<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="numdiff" xml:lang="en">
<refnamediv>
<refname>numdiff</refname>
- <refpurpose>numerical gradient estimation</refpurpose>
+ <refpurpose>numerical gradient estimation at one point</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<para>an external, Scilab function or list. See below for calling
sequence, see also <link linkend="external">external</link> for
details about external functions.
+ f: R<superscript>n</superscript> --> R<superscript>p</superscript>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>x</term>
<listitem>
- <para>a vector, the argument of the function
- <varname>fun</varname>.
+ <para>
+ a vector of the <code>n</code> coordinates of the single point at which the gradient is sought.
</para>
</listitem>
</varlistentry>
<term>dx</term>
<listitem>
<para>a vector, the finite difference step. Default value is
- <code>dx=sqrt(%eps)*(1+1d-3*abs(x))</code>.
+ <code>dx = sqrt(%eps)*(1+1d-3*abs(x))</code>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>g</term>
<listitem>
- <para>a vector, the estimated gradient.</para>
+ <para>
+ a matrix, the estimated gradient at the locus <varname>x</varname>.
+ </para>
</listitem>
</varlistentry>
</variablelist>
<title>Description</title>
<para>
Given a function <code>fun(x)</code> from
- <code>R^n</code> to <code>R^p</code> computes the matrix
- <varname>g</varname> such as
+ R<superscript>n</superscript> to R<superscript>p</superscript> computes the <code>p x n</code> matrix
+ <varname>g</varname> such that
</para>
<programlisting role="no-scilab-exec"><
]]></programlisting>
<para>using finite difference methods.
Uses an order 1 formula.
</para>
<para>
Without parameters, the function <varname>fun</varname> calling sequence is
- <code>y=fun(x)</code>, and <function>numdiff</function> can be called as
- <code>g=numdiff(fun,x)</code>. Else the function <varname>fun</varname> calling
+ <code>y = fun(x)</code>, with <varname>x</varname> ∈ R<superscript>n</superscript> and <varname>y</varname> ∈ R<superscript>p</superscript>, and <function>numdiff</function> can be called as
+ <code>g = numdiff(fun, x)</code>. Else the function <varname>fun</varname> calling
sequence must be <literal>y = fun(x, param_1, pararm_2, ..., param_q)</literal>.
If parameters <literal>param_1, param_2, ..., param_q</literal> exist then
<function>numdiff</function> can be called as follow
- <literal>g=numdiff(list(fun, param_1, param_2, ..., param_q), x)</literal>.
+ <literal>g = numdiff(list(fun, param_1, param_2, ..., param_q), x)</literal>.
</para>
<para>
See the
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-// example 1 (without parameters)
-// myfun is a function from R^2 to R: (x(1),x(2)) |--> myfun(x)
-function f=myfun(x)
- f=x(1)*x(1)+x(1)*x(2)
+ <programlisting role="example"><![CDATA[
+// Example 1 (without parameters)
+// myfun is a function from R^2 to R: (x(1), x(2)) |--> myfun(x)
+function f = myfun(x)
+ f = x(1)*x(1) + x(1)*x(2)
endfunction
-x=[5 8]
-g=numdiff(myfun,x)
+x = [5 8];
+g = numdiff(myfun, x)
+
+// The exact gradient (i.e first component = derivate with respect to x(1)
+// and second component = derivate with respect to x(2)) is:
+exact = [2*x(1)+x(2) x(1)]
-// The exact gradient (i.e derivate belong x(1): first component
-// and derivate belong x(2): second component) is
-exact=[2*x(1)+x(2) x(1)]
-//example 2 (with parameters)
-// myfun is a function from R to R: x(1) |--> myfun(x)
-// myfun contains 3 parameters: a, b, c
-function f=myfun(x,a,b,c)
- f=(x+a)^c+b
+// Example 2 (with parameters)
+// myfun is a function from R to R: x |--> myfun(x)
+// myfun contains 3 parameters: a, b and c
+function f = myfun(x, a, b, c)
+ f = (x+a)^c + b
endfunction
-a=3; b=4; c=2;
-x=1
-g2=numdiff(list(myfun,a,b,c),x)
+a = 3; b = 4; c = 2;
+x = 1;
+g2 = numdiff(list(myfun, a, b, c), x)
+
+// The exact gradient, i.e derivate with respiect to x, is:
+exact2 = c*(x+a)^(c-1)
+
+
+// Example 3 (f: R^3 --> R^3)
+// myfun is a function from R^2 to R^2: (x(1), x(2), x(3)) |--> (myfun(x)(1), myfun(x)(2), mfun(x)(3))
+function f = myfun(x)
+ f(1) = x(1) * x(1);
+ f(2) = x(1) * x(2) * x(3);
+ f(3) = 2*x(1) + 2*x(2) + 2*x(3);
+endfunction
+
+x = [5 8 10];
+g = numdiff(myfun, x)
+
+// The exact gradient is:
+// [ df_1/dx_1 df_1/dx_2 df_1/dx_3 ;
+// df_2/dx_1 df_2/dx_2 df_2/dx_3 ;
+// df_3/dx_1 df_3/dx_2 df_3/dx_3 ; ]
+exact3 = [2*x(1) 0 0 ; x(2)*x(3) x(1)*x(3) x(1)*x(2) ; 2 2 2]
-// The exact gradient, i.e derivate belong x(1), is :
-exact2=c*(x+a)^(c-1)
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
+ <link linkend="interp">interp</link>
+ </member>
+ <member>
+ <link linkend="interp2d">interp2d</link>
+ </member>
+ <member>
+ <link linkend="splin">splin</link>
+ </member>
+ <member>
+ <link linkend="eval_cshep2d">eval_cshep2d</link>
+ </member>
+ <member>
<link linkend="optim">optim</link>
</member>
<member>
+ <link linkend="diff">diff</link>
+ </member>
+ <member>
<link linkend="derivative">derivative</link>
</member>
<member>
projects / scilab.git / commitdiff