false">)</mo></mrow></math> über die Interpolations-Funktion.</li>
</ul>
<p>Handhabung</p>
<pre><code>KUSI [f] [x]
[f] ......... Funktionsmatrix Datei (F)
[x] ......... Funktionswert x
</code></pre>
<h2 id="kusf">KUSF</h2>
<p>Kubische Spline Funktion: Berechnung einer Funktionsmatrix <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">F</mi><mo mathvariant="bold-italic">=</mo><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mrow data-mjx-texclass="ORD"><mo mathvariant="bold-italic" stretchy="false">|</mo></mrow><mi mathvariant="bold-italic">p</mi><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo mathvariant="bold-italic" stretchy="false">)</mo><mo mathvariant="bold-italic" stretchy="false">)</mo></mrow></math> zu Koeffizientenmatrix <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">A</mi><mo mathvariant="bold-italic">=</mo><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">b</mi><mrow data-mjx-texclass="ORD"><mo mathvariant="bold-italic" stretchy="false">|</mo></mrow><mi mathvariant="bold-italic">c</mi><mrow data-mjx-texclass="ORD"><mo mathvariant="bold-italic" stretchy="false">|</mo></mrow><mi mathvariant="bold-italic">d</mi><mo mathvariant="bold-italic" stretchy="false">)</mo></mrow></math>, wobei</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>s</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo>+</mo><msub><mi>b</mi><mi>i</mi></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>c</mi><mi>i</mi></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msub><mi>d</mi><mi>i</mi></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo>;</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1.</mn></math></p>
<ul>
<li>Übernahme einer ASCII Funktionsmatrix Datei <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">F</mi></mrow></math>;</li>
<li>Übernahme der ASCII Koeffizientenmatrix Datei <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">A</mi></mrow></math> (<code>KUSI.txt</code>);</li>
<li>Ausgabe der ASCII Funktionsmatrix Datei <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">S</mi></mrow></math> (<code>KUSF.txt</code>).</li>
</ul>
<p>Handhabung</p>
<pre><code>KUSF [f] [a] [b] [d]
[f] ......... Funktionsmatrix Datei (F)
[a] ......... (x) Minimum
[b] ......... (x) Maximum
[d] ......... Intervall d
</code></pre>
<h2 id="nwti">NWTI</h2>
<p>Newton Interpolation: Berechnung des Koeffizientenvektors <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">a</mi></mrow></math> sowie <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">p</mi><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo mathvariant="bold-italic" stretchy="false">)</mo></mrow></math> zu einer (empirischen) Funktionsmatrix <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">F</mi><mo mathvariant="bold-italic">=</mo><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mrow data-mjx-texclass="ORD"><mo mathvariant="bold-italic" stretchy="false">|</mo></mrow><mi mathvariant="bold-italic">y</mi><mo mathvariant="bold-italic" stretchy="false">)</mo></mrow></math>, wobei</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mi>a</mi><mi>n</mi></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>·</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>n</mi><mo stretchy="false">)</mo><mo>.</mo></math></p>
<ul>
<li>Übernahme einer ASCII Funktionsmatrix Datei <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">F</mi></mrow></math>;</li>
<li>Ausgabe einer ASCII Koeffizientenvektor Datei <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">a</mi></mrow></math> (<code>nwti.txt</code>);</li>
<li>Berechnung von <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">p</mi><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo mathvariant="bold-italic" stretchy="false">)</mo></mrow></math> über das Interpolations-Polynom.</li>
</ul>
<p>Handhabung</p>
<pre><code>NWTI [f] [x]
[f] ......... Funktionsmatrix Datei (F)
[x] ......... Funktionswert x
</code></pre>
<h2 id="nwtp">NWTP</h2>
<p>Newton Interpolations Polynom: Berechnung einer Funktionsmatrix <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">F</mi><mo mathvariant="bold-italic">=</mo><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mrow data-mjx-texclass="ORD"><mo mathvariant="bold-italic" stretchy="false">|</mo></mrow><mi mathvariant="bold-italic">p</mi><mo mathvariant="bold-italic" stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo mathvariant="bold-italic" stretchy="false">)</mo><mo mathvariant="bold-italic" stretchy="false">)</mo></mrow></math> zu Koeffizientenvektor <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">a</mi></mrow></math>, wobei</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mi>a</mi><mi>n</mi></msub><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>·</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>·</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>n</mi><mo stretchy="false">)</mo><mo>.</mo></math></p>
<ul>
<li>Übernahme der ASCII Koeffizientenvektor Datei <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">a</mi></mrow></math> (<code>nwti.txt</code>);</li>
<li>Ausgabe der ASCII Funktionsmatrix Datei <math display="inline"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">F</mi></mrow></math> (<code>nwtp.txt</code>).</li>
</ul>
<p>Handhabung</p>
<pre><code>NWTP [a] [b] [d]
[a] ......... (x) Minimum
[b] ......... (x) Maximum
[d] ......... Intervall d
</code></pre>
<br>
<h2 id="references">References</h2>
<p>Meyberg, K., & Vachenauer, P. (2001). Integration. In <em>Höhere Mathematik 1</em>. Springer-Lehrbuch. Springer, Berlin, Heidelberg. <a href="https://doi.org/10.1007/978-3-642-56654-7_4">https://doi.org/10.1007/978-3-642-56654-7_4</a></p>
<hr WIDTH="75%">
<blockquote ALIGN="CENTER">
<center>
<a href="https://github.com/Schrausser/ConsoleApp_Integral">
ConsoleApp_Integral
</a>
<br><font color="#999999"><font size=-1>
<a
href="https://orcid.org/0000-0002-4924-8280">
Dietmar Gerald Schrausser
</a>
<p>22.07.2025</p>
</font></font>
</center>
</blockquote>
</body>
</html>