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<a name="Module:Scientific.Functions.Derivatives"><h1>Module Scientific.Functions.Derivatives</h1></a>
<p>This module provides automatic differentiation for functions with
any number of variables up to any order. An instance of the class
DerivVar represents the value of a function and the values of its partial
derivatives with respect to a list of variables. All common
mathematical operations and functions are available for these numbers.
There is no restriction on the type of the numbers fed into the
code; it works for real and complex numbers as well as for
any Python type that implements the necessary operations.</p>
<p>If only first-order derivatives are required, the module
FirstDerivatives should be used. It is compatible to this
one, but significantly faster.</p>
Example:
<p> <tt>print sin(DerivVar(2))</tt></p>
<p> produces the output</p>
<p> <tt>(0.909297426826, [-0.416146836547])</tt></p>
<p>The first number is the value of sin(2); the number in the following
list is the value of the derivative of sin(x) at x=2, i.e. cos(2).</p>
<p>When there is more than one variable, DerivVar must be called with
an integer second argument that specifies the number of the variable.</p>
Example:
<pre>
x = DerivVar(7., 0)
y = DerivVar(42., 1)
z = DerivVar(pi, 2)
print (sqrt(pow(x,2)+pow(y,2)+pow(z,2)))
</pre>
<p> produces the output</p>
<pre>
(42.6950770511, [0.163953328662, 0.98371997197, 0.0735820818365])
</pre>
<p>The numbers in the list are the partial derivatives with respect
to x, y, and z, respectively.</p>
<p>Higher-order derivatives are requested with an optional third
argument to DerivVar.</p>
Example:
<pre>
x = DerivVar(3., 0, 3)
y = DerivVar(5., 1, 3)
print sqrt(x*y)
</pre>
<p> produces the output</p>
<pre>
(3.87298334621,
[0.645497224368, 0.387298334621],
[[-0.107582870728, 0.0645497224368],
[0.0645497224368, -0.0387298334621]],
[[[0.053791435364, -0.0107582870728],
[-0.0107582870728, -0.00645497224368]],
[[-0.0107582870728, -0.00645497224368],
[-0.00645497224368, 0.0116189500386]]])
</pre>
The individual orders can be extracted by indexing:
<pre>
print sqrt(x*y)[0]
3.87298334621
print sqrt(x*y)[1]
[0.645497224368, 0.387298334621]
</pre>
<p>An n-th order derivative is represented by a nested list of
depth n.</p>
<p>When variables with different differentiation orders are mixed,
the result has the lower one of the two orders. An exception are
zeroth-order variables, which are treated as constants.</p>
<p>Caution: Higher-order derivatives are implemented by recursively
using DerivVars to represent derivatives. This makes the code
very slow for high orders.</p>
Note: It doesn't make sense to use multiple DerivVar objects with
different values for the same variable index in one calculation, but
there is no check for this. I.e.
<pre>
print DerivVar(3, 0)+DerivVar(5, 0)
</pre>
<p> produces</p>
<pre>
(8, [2])
</pre>
<p>but this result is meaningless.
</p>
<hr width=70%>
<h2>Functions</h2>
<ul>
<li> <p>
<a name="Function:Scientific.Functions.Derivatives.isDerivVar"><b><i>isDerivVar</i></b>(<i>x</i>)</a><br>
</p>
<p>Returns 1 if <i>x</i> is a DerivVar object.</p><li> <p>
<a name="Function:Scientific.Functions.Derivatives.DerivVector"><b><i>DerivVector</i></b>(<i>x</i>, <i>y</i>, <i>z</i>, <i>index</i>=<tt>0</tt>, <i>order</i>=<tt>1</tt>)</a><br>
</p>
<p>Returns a vector whose components are DerivVar objects.</p>
<p><dl>
<dt><i>x</i>, <i>y</i>, <i>z</i></dt>
<dd><p>
vector components (numbers)</p></dd>
<dt><i>index</i></dt>
<dd><p>
the DerivVar index for the x component. The y and z
components receive consecutive indices.</p></dd>
<dt><i>order</i></dt>
<dd><p>
the derivative order
</p></dd>
</dl>
</p></ul>
<hr width=70%>
<a name="Class:Scientific.Functions.Derivatives.DerivVar"><h2>Class DerivVar: Variable with derivatives</h2></a>
<p>Constructor: DerivVar(<i>value</i>, <i>index</i> = 0, <i>order</i> = 1)</p>
<p><dl>
<dt><i>value</i></dt>
<dd><p>
the numerical value of the variable</p></dd>
<dt><i>index</i></dt>
<dd><p>
the variable index (an integer), which serves to
distinguish between variables and as an index for
the derivative lists. Each explicitly created
instance of DerivVar must have a unique index.</p></dd>
<dt><i>order</i></dt>
<dd><p>
the derivative order</p></dd>
</dl>
</p>
<p>Indexing with an integer yields the derivatives of the corresponding
order.
</p>
<b>Methods:</b><br>
<ul>
<li> <b><i>toOrder</i></b>(<i>order</i>)
<p>Returns a DerivVar object with a lower derivative order.</p>
</ul>
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