333 lines
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333 lines
13 KiB
HTML
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<a href="Scientific-module.html">Package Scientific</a> ::
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<a href="Scientific.Functions-module.html">Package Functions</a> ::
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Module Derivatives
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>[<a href="frames.html" target="_top">frames</a
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<!-- ==================== MODULE DESCRIPTION ==================== -->
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<h1 class="epydoc">Module Derivatives</h1><p class="nomargin-top"></p>
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<p>Automatic differentiation for functions of any number of variables up
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to any order</p>
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<p>An instance of the class DerivVar represents the value of a function
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and the values of its partial <a name="index-derivatives"></a><i
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class="indexterm">derivatives</i> with respect to a list of variables.
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All common mathematical operations and functions are available for these
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numbers. There is no restriction on the type of the numbers fed into the
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code; it works for real and complex numbers as well as for any Python
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type that implements the necessary operations.</p>
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<p>If only first-order derivatives are required, the module
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FirstDerivatives should be used. It is compatible to this one, but
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significantly faster.</p>
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<p>Example:</p>
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<pre class="literalblock">
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print sin(DerivVar(2))
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</pre>
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<p>produces the output:</p>
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<pre class="literalblock">
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(0.909297426826, [-0.416146836547])
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</pre>
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<p>The first number is the value of sin(2); the number in the following
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list is the value of the derivative of sin(x) at x=2, i.e. cos(2).</p>
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<p>When there is more than one variable, DerivVar must be called with an
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integer second argument that specifies the number of the variable.</p>
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<p>Example:</p>
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<pre class="literalblock">
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>>>x = DerivVar(7., 0)
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>>>y = DerivVar(42., 1)
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>>>z = DerivVar(pi, 2)
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>>>print (sqrt(pow(x,2)+pow(y,2)+pow(z,2)))
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produces the output
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>>>(42.6950770511, [0.163953328662, 0.98371997197, 0.0735820818365])
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</pre>
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<p>The numbers in the list are the partial derivatives with respect to x,
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y, and z, respectively.</p>
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<p>Higher-order derivatives are requested with an optional third argument
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to DerivVar.</p>
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<p>Example:</p>
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<pre class="literalblock">
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>>>x = DerivVar(3., 0, 3)
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>>>y = DerivVar(5., 1, 3)
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>>>print sqrt(x*y)
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produces the output
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>>>(3.87298334621,
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>>> [0.645497224368, 0.387298334621],
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>>> [[-0.107582870728, 0.0645497224368],
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>>> [0.0645497224368, -0.0387298334621]],
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>>> [[[0.053791435364, -0.0107582870728],
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>>> [-0.0107582870728, -0.00645497224368]],
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>>> [[-0.0107582870728, -0.00645497224368],
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>>> [-0.00645497224368, 0.0116189500386]]])
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</pre>
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<p>The individual orders can be extracted by indexing:</p>
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<pre class="literalblock">
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>>>print sqrt(x*y)[0]
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>>>3.87298334621
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>>>print sqrt(x*y)[1]
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>>>[0.645497224368, 0.387298334621]
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</pre>
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<p>An n-th order derivative is represented by a nested list of depth
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n.</p>
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<p>When variables with different differentiation orders are mixed, the
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result has the lower one of the two orders. An exception are zeroth-order
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variables, which are treated as constants.</p>
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<p>Caution: Higher-order derivatives are implemented by recursively using
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DerivVars to represent derivatives. This makes the code very slow for
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high orders.</p>
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<p>Note: It doesn't make sense to use multiple DerivVar objects with
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different values for the same variable index in one calculation, but
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there is no check for this. I.e.:</p>
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<pre class="literalblock">
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>>>print DerivVar(3, 0)+DerivVar(5, 0)
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produces
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>>>(8, [2])
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</pre>
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<p>but this result is meaningless.</p>
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<!-- ==================== CLASSES ==================== -->
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<a name="section-Classes"></a>
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<table class="summary" border="1" cellpadding="3"
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cellspacing="0" width="100%" bgcolor="white">
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<tr bgcolor="#70b0f0" class="table-header">
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<span class="table-header">Classes</span></td>
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<span class="summary-type"> </span>
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</td><td class="summary">
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<a href="Scientific.Functions.Derivatives.DerivVar-class.html" class="summary-name">DerivVar</a><br />
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Numerical variable with automatic derivatives of arbitrary order
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</table>
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<!-- ==================== FUNCTIONS ==================== -->
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<a name="section-Functions"></a>
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<table class="summary" border="1" cellpadding="3"
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cellspacing="0" width="100%" bgcolor="white">
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<span class="table-header">Functions</span></td>
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</tr>
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<tr>
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<td width="15%" align="right" valign="top" class="summary">
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<span class="summary-type"><a href="Scientific.Geometry.Vector-class.html"
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class="link">Scientific.Geometry.Vector</a></span>
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</td><td class="summary">
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<table width="100%" cellpadding="0" cellspacing="0" border="0">
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<tr>
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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>,
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<span class="summary-sig-arg">y</span>,
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<span class="summary-sig-arg">z</span>,
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<span class="summary-sig-arg">index</span>=<span class="summary-sig-default">0</span>,
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<span class="summary-sig-arg">order</span>=<span class="summary-sig-default">1</span>)</span><br />
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Returns:
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a vector whose components are DerivVar objects</td>
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<td align="right" valign="top">
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</td>
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</table>
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</td>
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</tr>
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<tr>
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<td width="15%" align="right" valign="top" class="summary">
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<span class="summary-type"><code>bool</code></span>
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</td><td class="summary">
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<table width="100%" cellpadding="0" cellspacing="0" border="0">
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<tr>
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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 />
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Returns:
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True if x is a DerivVar object, False otherwise</td>
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<td align="right" valign="top">
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</td>
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</table>
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</table>
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<!-- ==================== FUNCTION DETAILS ==================== -->
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<a name="section-FunctionDetails"></a>
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<table class="details" border="1" cellpadding="3"
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cellspacing="0" width="100%" bgcolor="white">
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<tr bgcolor="#70b0f0" class="table-header">
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<td align="left" colspan="2" class="table-header">
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<span class="table-header">Function Details</span></td>
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</tr>
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</table>
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<a name="DerivVector"></a>
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<div>
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<table class="details" border="1" cellpadding="3"
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cellspacing="0" width="100%" bgcolor="white">
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<tr><td>
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<table width="100%" cellpadding="0" cellspacing="0" border="0">
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<tr valign="top"><td>
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<h3 class="epydoc"><span class="sig"><span class="sig-name">DerivVector</span>(<span class="sig-arg">x</span>,
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<span class="sig-arg">y</span>,
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<span class="sig-arg">z</span>,
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<span class="sig-arg">index</span>=<span class="sig-default">0</span>,
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<span class="sig-arg">order</span>=<span class="sig-default">1</span>)</span>
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</h3>
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</td><td align="right" valign="top"
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>
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</td>
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</tr></table>
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<dl class="fields">
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<dt>Parameters:</dt>
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<dd><ul class="nomargin-top">
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<li><strong class="pname"><code>x</code></strong> (number) - x component of the vector</li>
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<li><strong class="pname"><code>y</code></strong> (number) - y component of the vector</li>
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<li><strong class="pname"><code>z</code></strong> (number) - z component of the vector</li>
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<li><strong class="pname"><code>index</code></strong> (<code>int</code>) - the DerivVar index for the x component. The y and z components
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receive consecutive indices.</li>
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<li><strong class="pname"><code>order</code></strong> (<code>int</code>) - the derivative order</li>
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</ul></dd>
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<dt>Returns: <a href="Scientific.Geometry.Vector-class.html"
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class="link">Scientific.Geometry.Vector</a></dt>
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<dd>a vector whose components are DerivVar objects</dd>
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</dl>
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</td></tr></table>
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</div>
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<a name="isDerivVar"></a>
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<div>
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<table class="details" border="1" cellpadding="3"
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cellspacing="0" width="100%" bgcolor="white">
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<tr><td>
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<table width="100%" cellpadding="0" cellspacing="0" border="0">
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<tr valign="top"><td>
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<h3 class="epydoc"><span class="sig"><span class="sig-name">isDerivVar</span>(<span class="sig-arg">x</span>)</span>
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</h3>
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</td><td align="right" valign="top"
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>
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</td>
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</tr></table>
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<dl class="fields">
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<dt>Parameters:</dt>
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<dd><ul class="nomargin-top">
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<li><strong class="pname"><code>x</code></strong> - an arbitrary object</li>
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</ul></dd>
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<dt>Returns: <code>bool</code></dt>
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<dd>True if x is a DerivVar object, False otherwise</dd>
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</dl>
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