scilab/modules/differential_equations/help/en_US/intg.xml

   1 <?xml version="1.0" encoding="UTF-8"?>
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  14 <refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns5="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:id="intg" xml:lang="en">
  15     <refnamediv>
  16         <refname>intg</refname>
  17         <refpurpose>definite integral</refpurpose>
  18     </refnamediv>
  19     <refsynopsisdiv>
  20         <title>Calling Sequence</title>
  21         <synopsis>[v,err]=intg(a,b,f [,ea [,er])</synopsis>
  22     </refsynopsisdiv>
  23     <refsection>
  24         <title>Arguments</title>
  25         <variablelist>
  26             <varlistentry>
  27                 <term>a, b</term>
  28                 <listitem>
  29                     <para>real numbers.</para>
  30                 </listitem>
  31             </varlistentry>
  32             <varlistentry>
  33                 <term>f</term>
  34                 <listitem>
  35                     <para>external (function or list or string).</para>
  36                 </listitem>
  37             </varlistentry>
  38             <varlistentry>
  39                 <term>ea, er</term>
  40                 <listitem>
  41                     <para>real numbers.</para>
  42                 </listitem>
  43             </varlistentry>
  44             <varlistentry>
  45                 <term>ea</term>
  46                 <listitem>
  47                     <para>absolute error required on the result. Default value:
  48                         1.d-14.
  49                     </para>
  50                 </listitem>
  51             </varlistentry>
  52             <varlistentry>
  53                 <term>er</term>
  54                 <listitem>
  55                     <para>relative error required on the result. Default value:
  56                         1.d-8.
  57                     </para>
  58                 </listitem>
  59             </varlistentry>
  60             <varlistentry>
  61                 <term>err</term>
  62                 <listitem>
  63                     <para>estimated absolute error on the result.</para>
  64                 </listitem>
  65             </varlistentry>
  66         </variablelist>
  67     </refsection>
  68     <refsection>
  69         <title>Description</title>
  70         <para>
  71             <literal>intg(a,b,f)</literal> evaluates the definite integral from
  72             <literal>a</literal> to <literal>b</literal> of <literal>f(t)dt</literal>.
  73             The function <literal>f(t)</literal> should be continuous.
  74         </para>
  75         <para>The evaluation hopefully satisfies following claim for accuracy:
  76             <literal>abs(I-v)&lt;= max(ea,er*abs(I))</literal> where
  77             <literal>I</literal> stands for the exact value of the integral.
  78         </para>
  79         <para>
  80             <literal>f</literal> is an external :
  81         </para>
  82         <para>
  83             If <literal>f</literal> is function its definition must be as
  84             follows <literal> y = f(t) </literal>
  85         </para>
  86         <para>
  87             If <literal>f</literal> is a list the list must be as follows:
  88             <literal> list(f,x1,x2,...)</literal> where <literal>f</literal> is a
  89             function with calling sequence <literal>f(t,x1,x2,...)</literal>.
  90         </para>
  91         <para>
  92             If <literal>f</literal> is a string it refers to the name of a
  93             Fortran function or a C prodedure with a given calling sequence:
  94         </para>
  95         <para>
  96             In the fortran case the calling sequence should be <literal>double
  97                 precision function f(x)
  98             </literal>
  99             where <literal>x</literal> is also a
 100             double precision number.
 101         </para>
 102         <para>
 103             In the C case the calling sequence should be <literal>double
 104                 f(double *x)
 105             </literal>
 106             .
 107         </para>
 108     </refsection>
 109     <refsection>
 110         <title>Examples</title>
 111         <programlisting role="example"><![CDATA[
 112 // Function written in the Scilab language
 113 function y=f(x),y=x*sin(30*x)/sqrt(1-((x/(2*%pi))^2)),endfunction
 114 exact=-2.5432596188;
 115 I=intg(0,2*%pi,f)
 116 abs(exact-I)
 117
 118 // Function with an argument written in the Scilab language
 119 function y=f1(x,w),y=x*sin(w*x)/sqrt(1-((x/(2*%pi))^2)),endfunction
 120 I=intg(0,2*%pi,list(f1,30))
 121 abs(exact-I)
 122
 123
 124 // Function written in Fortran (a Fortran compiler is required)
 125 // define a Fortran function
 126 cd TMPDIR;
 127 F=['      double precision function ffun(x)'
 128    '      double precision x,pi'
 129    '      pi=3.14159265358979312d+0'
 130    '      ffun=x*sin(30.0d+0*x)/sqrt(1.0d+0-(x/(2.0d+0*pi))**2)'
 131    '      return'
 132    '      end'];
 133 mputl(F,fullfile(TMPDIR,'ffun.f'))
 134
 135 // compile the function
 136 l=ilib_for_link('ffun',fullfile(TMPDIR,'ffun.f'),[],'f');
 137
 138 // add the function to the working environment
 139 link(l,'ffun','f')
 140
 141 // integrate the function
 142 I=intg(0,2*%pi,'ffun')
 143 abs(exact-I)
 144
 145 // Function written in C (a C compiler is required)
 146 // define a C function
 147 C=['#include <math.h>'
 148    'double cfun(double *x)'
 149    '{'
 150    '  double y,pi=3.14159265358979312;'
 151    '  y=*x/(2.0e0*pi);'
 152    '  return *x*sin(30.0e0**x)/sqrt(1.0e0-y*y);'
 153    '}'];
 154 mputl(C,fullfile(TMPDIR,'cfun.c'))
 155
 156 // compile the function
 157 l=ilib_for_link('cfun',fullfile(TMPDIR,'cfun.c'),[],'c');
 158
 159 // add the function to the working environment
 160 link(l,'cfun','c')
 161
 162 // integrate the function
 163 I=intg(0,2*%pi,'cfun')
 164 abs(exact-I)
 165  ]]></programlisting>
 166     </refsection>
 167     <refsection role="see also">
 168         <title>See Also</title>
 169         <simplelist type="inline">
 170             <member>
 171                 <link linkend="intc">intc</link>
 172             </member>
 173             <member>
 174                 <link linkend="intl">intl</link>
 175             </member>
 176             <member>
 177                 <link linkend="inttrap">inttrap</link>
 178             </member>
 179             <member>
 180                 <link linkend="intsplin">intsplin</link>
 181             </member>
 182             <member>
 183                 <link linkend="ode">ode</link>
 184             </member>
 185         </simplelist>
 186     </refsection>
 187     <refsection>
 188         <title>Used Functions</title>
 189         <para>The associated routines can be found in SCI/modules/differential_equations/src/fortran directory
 190             :
 191         </para>
 192         <para>dqag0.f and dqags.f from quadpack</para>
 193     </refsection>
 194 </refentry>