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<a href="Scientific-module.html">Package Scientific</a> ::
<a href="Scientific.Functions-module.html">Package Functions</a> ::
Module Derivatives
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<!-- ==================== MODULE DESCRIPTION ==================== -->
<h1 class="epydoc">Module Derivatives</h1><p class="nomargin-top"></p>
<p>Automatic differentiation for functions of any number of variables up
to any order</p>
<p>An instance of the class DerivVar represents the value of a function
and the values of its partial <a name="index-derivatives"></a><i
class="indexterm">derivatives</i> 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>
<p>Example:</p>
<pre class="literalblock">
print sin(DerivVar(2))
</pre>
<p>produces the output:</p>
<pre class="literalblock">
(0.909297426826, [-0.416146836547])
</pre>
<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>
<p>Example:</p>
<pre class="literalblock">
>>>x = DerivVar(7., 0)
>>>y = DerivVar(42., 1)
>>>z = DerivVar(pi, 2)
>>>print (sqrt(pow(x,2)+pow(y,2)+pow(z,2)))
produces the output
>>>(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>
<p>Example:</p>
<pre class="literalblock">
>>>x = DerivVar(3., 0, 3)
>>>y = DerivVar(5., 1, 3)
>>>print sqrt(x*y)
produces the output
>>>(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>
<p>The individual orders can be extracted by indexing:</p>
<pre class="literalblock">
>>>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>
<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.:</p>
<pre class="literalblock">
>>>print DerivVar(3, 0)+DerivVar(5, 0)
produces
>>>(8, [2])
</pre>
<p>but this result is meaningless.</p>
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<span class="summary-type"> </span>
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<a href="Scientific.Functions.Derivatives.DerivVar-class.html" class="summary-name">DerivVar</a><br />
Numerical variable with automatic derivatives of arbitrary order
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<!-- ==================== FUNCTIONS ==================== -->
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<td><span class="summary-sig"><a href="Scientific.Functions.Derivatives-module.html#DerivVector" class="summary-sig-name">DerivVector</a>(<span class="summary-sig-arg">x</span>,
<span class="summary-sig-arg">y</span>,
<span class="summary-sig-arg">z</span>,
<span class="summary-sig-arg">index</span>=<span class="summary-sig-default">0</span>,
<span class="summary-sig-arg">order</span>=<span class="summary-sig-default">1</span>)</span><br />
Returns:
a vector whose components are DerivVar objects</td>
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</td>
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<span class="summary-type"><code>bool</code></span>
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<td><span class="summary-sig"><a href="Scientific.Functions.Derivatives-module.html#isDerivVar" class="summary-sig-name">isDerivVar</a>(<span class="summary-sig-arg">x</span>)</span><br />
Returns:
True if x is a DerivVar object, False otherwise</td>
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<a name="DerivVector"></a>
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<h3 class="epydoc"><span class="sig"><span class="sig-name">DerivVector</span>(<span class="sig-arg">x</span>,
<span class="sig-arg">y</span>,
<span class="sig-arg">z</span>,
<span class="sig-arg">index</span>=<span class="sig-default">0</span>,
<span class="sig-arg">order</span>=<span class="sig-default">1</span>)</span>
</h3>
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>
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<dl class="fields">
<dt>Parameters:</dt>
<dd><ul class="nomargin-top">
<li><strong class="pname"><code>x</code></strong> (number) - x component of the vector</li>
<li><strong class="pname"><code>y</code></strong> (number) - y component of the vector</li>
<li><strong class="pname"><code>z</code></strong> (number) - z component of the vector</li>
<li><strong class="pname"><code>index</code></strong> (<code>int</code>) - the DerivVar index for the x component. The y and z components
receive consecutive indices.</li>
<li><strong class="pname"><code>order</code></strong> (<code>int</code>) - the derivative order</li>
</ul></dd>
<dt>Returns: <a href="Scientific.Geometry.Vector-class.html"
class="link">Scientific.Geometry.Vector</a></dt>
<dd>a vector whose components are DerivVar objects</dd>
</dl>
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<a name="isDerivVar"></a>
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<h3 class="epydoc"><span class="sig"><span class="sig-name">isDerivVar</span>(<span class="sig-arg">x</span>)</span>
</h3>
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>
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<dl class="fields">
<dt>Parameters:</dt>
<dd><ul class="nomargin-top">
<li><strong class="pname"><code>x</code></strong> - an arbitrary object</li>
</ul></dd>
<dt>Returns: <code>bool</code></dt>
<dd>True if x is a DerivVar object, False otherwise</dd>
</dl>
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