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docs: stochastic euler
This commit is contained in:
Karsten Ahnert
parent
6b622cd2e1
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b40639d8c0
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to be used within <a class="link" href="../odeint_in_detail/integrate_functions.html" title="Integrate functions">integrate
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to be used within <a class="link" href="../odeint_in_detail/integrate_functions.html" title="Integrate functions">integrate
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functions</a>.
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functions</a>.
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</p>
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</p>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.description"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.description-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.h0"></a>
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<a class="link" href="controlled_stepper.html#boost_sandbox_numeric_odeint.concepts.controlled_stepper.description">Description</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.description"></a></span><a class="link" href="controlled_stepper.html#boost_sandbox_numeric_odeint.concepts.controlled_stepper.description">Description</a>
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</h5>
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</h5>
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<p>
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<p>
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A controlled stepper following this Controlled Stepper concept provides the
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A controlled stepper following this Controlled Stepper concept provides the
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with a given step-size <span class="emphasis"><em>dt</em></span>. Depending on an error estimate
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with a given step-size <span class="emphasis"><em>dt</em></span>. Depending on an error estimate
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of the solution the step might be rejected and a smaller step-size is suggested.
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of the solution the step might be rejected and a smaller step-size is suggested.
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</p>
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</p>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.notation"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.notation-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.h1"></a>
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<a class="link" href="controlled_stepper.html#boost_sandbox_numeric_odeint.concepts.controlled_stepper.notation">Notation</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.notation"></a></span><a class="link" href="controlled_stepper.html#boost_sandbox_numeric_odeint.concepts.controlled_stepper.notation">Notation</a>
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</h5>
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</h5>
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<p class="title"><b></b></p>
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</p></dd>
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</dl>
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</div>
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</div>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.valid_expressions"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.valid_expressions-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.h2"></a>
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<a class="link" href="controlled_stepper.html#boost_sandbox_numeric_odeint.concepts.controlled_stepper.valid_expressions">Valid
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<span><a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.valid_expressions"></a></span><a class="link" href="controlled_stepper.html#boost_sandbox_numeric_odeint.concepts.controlled_stepper.valid_expressions">Valid
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Expressions</a>
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Expressions</a>
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</h5>
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</h5>
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<div class="informaltable"><table class="table">
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<div class="informaltable"><table class="table">
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</tr>
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</tr>
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</tbody>
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</tbody>
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</table></div>
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</table></div>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.models"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.models-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.h3"></a>
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<a class="link" href="controlled_stepper.html#boost_sandbox_numeric_odeint.concepts.controlled_stepper.models">Models</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.controlled_stepper.models"></a></span><a class="link" href="controlled_stepper.html#boost_sandbox_numeric_odeint.concepts.controlled_stepper.models">Models</a>
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</h5>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" type="disc">
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<div class="itemizedlist"><ul class="itemizedlist" type="disc">
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<li class="listitem">
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<li class="listitem">
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to be used within <a class="link" href="../odeint_in_detail/integrate_functions.html" title="Integrate functions">integrate
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to be used within <a class="link" href="../odeint_in_detail/integrate_functions.html" title="Integrate functions">integrate
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functions</a>.
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functions</a>.
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</p>
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</p>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.description"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.description-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.h0"></a>
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<a class="link" href="dense_output_stepper.html#boost_sandbox_numeric_odeint.concepts.dense_output_stepper.description">Description</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.description"></a></span><a class="link" href="dense_output_stepper.html#boost_sandbox_numeric_odeint.concepts.dense_output_stepper.description">Description</a>
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</h5>
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</h5>
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<p>
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<p>
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A dense ouput stepper following this Dense Output Stepper concept provides
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A dense ouput stepper following this Dense Output Stepper concept provides
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to calculate the state <span class="emphasis"><em>x(t')</em></span> at any point <span class="emphasis"><em>t
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to calculate the state <span class="emphasis"><em>x(t')</em></span> at any point <span class="emphasis"><em>t
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<= t' <= t+dt</em></span>.
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<= t' <= t+dt</em></span>.
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</p>
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</p>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.associated_types"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.associated_types-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.h1"></a>
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<a class="link" href="dense_output_stepper.html#boost_sandbox_numeric_odeint.concepts.dense_output_stepper.associated_types">Associated
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<span><a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.associated_types"></a></span><a class="link" href="dense_output_stepper.html#boost_sandbox_numeric_odeint.concepts.dense_output_stepper.associated_types">Associated
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types</a>
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types</a>
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</h5>
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</h5>
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<div class="variablelist">
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<div class="variablelist">
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</p></dd>
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</p></dd>
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</dl>
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</dl>
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</div>
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</div>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.notation"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.notation-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.h2"></a>
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<a class="link" href="dense_output_stepper.html#boost_sandbox_numeric_odeint.concepts.dense_output_stepper.notation">Notation</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.notation"></a></span><a class="link" href="dense_output_stepper.html#boost_sandbox_numeric_odeint.concepts.dense_output_stepper.notation">Notation</a>
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</h5>
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</h5>
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<div class="variablelist">
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<div class="variablelist">
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<p class="title"><b></b></p>
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<p class="title"><b></b></p>
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</p></dd>
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</p></dd>
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</dl>
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</dl>
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</div>
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</div>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.valid_expressions"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.valid_expressions-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.h3"></a>
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<a class="link" href="dense_output_stepper.html#boost_sandbox_numeric_odeint.concepts.dense_output_stepper.valid_expressions">Valid
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<span><a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.valid_expressions"></a></span><a class="link" href="dense_output_stepper.html#boost_sandbox_numeric_odeint.concepts.dense_output_stepper.valid_expressions">Valid
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Expressions</a>
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Expressions</a>
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</h5>
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</h5>
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<div class="informaltable"><table class="table">
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<div class="informaltable"><table class="table">
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@ -284,9 +284,9 @@
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</tr>
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</tr>
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</tbody>
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</tbody>
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</table></div>
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</table></div>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.models"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.models-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.h4"></a>
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<a class="link" href="dense_output_stepper.html#boost_sandbox_numeric_odeint.concepts.dense_output_stepper.models">Models</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.dense_output_stepper.models"></a></span><a class="link" href="dense_output_stepper.html#boost_sandbox_numeric_odeint.concepts.dense_output_stepper.models">Models</a>
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</h5>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" type="disc">
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<div class="itemizedlist"><ul class="itemizedlist" type="disc">
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<li class="listitem">
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<li class="listitem">
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This concepts specifies the interface an error stepper has to fulfill to
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This concepts specifies the interface an error stepper has to fulfill to
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be used within a ControlledErrorStepper.
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be used within a ControlledErrorStepper.
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</p>
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</p>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.description"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.description-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.h0"></a>
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<a class="link" href="error_stepper.html#boost_sandbox_numeric_odeint.concepts.error_stepper.description">Description</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.error_stepper.description"></a></span><a class="link" href="error_stepper.html#boost_sandbox_numeric_odeint.concepts.error_stepper.description">Description</a>
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</h5>
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</h5>
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<p>
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<p>
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An error stepper following this Error Stepper concept is capable of doing
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An error stepper following this Error Stepper concept is capable of doing
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Symplectic System</a> or <a class="link" href="implicit_system.html" title="Implicit System">Implicit
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Symplectic System</a> or <a class="link" href="implicit_system.html" title="Implicit System">Implicit
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System</a>.
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System</a>.
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</p>
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</p>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.notation"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.notation-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.h1"></a>
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<a class="link" href="error_stepper.html#boost_sandbox_numeric_odeint.concepts.error_stepper.notation">Notation</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.error_stepper.notation"></a></span><a class="link" href="error_stepper.html#boost_sandbox_numeric_odeint.concepts.error_stepper.notation">Notation</a>
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</h5>
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</h5>
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<div class="variablelist">
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<div class="variablelist">
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<p class="title"><b></b></p>
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<p class="title"><b></b></p>
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</p></dd>
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</p></dd>
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</dl>
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</dl>
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</div>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.valid_expressions"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.valid_expressions-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.h2"></a>
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<a class="link" href="error_stepper.html#boost_sandbox_numeric_odeint.concepts.error_stepper.valid_expressions">Valid
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<span><a name="boost_sandbox_numeric_odeint.concepts.error_stepper.valid_expressions"></a></span><a class="link" href="error_stepper.html#boost_sandbox_numeric_odeint.concepts.error_stepper.valid_expressions">Valid
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Expressions</a>
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Expressions</a>
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</h5>
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</h5>
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<div class="informaltable"><table class="table">
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<div class="informaltable"><table class="table">
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</tr>
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</tr>
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</tbody>
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</tbody>
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</table></div>
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</table></div>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.models"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.models-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.error_stepper.h3"></a>
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<a class="link" href="error_stepper.html#boost_sandbox_numeric_odeint.concepts.error_stepper.models">Models</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.error_stepper.models"></a></span><a class="link" href="error_stepper.html#boost_sandbox_numeric_odeint.concepts.error_stepper.models">Models</a>
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</h5>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" type="disc">
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<div class="itemizedlist"><ul class="itemizedlist" type="disc">
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<li class="listitem">
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<a name="boost_sandbox_numeric_odeint.concepts.implicit_system"></a><a class="link" href="implicit_system.html" title="Implicit System">Implicit
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<a name="boost_sandbox_numeric_odeint.concepts.implicit_system"></a><a class="link" href="implicit_system.html" title="Implicit System">Implicit
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System</a>
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System</a>
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</h3></div></div></div>
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</h3></div></div></div>
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<a name="boost_sandbox_numeric_odeint.concepts.implicit_system.description"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.implicit_system.description-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.implicit_system.h0"></a>
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<a class="link" href="implicit_system.html#boost_sandbox_numeric_odeint.concepts.implicit_system.description">Description</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.implicit_system.description"></a></span><a class="link" href="implicit_system.html#boost_sandbox_numeric_odeint.concepts.implicit_system.description">Description</a>
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</h5>
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</h5>
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<p>
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<p>
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This concept describes how to define a ODE that can be solved by an implicit
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This concept describes how to define a ODE that can be solved by an implicit
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therefore, the <span class="emphasis"><em>state_type</em></span> implicit routines is <span class="emphasis"><em>ublas::vector</em></span>
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therefore, the <span class="emphasis"><em>state_type</em></span> implicit routines is <span class="emphasis"><em>ublas::vector</em></span>
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and the matrix is defined as <span class="emphasis"><em>ublas::matrix</em></span>.
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and the matrix is defined as <span class="emphasis"><em>ublas::matrix</em></span>.
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</p>
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</p>
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<a name="boost_sandbox_numeric_odeint.concepts.implicit_system.notation"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.implicit_system.notation-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.implicit_system.h1"></a>
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<a class="link" href="implicit_system.html#boost_sandbox_numeric_odeint.concepts.implicit_system.notation">Notation</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.implicit_system.notation"></a></span><a class="link" href="implicit_system.html#boost_sandbox_numeric_odeint.concepts.implicit_system.notation">Notation</a>
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</h5>
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</p></dd>
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</p></dd>
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</dl>
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</dl>
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</div>
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<a name="boost_sandbox_numeric_odeint.concepts.implicit_system.valid_expressions"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.implicit_system.valid_expressions-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.implicit_system.h2"></a>
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<a class="link" href="implicit_system.html#boost_sandbox_numeric_odeint.concepts.implicit_system.valid_expressions">Valid
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<span><a name="boost_sandbox_numeric_odeint.concepts.implicit_system.valid_expressions"></a></span><a class="link" href="implicit_system.html#boost_sandbox_numeric_odeint.concepts.implicit_system.valid_expressions">Valid
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Expressions</a>
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Expressions</a>
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</h5>
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</h5>
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<div class="informaltable"><table class="table">
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<div class="informaltable"><table class="table">
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<div class="titlepage"><div><div><h3 class="title">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_sandbox_numeric_odeint.concepts.observer"></a><a class="link" href="observer.html" title="Observer">Observer</a>
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<a name="boost_sandbox_numeric_odeint.concepts.observer"></a><a class="link" href="observer.html" title="Observer">Observer</a>
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</h3></div></div></div>
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</h3></div></div></div>
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<a name="boost_sandbox_numeric_odeint.concepts.observer.description"></a><h5>
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<h5>
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<a name="boost_sandbox_numeric_odeint.concepts.observer.description-heading"></a>
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<a name="boost_sandbox_numeric_odeint.concepts.observer.h0"></a>
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<a class="link" href="observer.html#boost_sandbox_numeric_odeint.concepts.observer.description">Description</a>
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<span><a name="boost_sandbox_numeric_odeint.concepts.observer.description"></a></span><a class="link" href="observer.html#boost_sandbox_numeric_odeint.concepts.observer.description">Description</a>
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</h5>
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The Observer concept is used in <code class="computeroutput"><span class="identifier">integrate</span></code>
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The Observer concept is used in <code class="computeroutput"><span class="identifier">integrate</span></code>
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or at user defined time points depending on which <code class="computeroutput"><span class="identifier">integrate</span></code>
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or at user defined time points depending on which <code class="computeroutput"><span class="identifier">integrate</span></code>
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function is used.
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function is used.
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</p>
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</p>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.observer.notation"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.observer.notation-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.observer.h1"></a>
|
||||||
<a class="link" href="observer.html#boost_sandbox_numeric_odeint.concepts.observer.notation">Notation</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.observer.notation"></a></span><a class="link" href="observer.html#boost_sandbox_numeric_odeint.concepts.observer.notation">Notation</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="variablelist">
|
<div class="variablelist">
|
||||||
<p class="title"><b></b></p>
|
<p class="title"><b></b></p>
|
||||||
@ -65,9 +65,9 @@
|
|||||||
</p></dd>
|
</p></dd>
|
||||||
</dl>
|
</dl>
|
||||||
</div>
|
</div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.observer.valid_expressions"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.observer.valid_expressions-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.observer.h2"></a>
|
||||||
<a class="link" href="observer.html#boost_sandbox_numeric_odeint.concepts.observer.valid_expressions">Valid
|
<span><a name="boost_sandbox_numeric_odeint.concepts.observer.valid_expressions"></a></span><a class="link" href="observer.html#boost_sandbox_numeric_odeint.concepts.observer.valid_expressions">Valid
|
||||||
Expressions</a>
|
Expressions</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="informaltable"><table class="table">
|
<div class="informaltable"><table class="table">
|
||||||
|
|||||||
@ -20,9 +20,9 @@
|
|||||||
<a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system"></a><a class="link" href="simple_symplectic_system.html" title="Simple Symplectic System">Simple
|
<a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system"></a><a class="link" href="simple_symplectic_system.html" title="Simple Symplectic System">Simple
|
||||||
Symplectic System</a>
|
Symplectic System</a>
|
||||||
</h3></div></div></div>
|
</h3></div></div></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.description"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.description-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.h0"></a>
|
||||||
<a class="link" href="simple_symplectic_system.html#boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.description">Description</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.description"></a></span><a class="link" href="simple_symplectic_system.html#boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.description">Description</a>
|
||||||
</h5>
|
</h5>
|
||||||
<p>
|
<p>
|
||||||
In most Hamiltonian systems the kinetic term is a quadratic term in the momentum
|
In most Hamiltonian systems the kinetic term is a quadratic term in the momentum
|
||||||
@ -44,9 +44,9 @@
|
|||||||
equation for <span class="emphasis"><em>p'</em></span> has to be provided to the symplectic
|
equation for <span class="emphasis"><em>p'</em></span> has to be provided to the symplectic
|
||||||
stepper. We call this concept <span class="emphasis"><em>Simple_Symplectic_System</em></span>
|
stepper. We call this concept <span class="emphasis"><em>Simple_Symplectic_System</em></span>
|
||||||
</p>
|
</p>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.notation"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.notation-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.h1"></a>
|
||||||
<a class="link" href="simple_symplectic_system.html#boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.notation">Notation</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.notation"></a></span><a class="link" href="simple_symplectic_system.html#boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.notation">Notation</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="variablelist">
|
<div class="variablelist">
|
||||||
<p class="title"><b></b></p>
|
<p class="title"><b></b></p>
|
||||||
@ -77,9 +77,9 @@
|
|||||||
</p></dd>
|
</p></dd>
|
||||||
</dl>
|
</dl>
|
||||||
</div>
|
</div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.valid_expressions"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.valid_expressions-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.h2"></a>
|
||||||
<a class="link" href="simple_symplectic_system.html#boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.valid_expressions">Valid
|
<span><a name="boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.valid_expressions"></a></span><a class="link" href="simple_symplectic_system.html#boost_sandbox_numeric_odeint.concepts.simple_symplectic_system.valid_expressions">Valid
|
||||||
Expressions</a>
|
Expressions</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="informaltable"><table class="table">
|
<div class="informaltable"><table class="table">
|
||||||
|
|||||||
@ -40,9 +40,9 @@
|
|||||||
and no algebra/operations are used.
|
and no algebra/operations are used.
|
||||||
</p></td></tr>
|
</p></td></tr>
|
||||||
</table></div>
|
</table></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.description"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.description-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.h0"></a>
|
||||||
<a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.description">Description</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.description"></a></span><a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.description">Description</a>
|
||||||
</h5>
|
</h5>
|
||||||
<p>
|
<p>
|
||||||
The <code class="computeroutput"><span class="identifier">State</span></code>, <code class="computeroutput"><span class="identifier">Algebra</span></code> and <code class="computeroutput"><span class="identifier">Operations</span></code>
|
The <code class="computeroutput"><span class="identifier">State</span></code>, <code class="computeroutput"><span class="identifier">Algebra</span></code> and <code class="computeroutput"><span class="identifier">Operations</span></code>
|
||||||
@ -82,9 +82,9 @@
|
|||||||
<div class="titlepage"><div><div><h4 class="title">
|
<div class="titlepage"><div><div><h4 class="title">
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations"></a><a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations" title="Operations">Operations</a>
|
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations"></a><a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations" title="Operations">Operations</a>
|
||||||
</h4></div></div></div>
|
</h4></div></div></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.notation"></a><h6>
|
<h6>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.notation-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.h0"></a>
|
||||||
<a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.notation">Notation</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.notation"></a></span><a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.notation">Notation</a>
|
||||||
</h6>
|
</h6>
|
||||||
<div class="variablelist">
|
<div class="variablelist">
|
||||||
<p class="title"><b></b></p>
|
<p class="title"><b></b></p>
|
||||||
@ -142,9 +142,9 @@
|
|||||||
</p></dd>
|
</p></dd>
|
||||||
</dl>
|
</dl>
|
||||||
</div>
|
</div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.valid_expressions"></a><h6>
|
<h6>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.valid_expressions-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.h1"></a>
|
||||||
<a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.valid_expressions">Valid
|
<span><a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.valid_expressions"></a></span><a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.operations.valid_expressions">Valid
|
||||||
Expressions</a>
|
Expressions</a>
|
||||||
</h6>
|
</h6>
|
||||||
<div class="informaltable"><table class="table">
|
<div class="informaltable"><table class="table">
|
||||||
@ -414,9 +414,9 @@
|
|||||||
<div class="titlepage"><div><div><h4 class="title">
|
<div class="titlepage"><div><div><h4 class="title">
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra"></a><a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra" title="Algebra">Algebra</a>
|
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra"></a><a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra" title="Algebra">Algebra</a>
|
||||||
</h4></div></div></div>
|
</h4></div></div></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.notation"></a><h6>
|
<h6>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.notation-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.h0"></a>
|
||||||
<a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.notation">Notation</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.notation"></a></span><a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.notation">Notation</a>
|
||||||
</h6>
|
</h6>
|
||||||
<div class="variablelist">
|
<div class="variablelist">
|
||||||
<p class="title"><b></b></p>
|
<p class="title"><b></b></p>
|
||||||
@ -452,9 +452,9 @@
|
|||||||
</p></dd>
|
</p></dd>
|
||||||
</dl>
|
</dl>
|
||||||
</div>
|
</div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.valid_expressions"></a><h6>
|
<h6>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.valid_expressions-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.h1"></a>
|
||||||
<a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.valid_expressions">Valid
|
<span><a name="boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.valid_expressions"></a></span><a class="link" href="state_algebra_operations.html#boost_sandbox_numeric_odeint.concepts.state_algebra_operations.algebra.valid_expressions">Valid
|
||||||
Expressions</a>
|
Expressions</a>
|
||||||
</h6>
|
</h6>
|
||||||
<div class="informaltable"><table class="table">
|
<div class="informaltable"><table class="table">
|
||||||
|
|||||||
@ -20,9 +20,9 @@
|
|||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_wrapper"></a><a class="link" href="state_wrapper.html" title="State Wrapper">State
|
<a name="boost_sandbox_numeric_odeint.concepts.state_wrapper"></a><a class="link" href="state_wrapper.html" title="State Wrapper">State
|
||||||
Wrapper</a>
|
Wrapper</a>
|
||||||
</h3></div></div></div>
|
</h3></div></div></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.description"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.description-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.h0"></a>
|
||||||
<a class="link" href="state_wrapper.html#boost_sandbox_numeric_odeint.concepts.state_wrapper.description">Description</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.description"></a></span><a class="link" href="state_wrapper.html#boost_sandbox_numeric_odeint.concepts.state_wrapper.description">Description</a>
|
||||||
</h5>
|
</h5>
|
||||||
<p>
|
<p>
|
||||||
The <code class="computeroutput"><span class="identifier">State</span> <span class="identifier">Wrapper</span></code>
|
The <code class="computeroutput"><span class="identifier">State</span> <span class="identifier">Wrapper</span></code>
|
||||||
@ -30,9 +30,9 @@
|
|||||||
intermediate results within the stepper's <code class="computeroutput"><span class="identifier">do_step</span></code>
|
intermediate results within the stepper's <code class="computeroutput"><span class="identifier">do_step</span></code>
|
||||||
methods.
|
methods.
|
||||||
</p>
|
</p>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.notation"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.notation-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.h1"></a>
|
||||||
<a class="link" href="state_wrapper.html#boost_sandbox_numeric_odeint.concepts.state_wrapper.notation">Notation</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.notation"></a></span><a class="link" href="state_wrapper.html#boost_sandbox_numeric_odeint.concepts.state_wrapper.notation">Notation</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="variablelist">
|
<div class="variablelist">
|
||||||
<p class="title"><b></b></p>
|
<p class="title"><b></b></p>
|
||||||
@ -56,9 +56,9 @@
|
|||||||
</p></dd>
|
</p></dd>
|
||||||
</dl>
|
</dl>
|
||||||
</div>
|
</div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.valid_expressions"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.valid_expressions-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.h2"></a>
|
||||||
<a class="link" href="state_wrapper.html#boost_sandbox_numeric_odeint.concepts.state_wrapper.valid_expressions">Valid
|
<span><a name="boost_sandbox_numeric_odeint.concepts.state_wrapper.valid_expressions"></a></span><a class="link" href="state_wrapper.html#boost_sandbox_numeric_odeint.concepts.state_wrapper.valid_expressions">Valid
|
||||||
Expressions</a>
|
Expressions</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="informaltable"><table class="table">
|
<div class="informaltable"><table class="table">
|
||||||
|
|||||||
@ -24,9 +24,9 @@
|
|||||||
be used within the <a class="link" href="../odeint_in_detail/integrate_functions.html" title="Integrate functions">integrate
|
be used within the <a class="link" href="../odeint_in_detail/integrate_functions.html" title="Integrate functions">integrate
|
||||||
functions</a>.
|
functions</a>.
|
||||||
</p>
|
</p>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.stepper.description"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.stepper.description-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.stepper.h0"></a>
|
||||||
<a class="link" href="stepper.html#boost_sandbox_numeric_odeint.concepts.stepper.description">Description</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.stepper.description"></a></span><a class="link" href="stepper.html#boost_sandbox_numeric_odeint.concepts.stepper.description">Description</a>
|
||||||
</h5>
|
</h5>
|
||||||
<p>
|
<p>
|
||||||
The basic stepper concept. A basic stepper following this Stepper concept
|
The basic stepper concept. A basic stepper following this Stepper concept
|
||||||
@ -40,9 +40,9 @@
|
|||||||
Symplectic System</a> or <a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
Symplectic System</a> or <a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
||||||
System</a>. Note that all error steppers are also basic steppers.
|
System</a>. Note that all error steppers are also basic steppers.
|
||||||
</p>
|
</p>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.stepper.refinement_of"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.stepper.refinement_of-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.stepper.h1"></a>
|
||||||
<a class="link" href="stepper.html#boost_sandbox_numeric_odeint.concepts.stepper.refinement_of">Refinement
|
<span><a name="boost_sandbox_numeric_odeint.concepts.stepper.refinement_of"></a></span><a class="link" href="stepper.html#boost_sandbox_numeric_odeint.concepts.stepper.refinement_of">Refinement
|
||||||
of</a>
|
of</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||||
@ -53,9 +53,9 @@
|
|||||||
CopyConstructable
|
CopyConstructable
|
||||||
</li>
|
</li>
|
||||||
</ul></div>
|
</ul></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.stepper.notation"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.stepper.notation-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.stepper.h2"></a>
|
||||||
<a class="link" href="stepper.html#boost_sandbox_numeric_odeint.concepts.stepper.notation">Notation</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.stepper.notation"></a></span><a class="link" href="stepper.html#boost_sandbox_numeric_odeint.concepts.stepper.notation">Notation</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="variablelist">
|
<div class="variablelist">
|
||||||
<p class="title"><b></b></p>
|
<p class="title"><b></b></p>
|
||||||
@ -95,9 +95,9 @@
|
|||||||
</p></dd>
|
</p></dd>
|
||||||
</dl>
|
</dl>
|
||||||
</div>
|
</div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.stepper.valid_expressions"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.stepper.valid_expressions-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.stepper.h3"></a>
|
||||||
<a class="link" href="stepper.html#boost_sandbox_numeric_odeint.concepts.stepper.valid_expressions">Valid
|
<span><a name="boost_sandbox_numeric_odeint.concepts.stepper.valid_expressions"></a></span><a class="link" href="stepper.html#boost_sandbox_numeric_odeint.concepts.stepper.valid_expressions">Valid
|
||||||
Expressions</a>
|
Expressions</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="informaltable"><table class="table">
|
<div class="informaltable"><table class="table">
|
||||||
@ -187,9 +187,9 @@
|
|||||||
</tr>
|
</tr>
|
||||||
</tbody>
|
</tbody>
|
||||||
</table></div>
|
</table></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.stepper.models"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.stepper.models-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.stepper.h4"></a>
|
||||||
<a class="link" href="stepper.html#boost_sandbox_numeric_odeint.concepts.stepper.models">Models</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.stepper.models"></a></span><a class="link" href="stepper.html#boost_sandbox_numeric_odeint.concepts.stepper.models">Models</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||||
<li class="listitem">
|
<li class="listitem">
|
||||||
|
|||||||
@ -20,9 +20,9 @@
|
|||||||
<a name="boost_sandbox_numeric_odeint.concepts.symplectic_system"></a><a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
|
<a name="boost_sandbox_numeric_odeint.concepts.symplectic_system"></a><a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
|
||||||
System</a>
|
System</a>
|
||||||
</h3></div></div></div>
|
</h3></div></div></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.description"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.description-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.h0"></a>
|
||||||
<a class="link" href="symplectic_system.html#boost_sandbox_numeric_odeint.concepts.symplectic_system.description">Description</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.description"></a></span><a class="link" href="symplectic_system.html#boost_sandbox_numeric_odeint.concepts.symplectic_system.description">Description</a>
|
||||||
</h5>
|
</h5>
|
||||||
<p>
|
<p>
|
||||||
This concept describes how to define a symplectic system written with generalized
|
This concept describes how to define a symplectic system written with generalized
|
||||||
@ -58,9 +58,9 @@
|
|||||||
std::pair of functions or a functors. Symplectic systems are used in symplectic
|
std::pair of functions or a functors. Symplectic systems are used in symplectic
|
||||||
steppers like <code class="computeroutput"><span class="identifier">symplectic_rkn_sb3a_mclachlan</span></code>.
|
steppers like <code class="computeroutput"><span class="identifier">symplectic_rkn_sb3a_mclachlan</span></code>.
|
||||||
</p>
|
</p>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.notation"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.notation-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.h1"></a>
|
||||||
<a class="link" href="symplectic_system.html#boost_sandbox_numeric_odeint.concepts.symplectic_system.notation">Notation</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.notation"></a></span><a class="link" href="symplectic_system.html#boost_sandbox_numeric_odeint.concepts.symplectic_system.notation">Notation</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="variablelist">
|
<div class="variablelist">
|
||||||
<p class="title"><b></b></p>
|
<p class="title"><b></b></p>
|
||||||
@ -107,9 +107,9 @@
|
|||||||
</p></dd>
|
</p></dd>
|
||||||
</dl>
|
</dl>
|
||||||
</div>
|
</div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.valid_expressions"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.valid_expressions-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.h2"></a>
|
||||||
<a class="link" href="symplectic_system.html#boost_sandbox_numeric_odeint.concepts.symplectic_system.valid_expressions">Valid
|
<span><a name="boost_sandbox_numeric_odeint.concepts.symplectic_system.valid_expressions"></a></span><a class="link" href="symplectic_system.html#boost_sandbox_numeric_odeint.concepts.symplectic_system.valid_expressions">Valid
|
||||||
expressions</a>
|
expressions</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="informaltable"><table class="table">
|
<div class="informaltable"><table class="table">
|
||||||
|
|||||||
@ -19,9 +19,9 @@
|
|||||||
<div class="titlepage"><div><div><h3 class="title">
|
<div class="titlepage"><div><div><h3 class="title">
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.system"></a><a class="link" href="system.html" title="System">System</a>
|
<a name="boost_sandbox_numeric_odeint.concepts.system"></a><a class="link" href="system.html" title="System">System</a>
|
||||||
</h3></div></div></div>
|
</h3></div></div></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.system.description"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.system.description-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.system.h0"></a>
|
||||||
<a class="link" href="system.html#boost_sandbox_numeric_odeint.concepts.system.description">Description</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.system.description"></a></span><a class="link" href="system.html#boost_sandbox_numeric_odeint.concepts.system.description">Description</a>
|
||||||
</h5>
|
</h5>
|
||||||
<p>
|
<p>
|
||||||
The System concept models the algorithmic implementation of the rhs. of the
|
The System concept models the algorithmic implementation of the rhs. of the
|
||||||
@ -34,9 +34,9 @@
|
|||||||
System</a> and <a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
System</a> and <a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
||||||
System</a>.
|
System</a>.
|
||||||
</p>
|
</p>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.system.notation"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.system.notation-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.system.h1"></a>
|
||||||
<a class="link" href="system.html#boost_sandbox_numeric_odeint.concepts.system.notation">Notation</a>
|
<span><a name="boost_sandbox_numeric_odeint.concepts.system.notation"></a></span><a class="link" href="system.html#boost_sandbox_numeric_odeint.concepts.system.notation">Notation</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="variablelist">
|
<div class="variablelist">
|
||||||
<p class="title"><b></b></p>
|
<p class="title"><b></b></p>
|
||||||
@ -75,9 +75,9 @@
|
|||||||
</p></dd>
|
</p></dd>
|
||||||
</dl>
|
</dl>
|
||||||
</div>
|
</div>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.system.valid_expressions"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.concepts.system.valid_expressions-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.concepts.system.h2"></a>
|
||||||
<a class="link" href="system.html#boost_sandbox_numeric_odeint.concepts.system.valid_expressions">Valid
|
<span><a name="boost_sandbox_numeric_odeint.concepts.system.valid_expressions"></a></span><a class="link" href="system.html#boost_sandbox_numeric_odeint.concepts.system.valid_expressions">Valid
|
||||||
expressions</a>
|
expressions</a>
|
||||||
</h5>
|
</h5>
|
||||||
<div class="informaltable"><table class="table">
|
<div class="informaltable"><table class="table">
|
||||||
|
|||||||
@ -30,7 +30,6 @@
|
|||||||
= (p,-q+γ p)</em></span>:
|
= (p,-q+γ p)</em></span>:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">/* The type of container used to hold the state vector */</span>
|
<pre class="programlisting"><span class="comment">/* The type of container used to hold the state vector */</span>
|
||||||
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||||
@ -60,7 +59,6 @@
|
|||||||
start:
|
start:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">(</span><span class="number">2</span><span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">(</span><span class="number">2</span><span class="special">);</span>
|
||||||
<span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> <span class="comment">// start at x=1.0, p=0.0</span>
|
<span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> <span class="comment">// start at x=1.0, p=0.0</span>
|
||||||
@ -75,7 +73,6 @@
|
|||||||
stepper (5th order) and uses adaptive stepsize.
|
stepper (5th order) and uses adaptive stepsize.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">size_t</span> <span class="identifier">steps</span> <span class="special">=</span> <span class="identifier">integrate</span><span class="special">(</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span>
|
<pre class="programlisting"><span class="identifier">size_t</span> <span class="identifier">steps</span> <span class="special">=</span> <span class="identifier">integrate</span><span class="special">(</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span>
|
||||||
<span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="number">0.1</span> <span class="special">);</span>
|
<span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="number">0.1</span> <span class="special">);</span>
|
||||||
@ -96,7 +93,6 @@
|
|||||||
rhs must then be implemented as a functor having defined the ()-operator:
|
rhs must then be implemented as a functor having defined the ()-operator:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">/* The rhs of x' = f(x) defined as a class */</span>
|
<pre class="programlisting"><span class="comment">/* The rhs of x' = f(x) defined as a class */</span>
|
||||||
<span class="keyword">class</span> <span class="identifier">harm_osc</span> <span class="special">{</span>
|
<span class="keyword">class</span> <span class="identifier">harm_osc</span> <span class="special">{</span>
|
||||||
@ -119,7 +115,6 @@
|
|||||||
which can be used via
|
which can be used via
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">harm_osc</span> <span class="identifier">ho</span><span class="special">(</span><span class="number">0.15</span><span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">harm_osc</span> <span class="identifier">ho</span><span class="special">(</span><span class="number">0.15</span><span class="special">);</span>
|
||||||
<span class="identifier">steps</span> <span class="special">=</span> <span class="identifier">integrate</span><span class="special">(</span> <span class="identifier">ho</span> <span class="special">,</span>
|
<span class="identifier">steps</span> <span class="special">=</span> <span class="identifier">integrate</span><span class="special">(</span> <span class="identifier">ho</span> <span class="special">,</span>
|
||||||
@ -134,7 +129,6 @@
|
|||||||
to provide a reasonable observer. An example is
|
to provide a reasonable observer. An example is
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">push_back_state_and_time</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">push_back_state_and_time</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -158,7 +152,6 @@
|
|||||||
pass this container to the integration function:
|
pass this container to the integration function:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">vector</span><span class="special"><</span><span class="identifier">state_type</span><span class="special">></span> <span class="identifier">x_vec</span><span class="special">;</span>
|
<pre class="programlisting"><span class="identifier">vector</span><span class="special"><</span><span class="identifier">state_type</span><span class="special">></span> <span class="identifier">x_vec</span><span class="special">;</span>
|
||||||
<span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">times</span><span class="special">;</span>
|
<span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">times</span><span class="special">;</span>
|
||||||
|
|||||||
@ -25,7 +25,6 @@
|
|||||||
library. It can be include by
|
library. It can be include by
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">numeric</span><span class="special">/</span><span class="identifier">odeint</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">numeric</span><span class="special">/</span><span class="identifier">odeint</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||||
</pre>
|
</pre>
|
||||||
|
|||||||
@ -35,9 +35,9 @@
|
|||||||
Output Stepper</a>. Depending on the abilities of the stepper, the integrate
|
Output Stepper</a>. Depending on the abilities of the stepper, the integrate
|
||||||
functions make use of step-size control or dense output.
|
functions make use of step-size control or dense output.
|
||||||
</p>
|
</p>
|
||||||
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.equidistant_observer_calls"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.equidistant_observer_calls-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.h0"></a>
|
||||||
<a class="link" href="integrate_functions.html#boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.equidistant_observer_calls">Equidistant
|
<span><a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.equidistant_observer_calls"></a></span><a class="link" href="integrate_functions.html#boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.equidistant_observer_calls">Equidistant
|
||||||
observer calls</a>
|
observer calls</a>
|
||||||
</h5>
|
</h5>
|
||||||
<p>
|
<p>
|
||||||
@ -109,9 +109,9 @@
|
|||||||
dt</em></span>.
|
dt</em></span>.
|
||||||
</li>
|
</li>
|
||||||
</ul></div>
|
</ul></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_each_step"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_each_step-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.h1"></a>
|
||||||
<a class="link" href="integrate_functions.html#boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_each_step">Observer
|
<span><a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_each_step"></a></span><a class="link" href="integrate_functions.html#boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_each_step">Observer
|
||||||
calls at each step</a>
|
calls at each step</a>
|
||||||
</h5>
|
</h5>
|
||||||
<p>
|
<p>
|
||||||
@ -179,9 +179,9 @@
|
|||||||
Stepper</a> above. No dense output is used.
|
Stepper</a> above. No dense output is used.
|
||||||
</li>
|
</li>
|
||||||
</ul></div>
|
</ul></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_given_time_points"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_given_time_points-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.h2"></a>
|
||||||
<a class="link" href="integrate_functions.html#boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_given_time_points">Observer
|
<span><a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_given_time_points"></a></span><a class="link" href="integrate_functions.html#boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_given_time_points">Observer
|
||||||
calls at given time points</a>
|
calls at given time points</a>
|
||||||
</h5>
|
</h5>
|
||||||
<p>
|
<p>
|
||||||
@ -254,9 +254,9 @@
|
|||||||
<span class="emphasis"><em>x(t)</em></span> at the time points from the sequence.
|
<span class="emphasis"><em>x(t)</em></span> at the time points from the sequence.
|
||||||
</li>
|
</li>
|
||||||
</ul></div>
|
</ul></div>
|
||||||
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.integrate_a_given_number_of_steps"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.integrate_a_given_number_of_steps-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.h3"></a>
|
||||||
<a class="link" href="integrate_functions.html#boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.integrate_a_given_number_of_steps">Integrate
|
<span><a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.integrate_a_given_number_of_steps"></a></span><a class="link" href="integrate_functions.html#boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.integrate_a_given_number_of_steps">Integrate
|
||||||
a given number of steps</a>
|
a given number of steps</a>
|
||||||
</h5>
|
</h5>
|
||||||
<p>
|
<p>
|
||||||
@ -311,9 +311,9 @@
|
|||||||
This function returns the end time <code class="computeroutput"><span class="identifier">t0</span>
|
This function returns the end time <code class="computeroutput"><span class="identifier">t0</span>
|
||||||
<span class="special">+</span> <span class="identifier">n</span><span class="special">*</span><span class="identifier">dt</span></code>.
|
<span class="special">+</span> <span class="identifier">n</span><span class="special">*</span><span class="identifier">dt</span></code>.
|
||||||
</p>
|
</p>
|
||||||
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.convenience_integrate_function"></a><h5>
|
<h5>
|
||||||
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.convenience_integrate_function-heading"></a>
|
<a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.h4"></a>
|
||||||
<a class="link" href="integrate_functions.html#boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.convenience_integrate_function">Convenience
|
<span><a name="boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.convenience_integrate_function"></a></span><a class="link" href="integrate_functions.html#boost_sandbox_numeric_odeint.odeint_in_detail.integrate_functions.convenience_integrate_function">Convenience
|
||||||
integrate function</a>
|
integrate function</a>
|
||||||
</h5>
|
</h5>
|
||||||
<p>
|
<p>
|
||||||
|
|||||||
@ -399,7 +399,6 @@
|
|||||||
as resizeable as it is a dynamically sized array:
|
as resizeable as it is a dynamically sized array:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">class</span> <span class="identifier">my_vector</span> <span class="special">:</span> <span class="keyword">public</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span>
|
<pre class="programlisting"><span class="keyword">class</span> <span class="identifier">my_vector</span> <span class="special">:</span> <span class="keyword">public</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -534,7 +533,6 @@
|
|||||||
specializations:
|
specializations:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">list</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">list</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||||
|
|
||||||
@ -627,7 +625,6 @@
|
|||||||
instances of state types:
|
instances of state types:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">template</span><span class="special"><></span>
|
<pre class="programlisting"><span class="keyword">template</span><span class="special"><></span>
|
||||||
<span class="keyword">struct</span> <span class="identifier">state_wrapper</span><span class="special"><</span> <span class="identifier">gsl_vector</span><span class="special">*</span> <span class="special">></span>
|
<span class="keyword">struct</span> <span class="identifier">state_wrapper</span><span class="special"><</span> <span class="identifier">gsl_vector</span><span class="special">*</span> <span class="special">></span>
|
||||||
@ -699,7 +696,6 @@
|
|||||||
support iterators, so we first implement a gsl iterator:
|
support iterators, so we first implement a gsl iterator:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">/*
|
<pre class="programlisting"><span class="comment">/*
|
||||||
* defines an iterator for gsl_vector
|
* defines an iterator for gsl_vector
|
||||||
@ -1061,7 +1057,6 @@
|
|||||||
function so all that has to be done in this case is to declare resizability:
|
function so all that has to be done in this case is to declare resizability:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||||
|
|
||||||
@ -1085,7 +1080,6 @@
|
|||||||
code shows the corresponding definitions:
|
code shows the corresponding definitions:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
|
<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -1118,7 +1112,6 @@
|
|||||||
type looks as follows:
|
type looks as follows:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">class</span> <span class="identifier">point3D</span> <span class="special">:</span>
|
<pre class="programlisting"><span class="keyword">class</span> <span class="identifier">point3D</span> <span class="special">:</span>
|
||||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">additive1</span><span class="special"><</span> <span class="identifier">point3D</span> <span class="special">,</span>
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">additive1</span><span class="special"><</span> <span class="identifier">point3D</span> <span class="special">,</span>
|
||||||
@ -1173,7 +1166,6 @@
|
|||||||
type:
|
type:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">// only required for steppers with error control</span>
|
<pre class="programlisting"><span class="comment">// only required for steppers with error control</span>
|
||||||
<span class="identifier">point3D</span> <span class="keyword">operator</span><span class="special">/(</span> <span class="keyword">const</span> <span class="identifier">point3D</span> <span class="special">&</span><span class="identifier">p1</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">point3D</span> <span class="special">&</span><span class="identifier">p2</span> <span class="special">)</span>
|
<span class="identifier">point3D</span> <span class="keyword">operator</span><span class="special">/(</span> <span class="keyword">const</span> <span class="identifier">point3D</span> <span class="special">&</span><span class="identifier">p1</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">point3D</span> <span class="special">&</span><span class="identifier">p2</span> <span class="special">)</span>
|
||||||
@ -1193,7 +1185,6 @@
|
|||||||
implementing a reduction over the state type:
|
implementing a reduction over the state type:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">numeric</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">odeint</span> <span class="special">{</span>
|
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">numeric</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">odeint</span> <span class="special">{</span>
|
||||||
<span class="comment">// specialization of vector_space_reduce, only required for steppers with error control</span>
|
<span class="comment">// specialization of vector_space_reduce, only required for steppers with error control</span>
|
||||||
@ -1221,7 +1212,6 @@
|
|||||||
easily perform the integration on a lorenz system by using the <code class="computeroutput"><span class="identifier">vector_space_algebra</span></code> again:
|
easily perform the integration on a lorenz system by using the <code class="computeroutput"><span class="identifier">vector_space_algebra</span></code> again:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">10.0</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">10.0</span><span class="special">;</span>
|
||||||
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">R</span> <span class="special">=</span> <span class="number">28.0</span><span class="special">;</span>
|
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">R</span> <span class="special">=</span> <span class="number">28.0</span><span class="special">;</span>
|
||||||
|
|||||||
@ -132,7 +132,6 @@
|
|||||||
is the Runge Kutta stepper of fourth order:
|
is the Runge Kutta stepper of fourth order:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">rk</span><span class="special">;</span>
|
<pre class="programlisting"><span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">rk</span><span class="special">;</span>
|
||||||
<span class="identifier">rk</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">sys1</span> <span class="special">,</span> <span class="identifier">inout</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span> <span class="comment">// In-place transformation of inout</span>
|
<span class="identifier">rk</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">sys1</span> <span class="special">,</span> <span class="identifier">inout</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span> <span class="comment">// In-place transformation of inout</span>
|
||||||
@ -177,7 +176,6 @@
|
|||||||
how the FSAL steppers can be used is
|
how the FSAL steppers can be used is
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">rk</span><span class="special">;</span>
|
<pre class="programlisting"><span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">rk</span><span class="special">;</span>
|
||||||
<span class="identifier">rk</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">sys1</span> <span class="special">,</span> <span class="identifier">in</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">out</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
<span class="identifier">rk</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">sys1</span> <span class="special">,</span> <span class="identifier">in</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">out</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
||||||
@ -243,7 +241,6 @@
|
|||||||
again simple C-functions of functors. An example is the harmonic oscillator:
|
again simple C-functions of functors. An example is the harmonic oscillator:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="number">1</span> <span class="special">></span> <span class="identifier">vector_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="number">1</span> <span class="special">></span> <span class="identifier">vector_type</span><span class="special">;</span>
|
||||||
|
|
||||||
@ -272,7 +269,6 @@
|
|||||||
example for the harmonic oscillator is now:
|
example for the harmonic oscillator is now:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">pair</span><span class="special"><</span> <span class="identifier">vector_type</span> <span class="special">,</span> <span class="identifier">vector_type</span> <span class="special">></span> <span class="identifier">x</span><span class="special">;</span>
|
<pre class="programlisting"><span class="identifier">pair</span><span class="special"><</span> <span class="identifier">vector_type</span> <span class="special">,</span> <span class="identifier">vector_type</span> <span class="special">></span> <span class="identifier">x</span><span class="special">;</span>
|
||||||
<span class="identifier">x</span><span class="special">.</span><span class="identifier">first</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">second</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
<span class="identifier">x</span><span class="special">.</span><span class="identifier">first</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">second</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
||||||
@ -286,7 +282,6 @@
|
|||||||
two public method:
|
two public method:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">harm_osc</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">harm_osc</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -304,7 +299,6 @@
|
|||||||
<p>
|
<p>
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">harm_osc</span> <span class="identifier">h</span><span class="special">;</span>
|
<pre class="programlisting"><span class="identifier">harm_osc</span> <span class="identifier">h</span><span class="special">;</span>
|
||||||
<span class="identifier">rkn</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">bind</span><span class="special">(</span> <span class="special">&</span><span class="identifier">harm_osc</span><span class="special">::</span><span class="identifier">f1</span> <span class="special">,</span> <span class="identifier">h</span> <span class="special">,</span> <span class="identifier">_1</span> <span class="special">,</span> <span class="identifier">_2</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">bind</span><span class="special">(</span> <span class="special">&</span><span class="identifier">harm_osc</span><span class="special">::</span><span class="identifier">f2</span> <span class="special">,</span> <span class="identifier">h</span> <span class="special">,</span> <span class="identifier">_1</span> <span class="special">,</span> <span class="identifier">_2</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span>
|
<span class="identifier">rkn</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">bind</span><span class="special">(</span> <span class="special">&</span><span class="identifier">harm_osc</span><span class="special">::</span><span class="identifier">f1</span> <span class="special">,</span> <span class="identifier">h</span> <span class="special">,</span> <span class="identifier">_1</span> <span class="special">,</span> <span class="identifier">_2</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">bind</span><span class="special">(</span> <span class="special">&</span><span class="identifier">harm_osc</span><span class="special">::</span><span class="identifier">f2</span> <span class="special">,</span> <span class="identifier">h</span> <span class="special">,</span> <span class="identifier">_1</span> <span class="special">,</span> <span class="identifier">_2</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span>
|
||||||
@ -324,7 +318,6 @@
|
|||||||
can be written as
|
can be written as
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">pair</span><span class="special"><</span> <span class="identifier">vector_type</span> <span class="special">,</span> <span class="identifier">vector_type</span> <span class="special">></span> <span class="identifier">x</span><span class="special">;</span>
|
<pre class="programlisting"><span class="identifier">pair</span><span class="special"><</span> <span class="identifier">vector_type</span> <span class="special">,</span> <span class="identifier">vector_type</span> <span class="special">></span> <span class="identifier">x</span><span class="special">;</span>
|
||||||
<span class="identifier">x</span><span class="special">.</span><span class="identifier">first</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">second</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
<span class="identifier">x</span><span class="special">.</span><span class="identifier">first</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">second</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
||||||
@ -348,7 +341,6 @@
|
|||||||
takes q and p without combining them into a pair:
|
takes q and p without combining them into a pair:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">rkn</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">harm_osc_f1</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">q</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">p</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">rkn</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">harm_osc_f1</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">q</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">p</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
||||||
<span class="identifier">rkn</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">harm_osc_f1</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">q</span> <span class="special">,</span> <span class="identifier">p</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
<span class="identifier">rkn</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">harm_osc_f1</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">q</span> <span class="special">,</span> <span class="identifier">p</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
||||||
@ -414,7 +406,6 @@
|
|||||||
steps is instantiated and initialized;
|
steps is instantiated and initialized;
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">adams_bashforth_moulton</span><span class="special"><</span> <span class="number">5</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">abm</span><span class="special">;</span>
|
<pre class="programlisting"><span class="identifier">adams_bashforth_moulton</span><span class="special"><</span> <span class="number">5</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">abm</span><span class="special">;</span>
|
||||||
<span class="identifier">abm</span><span class="special">.</span><span class="identifier">initialize</span><span class="special">(</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">inout</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
<span class="identifier">abm</span><span class="special">.</span><span class="identifier">initialize</span><span class="special">(</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">inout</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
||||||
@ -432,7 +423,6 @@
|
|||||||
state of the Adams-Bashforth-Stepper:
|
state of the Adams-Bashforth-Stepper:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">abm</span><span class="special">.</span><span class="identifier">initialize</span><span class="special">(</span> <span class="identifier">runge_kutta_fehlberg78</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">>()</span> <span class="special">,</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">inout</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">abm</span><span class="special">.</span><span class="identifier">initialize</span><span class="special">(</span> <span class="identifier">runge_kutta_fehlberg78</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">>()</span> <span class="special">,</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">inout</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
||||||
</pre>
|
</pre>
|
||||||
@ -695,7 +685,6 @@
|
|||||||
An example is
|
An example is
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">dense_output_runge_kutta</span><span class="special"><</span> <span class="identifier">controlled_runge_kutta</span><span class="special"><</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="special">></span> <span class="special">></span> <span class="identifier">dense</span><span class="special">;</span>
|
<pre class="programlisting"><span class="identifier">dense_output_runge_kutta</span><span class="special"><</span> <span class="identifier">controlled_runge_kutta</span><span class="special"><</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="special">></span> <span class="special">></span> <span class="identifier">dense</span><span class="special">;</span>
|
||||||
<span class="identifier">dense</span><span class="special">.</span><span class="identifier">initialize</span><span class="special">(</span> <span class="identifier">in</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
<span class="identifier">dense</span><span class="special">.</span><span class="identifier">initialize</span><span class="special">(</span> <span class="identifier">in</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
||||||
@ -721,7 +710,6 @@
|
|||||||
exist:
|
exist:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">odeint</span><span class="special">::</span><span class="identifier">result_of</span><span class="special">::</span><span class="identifier">make_dense_output</span><span class="special"><</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">dense2</span> <span class="special">=</span> <span class="identifier">make_dense_output</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">>()</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">odeint</span><span class="special">::</span><span class="identifier">result_of</span><span class="special">::</span><span class="identifier">make_dense_output</span><span class="special"><</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">dense2</span> <span class="special">=</span> <span class="identifier">make_dense_output</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">>()</span> <span class="special">);</span>
|
||||||
</pre>
|
</pre>
|
||||||
@ -739,7 +727,6 @@
|
|||||||
for easy use with the integrate functions:
|
for easy use with the integrate functions:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">make_dense_output</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">>()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">inout</span> <span class="special">,</span> <span class="identifier">t_start</span> <span class="special">,</span> <span class="identifier">t_end</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">make_dense_output</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">>()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">inout</span> <span class="special">,</span> <span class="identifier">t_start</span> <span class="special">,</span> <span class="identifier">t_end</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
||||||
</pre>
|
</pre>
|
||||||
@ -808,7 +795,6 @@
|
|||||||
4 internal derivatives of the solution at times <code class="computeroutput"><span class="special">(</span><span class="identifier">t</span><span class="special">-</span><span class="identifier">dt</span><span class="special">,</span><span class="identifier">t</span><span class="special">-</span><span class="number">2d</span><span class="identifier">t</span><span class="special">,</span><span class="identifier">t</span><span class="special">-</span><span class="number">3d</span><span class="identifier">t</span><span class="special">,</span><span class="identifier">t</span><span class="special">-</span><span class="number">4d</span><span class="identifier">t</span><span class="special">)</span></code>.
|
4 internal derivatives of the solution at times <code class="computeroutput"><span class="special">(</span><span class="identifier">t</span><span class="special">-</span><span class="identifier">dt</span><span class="special">,</span><span class="identifier">t</span><span class="special">-</span><span class="number">2d</span><span class="identifier">t</span><span class="special">,</span><span class="identifier">t</span><span class="special">-</span><span class="number">3d</span><span class="identifier">t</span><span class="special">,</span><span class="identifier">t</span><span class="special">-</span><span class="number">4d</span><span class="identifier">t</span><span class="special">)</span></code>.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">adams_bashforth_moulton</span><span class="special"><</span> <span class="number">4</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">stepper</span><span class="special">;</span>
|
<pre class="programlisting"><span class="identifier">adams_bashforth_moulton</span><span class="special"><</span> <span class="number">4</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">stepper</span><span class="special">;</span>
|
||||||
<span class="identifier">stepper</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span> <span class="comment">// make one step with the classical Runge-Kutta stepper and initialize the first internal state at time t-4dt</span>
|
<span class="identifier">stepper</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span> <span class="comment">// make one step with the classical Runge-Kutta stepper and initialize the first internal state at time t-4dt</span>
|
||||||
@ -1939,39 +1925,89 @@
|
|||||||
<span class="emphasis"><em>x(t+Δ t) = x(t) + Δ t f(x(t)) + Δ t<sup>1/2</sup> g(x) ξ(t)</em></span>
|
<span class="emphasis"><em>x(t+Δ t) = x(t) + Δ t f(x(t)) + Δ t<sup>1/2</sup> g(x) ξ(t)</em></span>
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
where ξ(t) is a independent and identically distributed normal random variable.
|
where ξ(t) is an independent normal distributed random variable.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
Now we will implement this method. We will call the stepper <code class="computeroutput"><span class="identifier">stochastic_euler</span></code>. Its class definition
|
Now we will implement this method. We will call the stepper <code class="computeroutput"><span class="identifier">stochastic_euler</span></code>. It models the __stepper_concept.
|
||||||
looks like
|
The class definition looks like
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">class</span> <span class="identifier">stochastic_euler</span>
|
<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span> <span class="identifier">size_t</span> <span class="identifier">N</span> <span class="special">></span> <span class="keyword">class</span> <span class="identifier">stochastic_euler</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
<span class="keyword">public</span><span class="special">:</span>
|
<span class="keyword">public</span><span class="special">:</span>
|
||||||
|
|
||||||
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">N</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||||
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">deriv_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">N</span> <span class="special">></span> <span class="identifier">deriv_type</span><span class="special">;</span>
|
||||||
<span class="keyword">typedef</span> <span class="keyword">double</span> <span class="identifier">value_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="keyword">double</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||||
<span class="keyword">typedef</span> <span class="keyword">double</span> <span class="identifier">time_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="keyword">double</span> <span class="identifier">time_type</span><span class="special">;</span>
|
||||||
<span class="keyword">typedef</span> <span class="keyword">unsigned</span> <span class="keyword">short</span> <span class="identifier">order_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="keyword">unsigned</span> <span class="keyword">short</span> <span class="identifier">order_type</span><span class="special">;</span>
|
||||||
|
|
||||||
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">odeint</span><span class="special">::</span><span class="identifier">stepper_tag</span> <span class="identifier">stepper_category</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">odeint</span><span class="special">::</span><span class="identifier">stepper_tag</span> <span class="identifier">stepper_category</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="keyword">static</span> <span class="identifier">order_type</span> <span class="identifier">order</span><span class="special">(</span> <span class="keyword">void</span> <span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="number">1</span><span class="special">;</span> <span class="special">}</span>
|
||||||
|
|
||||||
|
<span class="comment">// ...</span>
|
||||||
<span class="special">};</span>
|
<span class="special">};</span>
|
||||||
</pre>
|
</pre>
|
||||||
<p>
|
<p>
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
System function
|
The types are needed in order to fulfill the stepper concept. As internal
|
||||||
|
state and deriv type we use simple arrays in the stochastic euler, they
|
||||||
|
are needed for the temporaries. The stepper has the order one which is
|
||||||
|
returned from the <code class="computeroutput"><span class="identifier">order</span><span class="special">()</span></code> function.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
steppers body
|
The system functions needs to calculate the deterministic and the stochastic
|
||||||
|
part of our stochastic differential equation. So it might be suitable that
|
||||||
|
the system function itself is a pair of functions, one for computing the
|
||||||
|
deterministic and one for the stochastic part. The first element of the
|
||||||
|
pair simply computes the deterministic part while the second the stochastic
|
||||||
|
one. Then, the second part also needs to calculate the random numbers in
|
||||||
|
order to simulate the stochastic process. We can now implement the do_step
|
||||||
|
method
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
do_step method
|
|
||||||
</p>
|
</p>
|
||||||
|
<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span> <span class="identifier">size_t</span> <span class="identifier">N</span> <span class="special">></span> <span class="keyword">class</span> <span class="identifier">stochastic_euler</span>
|
||||||
|
<span class="special">{</span>
|
||||||
|
<span class="keyword">public</span><span class="special">:</span>
|
||||||
|
|
||||||
|
<span class="comment">// ...</span>
|
||||||
|
|
||||||
|
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">System</span> <span class="special">></span>
|
||||||
|
<span class="keyword">void</span> <span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">System</span> <span class="identifier">system</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">time_type</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">time_type</span> <span class="identifier">dt</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||||
|
<span class="special">{</span>
|
||||||
|
<span class="identifier">deriv_type</span> <span class="identifier">det</span> <span class="special">,</span> <span class="identifier">stoch</span> <span class="special">;</span>
|
||||||
|
<span class="identifier">system</span><span class="special">.</span><span class="identifier">first</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">det</span> <span class="special">);</span>
|
||||||
|
<span class="identifier">system</span><span class="special">.</span><span class="identifier">second</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">stoch</span> <span class="special">);</span>
|
||||||
|
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">x</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
|
||||||
|
<span class="identifier">x</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">+=</span> <span class="identifier">dt</span> <span class="special">*</span> <span class="identifier">det</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">sqrt</span><span class="special">(</span> <span class="identifier">dt</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">stoch</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
|
||||||
|
<span class="special">}</span>
|
||||||
|
<span class="special">};</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
This is all. It is quite simple and the stochastic euler stepper implement
|
||||||
|
here is quite general. Of course it can be enhanced, for example
|
||||||
|
</p>
|
||||||
|
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||||
|
<li class="listitem">
|
||||||
|
use of operations and algebras as well as the resizing mechanism for
|
||||||
|
maximal flexibility and portability
|
||||||
|
</li>
|
||||||
|
<li class="listitem">
|
||||||
|
use of <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span></code> for the system functions
|
||||||
|
</li>
|
||||||
|
<li class="listitem">
|
||||||
|
use of <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">range</span></code> for the state type in the
|
||||||
|
<code class="computeroutput"><span class="identifier">do_step</span></code> method
|
||||||
|
</li>
|
||||||
|
<li class="listitem">
|
||||||
|
...
|
||||||
|
</li>
|
||||||
|
</ul></div>
|
||||||
<p>
|
<p>
|
||||||
examples, with integrate function
|
examples, with integrate function
|
||||||
</p>
|
</p>
|
||||||
|
|||||||
@ -61,7 +61,6 @@
|
|||||||
The implementation of the Lorenz system is
|
The implementation of the Lorenz system is
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">10.0</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">10.0</span><span class="special">;</span>
|
||||||
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">R</span> <span class="special">=</span> <span class="number">28.0</span><span class="special">;</span>
|
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">R</span> <span class="special">=</span> <span class="number">28.0</span><span class="special">;</span>
|
||||||
@ -84,7 +83,6 @@
|
|||||||
the perturbation could look like:
|
the perturbation could look like:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
|
||||||
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">num_of_lyap</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
|
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">num_of_lyap</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
|
||||||
@ -116,7 +114,6 @@
|
|||||||
inside a functor with templatized arguments:
|
inside a functor with templatized arguments:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -154,7 +151,6 @@
|
|||||||
current state lies on the attractor, hence it would look like
|
current state lies on the attractor, hence it would look like
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">;</span>
|
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">;</span>
|
||||||
<span class="comment">// initialize x</span>
|
<span class="comment">// initialize x</span>
|
||||||
@ -170,7 +166,6 @@
|
|||||||
by pointer to the state and an appropriate length, something similar to
|
by pointer to the state and an appropriate length, something similar to
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">lorenz</span><span class="special">(</span> <span class="keyword">double</span><span class="special">*</span> <span class="identifier">x</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">*</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span><span class="special">,</span> <span class="keyword">void</span><span class="special">*</span> <span class="identifier">params</span> <span class="special">)</span>
|
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">lorenz</span><span class="special">(</span> <span class="keyword">double</span><span class="special">*</span> <span class="identifier">x</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">*</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span><span class="special">,</span> <span class="keyword">void</span><span class="special">*</span> <span class="identifier">params</span> <span class="special">)</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -188,7 +183,6 @@
|
|||||||
the system has to by changed:
|
the system has to by changed:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -211,7 +205,6 @@
|
|||||||
with an suited range:
|
with an suited range:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">// perform 10000 transient steps</span>
|
<pre class="programlisting"><span class="comment">// perform 10000 transient steps</span>
|
||||||
<span class="identifier">integrate_n_steps</span><span class="special">(</span> <span class="identifier">rk4</span> <span class="special">,</span> <span class="identifier">lorenz</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="identifier">n</span> <span class="special">)</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="number">10000</span> <span class="special">);</span>
|
<span class="identifier">integrate_n_steps</span><span class="special">(</span> <span class="identifier">rk4</span> <span class="special">,</span> <span class="identifier">lorenz</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="identifier">n</span> <span class="special">)</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="number">10000</span> <span class="special">);</span>
|
||||||
@ -242,7 +235,6 @@
|
|||||||
</li>
|
</li>
|
||||||
</ol></div>
|
</ol></div>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">fill</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()+</span><span class="identifier">n</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">fill</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()+</span><span class="identifier">n</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span>
|
||||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">num_of_lyap</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> <span class="identifier">x</span><span class="special">[</span><span class="identifier">n</span><span class="special">+</span><span class="identifier">n</span><span class="special">*</span><span class="identifier">i</span><span class="special">+</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
|
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">num_of_lyap</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> <span class="identifier">x</span><span class="special">[</span><span class="identifier">n</span><span class="special">+</span><span class="identifier">n</span><span class="special">*</span><span class="identifier">i</span><span class="special">+</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
|
||||||
|
|||||||
@ -52,7 +52,6 @@
|
|||||||
to just define a function, e.g:
|
to just define a function, e.g:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">/* The type of container used to hold the state vector */</span>
|
<pre class="programlisting"><span class="comment">/* The type of container used to hold the state vector */</span>
|
||||||
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||||
@ -79,7 +78,6 @@
|
|||||||
parameter structure as above:
|
parameter structure as above:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">/* The rhs of x' = f(x) defined as a class */</span>
|
<pre class="programlisting"><span class="comment">/* The rhs of x' = f(x) defined as a class */</span>
|
||||||
<span class="keyword">class</span> <span class="identifier">harm_osc</span> <span class="special">{</span>
|
<span class="keyword">class</span> <span class="identifier">harm_osc</span> <span class="special">{</span>
|
||||||
@ -1138,7 +1136,6 @@
|
|||||||
<span class="special">)</span></code> function from odeint:
|
<span class="special">)</span></code> function from odeint:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">stepper</span><span class="special">;</span>
|
<pre class="programlisting"><span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">stepper</span><span class="special">;</span>
|
||||||
<span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">stepper</span> <span class="special">,</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">);</span>
|
<span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">stepper</span> <span class="special">,</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">);</span>
|
||||||
@ -1155,7 +1152,6 @@
|
|||||||
method which can used directly. So, you write down the above example as
|
method which can used directly. So, you write down the above example as
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">0.01</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">0.01</span><span class="special">;</span>
|
||||||
<span class="keyword">for</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">t</span><span class="special">=</span><span class="number">0.0</span> <span class="special">;</span> <span class="identifier">t</span><span class="special"><</span><span class="number">10.0</span> <span class="special">;</span> <span class="identifier">t</span><span class="special">+=</span> <span class="identifier">dt</span> <span class="special">)</span>
|
<span class="keyword">for</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">t</span><span class="special">=</span><span class="number">0.0</span> <span class="special">;</span> <span class="identifier">t</span><span class="special"><</span><span class="number">10.0</span> <span class="special">;</span> <span class="identifier">t</span><span class="special">+=</span> <span class="identifier">dt</span> <span class="special">)</span>
|
||||||
@ -1179,7 +1175,6 @@
|
|||||||
with 4th order error estimation and coefficients introduced by Cash-Karp.
|
with 4th order error estimation and coefficients introduced by Cash-Karp.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta_cash_karp54</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">error_stepper_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta_cash_karp54</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">error_stepper_type</span><span class="special">;</span>
|
||||||
</pre>
|
</pre>
|
||||||
@ -1192,7 +1187,6 @@
|
|||||||
stepper exists which can be used via
|
stepper exists which can be used via
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">controlled_runge_kutta</span><span class="special"><</span> <span class="identifier">error_stepper_type</span> <span class="special">></span> <span class="identifier">controlled_stepper_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">controlled_runge_kutta</span><span class="special"><</span> <span class="identifier">error_stepper_type</span> <span class="special">></span> <span class="identifier">controlled_stepper_type</span><span class="special">;</span>
|
||||||
<span class="identifier">controlled_stepper_type</span> <span class="identifier">controlled_stepper</span><span class="special">;</span>
|
<span class="identifier">controlled_stepper_type</span> <span class="identifier">controlled_stepper</span><span class="special">;</span>
|
||||||
@ -1219,7 +1213,6 @@
|
|||||||
by using <code class="computeroutput"><span class="identifier">make_controlled</span></code>:
|
by using <code class="computeroutput"><span class="identifier">make_controlled</span></code>:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">integrate_adaptive</span><span class="special">(</span> <span class="identifier">make_controlled</span><span class="special"><</span> <span class="identifier">error_stepper_type</span> <span class="special">>(</span> <span class="number">1.0e-10</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">integrate_adaptive</span><span class="special">(</span> <span class="identifier">make_controlled</span><span class="special"><</span> <span class="identifier">error_stepper_type</span> <span class="special">>(</span> <span class="number">1.0e-10</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">);</span>
|
||||||
</pre>
|
</pre>
|
||||||
@ -1235,7 +1228,6 @@
|
|||||||
is
|
is
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">integrate_adaptive</span><span class="special">(</span> <span class="identifier">make_controlled</span><span class="special">(</span> <span class="number">1.0e-10</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">error_stepper_type</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">integrate_adaptive</span><span class="special">(</span> <span class="identifier">make_controlled</span><span class="special">(</span> <span class="number">1.0e-10</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">error_stepper_type</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">);</span>
|
||||||
</pre>
|
</pre>
|
||||||
@ -1251,7 +1243,6 @@
|
|||||||
with all parameters is therefore
|
with all parameters is therefore
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">abs_err</span> <span class="special">=</span> <span class="number">1.0e-10</span> <span class="special">,</span> <span class="identifier">rel_err</span> <span class="special">=</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">a_x</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">,</span> <span class="identifier">a_dxdt</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">abs_err</span> <span class="special">=</span> <span class="number">1.0e-10</span> <span class="special">,</span> <span class="identifier">rel_err</span> <span class="special">=</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">a_x</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">,</span> <span class="identifier">a_dxdt</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
|
||||||
<span class="identifier">controlled_stepper_type</span> <span class="identifier">controlled_stepper</span><span class="special">(</span> <span class="identifier">default_error_checker</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">>(</span> <span class="identifier">abs_err</span> <span class="special">,</span> <span class="identifier">rel_err</span> <span class="special">,</span> <span class="identifier">a_x</span> <span class="special">,</span> <span class="identifier">a_dxdt</span> <span class="special">)</span> <span class="special">);</span>
|
<span class="identifier">controlled_stepper_type</span> <span class="identifier">controlled_stepper</span><span class="special">(</span> <span class="identifier">default_error_checker</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">>(</span> <span class="identifier">abs_err</span> <span class="special">,</span> <span class="identifier">rel_err</span> <span class="special">,</span> <span class="identifier">a_x</span> <span class="special">,</span> <span class="identifier">a_dxdt</span> <span class="special">)</span> <span class="special">);</span>
|
||||||
|
|||||||
@ -98,7 +98,6 @@
|
|||||||
space as well as the velocity. Therefore, we use the operators from <a href="http://www.boost.org/doc/libs/release/doc/html/operators.html" target="_top">Boost.Operators</a>:
|
space as well as the velocity. Therefore, we use the operators from <a href="http://www.boost.org/doc/libs/release/doc/html/operators.html" target="_top">Boost.Operators</a>:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">/*the point type */</span>
|
<pre class="programlisting"><span class="comment">/*the point type */</span>
|
||||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">T</span> <span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Dim</span> <span class="special">></span>
|
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">T</span> <span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Dim</span> <span class="special">></span>
|
||||||
@ -136,7 +135,6 @@
|
|||||||
type we use <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">tr1</span><span class="special">::</span><span class="identifier">array</span></code>
|
type we use <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">tr1</span><span class="special">::</span><span class="identifier">array</span></code>
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">// we simulate 5 planets and the sun</span>
|
<pre class="programlisting"><span class="comment">// we simulate 5 planets and the sun</span>
|
||||||
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">6</span><span class="special">;</span>
|
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">6</span><span class="special">;</span>
|
||||||
@ -159,7 +157,6 @@
|
|||||||
As system function we have to provide <span class="emphasis"><em>f(p)</em></span> and <span class="emphasis"><em>f(q)</em></span>:
|
As system function we have to provide <span class="emphasis"><em>f(p)</em></span> and <span class="emphasis"><em>f(q)</em></span>:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">gravitational_constant</span> <span class="special">=</span> <span class="number">2.95912208286e-4</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">gravitational_constant</span> <span class="special">=</span> <span class="number">2.95912208286e-4</span><span class="special">;</span>
|
||||||
|
|
||||||
@ -179,7 +176,6 @@
|
|||||||
<p>
|
<p>
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">solar_system_momentum</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">solar_system_momentum</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -221,7 +217,6 @@
|
|||||||
apply here:
|
apply here:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">symplectic_rkn_sb3a_mclachlan</span><span class="special"><</span> <span class="identifier">container_type</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">symplectic_rkn_sb3a_mclachlan</span><span class="special"><</span> <span class="identifier">container_type</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
||||||
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">100.0</span><span class="special">;</span>
|
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">100.0</span><span class="special">;</span>
|
||||||
@ -246,7 +241,6 @@
|
|||||||
The reference wrapper is also passed, but this is not a problem at all:
|
The reference wrapper is also passed, but this is not a problem at all:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">streaming_observer</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">streaming_observer</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
|
|||||||
@ -63,7 +63,6 @@
|
|||||||
code is very simple
|
code is very simple
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="identifier">complex</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="special">,</span> <span class="number">1</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="identifier">complex</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="special">,</span> <span class="number">1</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||||
|
|
||||||
@ -89,7 +88,6 @@
|
|||||||
function. In this cast the Stuart-Landau oscillator looks like
|
function. In this cast the Stuart-Landau oscillator looks like
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">eta</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">eta</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
|
||||||
<span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
|
<span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
|
||||||
@ -111,7 +109,6 @@
|
|||||||
Integration is also very easy:
|
Integration is also very easy:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="special">{{</span> <span class="identifier">complex</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">>(</span> <span class="number">1.0</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">}};</span>
|
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="special">{{</span> <span class="identifier">complex</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">>(</span> <span class="number">1.0</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">}};</span>
|
||||||
|
|
||||||
@ -156,7 +153,6 @@
|
|||||||
components. Here, is how this function looks like
|
components. Here, is how this function looks like
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">container_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">container_type</span><span class="special">;</span>
|
||||||
|
|
||||||
@ -208,7 +204,6 @@
|
|||||||
All this can be easily done with the following piece of code:
|
All this can be easily done with the following piece of code:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">64</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">64</span><span class="special">;</span>
|
||||||
<span class="identifier">container_type</span> <span class="identifier">q</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">p</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span>
|
<span class="identifier">container_type</span> <span class="identifier">q</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">p</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span>
|
||||||
@ -235,7 +230,6 @@
|
|||||||
The observer is more or less trivial
|
The observer is more or less trivial
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">streaming_observer</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">streaming_observer</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -304,7 +298,6 @@
|
|||||||
also need to store the individual frequencies of each oscillator.
|
also need to store the individual frequencies of each oscillator.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">container_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">container_type</span><span class="special">;</span>
|
||||||
|
|
||||||
@ -367,7 +360,6 @@
|
|||||||
and we record <span class="emphasis"><em>Z</em></span> for different values of <span class="emphasis"><em>ε</em></span>.
|
and we record <span class="emphasis"><em>Z</em></span> for different values of <span class="emphasis"><em>ε</em></span>.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">statistics_observer</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">statistics_observer</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -400,7 +392,6 @@
|
|||||||
2</em></span>.
|
2</em></span>.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">16384</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">16384</span><span class="special">;</span>
|
||||||
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span>
|
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span>
|
||||||
@ -458,7 +449,6 @@
|
|||||||
all quantities
|
all quantities
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">// first increase fusion macro variables</span>
|
<pre class="programlisting"><span class="comment">// first increase fusion macro variables</span>
|
||||||
<span class="preprocessor">#define</span> <span class="identifier">BOOST_FUSION_INVOKE_MAX_ARITY</span> <span class="number">15</span>
|
<span class="preprocessor">#define</span> <span class="identifier">BOOST_FUSION_INVOKE_MAX_ARITY</span> <span class="number">15</span>
|
||||||
@ -500,7 +490,6 @@
|
|||||||
equivalent to the example in the first section of this tutorial
|
equivalent to the example in the first section of this tutorial
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">oscillator</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">oscillator</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -524,7 +513,6 @@
|
|||||||
Furthermore, the basic calculations are now performed by the <code class="computeroutput"><span class="identifier">fusion_algebra</span></code> which must also be given.
|
Furthermore, the basic calculations are now performed by the <code class="computeroutput"><span class="identifier">fusion_algebra</span></code> which must also be given.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">deriv_type</span> <span class="special">,</span> <span class="identifier">time_type</span> <span class="special">,</span> <span class="identifier">fusion_algebra</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">deriv_type</span> <span class="special">,</span> <span class="identifier">time_type</span> <span class="special">,</span> <span class="identifier">fusion_algebra</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
||||||
|
|
||||||
@ -539,7 +527,6 @@
|
|||||||
the observer is defined a bit different
|
the observer is defined a bit different
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">streaming_observer</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">streaming_observer</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -611,7 +598,6 @@
|
|||||||
The definition of the system is
|
The definition of the system is
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||||
|
|
||||||
@ -705,7 +691,6 @@
|
|||||||
Here is a simple example:
|
Here is a simple example:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">mpf_class</span> <span class="identifier">value_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">mpf_class</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||||
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="number">3</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="number">3</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||||
@ -730,7 +715,6 @@
|
|||||||
which can be used
|
which can be used
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">precision</span> <span class="special">=</span> <span class="number">1024</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">precision</span> <span class="special">=</span> <span class="number">1024</span><span class="special">;</span>
|
||||||
<span class="identifier">mpf_set_default_prec</span><span class="special">(</span> <span class="identifier">precision</span> <span class="special">);</span>
|
<span class="identifier">mpf_set_default_prec</span><span class="special">(</span> <span class="identifier">precision</span> <span class="special">);</span>
|
||||||
@ -792,7 +776,6 @@
|
|||||||
is kept which should be changed whenever the states' size change.
|
is kept which should be changed whenever the states' size change.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">coord_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">coord_type</span><span class="special">;</span>
|
||||||
<span class="keyword">typedef</span> <span class="identifier">pair</span><span class="special"><</span> <span class="identifier">coord_type</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="identifier">pair</span><span class="special"><</span> <span class="identifier">coord_type</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||||
@ -878,7 +861,6 @@
|
|||||||
of 60.
|
of 60.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">//start with 60 sites</span>
|
<pre class="programlisting"><span class="comment">//start with 60 sites</span>
|
||||||
<span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">N_start</span> <span class="special">=</span> <span class="number">60</span><span class="special">;</span>
|
<span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">N_start</span> <span class="special">=</span> <span class="number">60</span><span class="special">;</span>
|
||||||
@ -914,7 +896,6 @@
|
|||||||
the start index of the potential changes in this case.
|
the start index of the potential changes in this case.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
||||||
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span>
|
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span>
|
||||||
@ -947,7 +928,6 @@
|
|||||||
, <code class="computeroutput"><span class="identifier">p</span></code> and <code class="computeroutput"><span class="identifier">distr</span></code>.
|
, <code class="computeroutput"><span class="identifier">p</span></code> and <code class="computeroutput"><span class="identifier">distr</span></code>.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">do_resize</span><span class="special">(</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">distr</span> <span class="special">,</span> <span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">N</span> <span class="special">)</span>
|
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">do_resize</span><span class="special">(</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">distr</span> <span class="special">,</span> <span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">N</span> <span class="special">)</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
|
|||||||
@ -45,7 +45,6 @@
|
|||||||
is needed. Here is the definition of the above example
|
is needed. Here is the definition of the above example
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">vector_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">vector_type</span><span class="special">;</span>
|
||||||
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">matrix_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">matrix_type</span><span class="special">;</span>
|
||||||
@ -82,7 +81,6 @@
|
|||||||
just templatize the <code class="computeroutput"><span class="keyword">operator</span><span class="special">()</span></code>:
|
just templatize the <code class="computeroutput"><span class="keyword">operator</span><span class="special">()</span></code>:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">vector_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">vector_type</span><span class="special">;</span>
|
||||||
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">matrix_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">matrix_type</span><span class="special">;</span>
|
||||||
@ -118,7 +116,6 @@
|
|||||||
all the other stepper:
|
all the other stepper:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">vector_type</span> <span class="identifier">x</span><span class="special">(</span> <span class="number">3</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">vector_type</span> <span class="identifier">x</span><span class="special">(</span> <span class="number">3</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
|
||||||
|
|
||||||
@ -135,7 +132,6 @@
|
|||||||
5 method with step size control and dense output yields 1531 steps.
|
5 method with step size control and dense output yields 1531 steps.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">vector_type</span> <span class="identifier">x2</span><span class="special">(</span> <span class="number">3</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">vector_type</span> <span class="identifier">x2</span><span class="special">(</span> <span class="number">3</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
|
||||||
|
|
||||||
|
|||||||
@ -87,7 +87,6 @@
|
|||||||
website.
|
website.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">//change this to float if your device does not support double computation</span>
|
<pre class="programlisting"><span class="comment">//change this to float if your device does not support double computation</span>
|
||||||
<span class="keyword">typedef</span> <span class="keyword">double</span> <span class="identifier">value_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="keyword">double</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||||
@ -138,7 +137,6 @@
|
|||||||
The mean field is calculated in a class <code class="computeroutput"><span class="identifier">mean_field_calculator</span></code>
|
The mean field is calculated in a class <code class="computeroutput"><span class="identifier">mean_field_calculator</span></code>
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">mean_field_calculator</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">mean_field_calculator</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -192,7 +190,6 @@
|
|||||||
GPU as well as on the CPU. The line
|
GPU as well as on the CPU. The line
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">value_type</span> <span class="identifier">sin_sum</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">reduce</span><span class="special">(</span>
|
<pre class="programlisting"><span class="identifier">value_type</span> <span class="identifier">sin_sum</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">reduce</span><span class="special">(</span>
|
||||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_transform_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">sin_functor</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span>
|
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_transform_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">sin_functor</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span>
|
||||||
@ -208,7 +205,6 @@
|
|||||||
The system function is defined via
|
The system function is defined via
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">class</span> <span class="identifier">phase_oscillator_ensemble</span>
|
<pre class="programlisting"><span class="keyword">class</span> <span class="identifier">phase_oscillator_ensemble</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -264,7 +260,6 @@
|
|||||||
thrust algebra:
|
thrust algebra:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">thrust_algebra</span> <span class="special">,</span> <span class="identifier">thrust_operations</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">thrust_algebra</span> <span class="special">,</span> <span class="identifier">thrust_operations</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
||||||
</pre>
|
</pre>
|
||||||
@ -274,7 +269,6 @@
|
|||||||
Of course, you can also use a controlled or dense output stepper, e.g.
|
Of course, you can also use a controlled or dense output stepper, e.g.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">thrust_algebra</span> <span class="special">,</span> <span class="identifier">thrust_operations</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">thrust_algebra</span> <span class="special">,</span> <span class="identifier">thrust_operations</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
||||||
</pre>
|
</pre>
|
||||||
@ -285,14 +279,12 @@
|
|||||||
an instance of the rhs class and using an integration function:
|
an instance of the rhs class and using an integration function:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">phase_oscillator_ensemble</span> <span class="identifier">ensemble</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">phase_oscillator_ensemble</span> <span class="identifier">ensemble</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
|
||||||
</pre>
|
</pre>
|
||||||
<p>
|
<p>
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">size_t</span> <span class="identifier">steps1</span> <span class="special">=</span> <span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">make_controlled</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">ensemble</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">t_transients</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">size_t</span> <span class="identifier">steps1</span> <span class="special">=</span> <span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">make_controlled</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">ensemble</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">t_transients</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
||||||
</pre>
|
</pre>
|
||||||
@ -334,7 +326,6 @@
|
|||||||
in front. The full system class is:
|
in front. The full system class is:
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">//change this to host_vector< ... > of you want to run on CPU</span>
|
<pre class="programlisting"><span class="comment">//change this to host_vector< ... > of you want to run on CPU</span>
|
||||||
<span class="keyword">typedef</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">device_vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
<span class="keyword">typedef</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">device_vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||||
@ -423,7 +414,6 @@
|
|||||||
current state, hence the phase of each oscillator.
|
current state, hence the phase of each oscillator.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="comment">// create initial conditions and omegas on host:</span>
|
<pre class="programlisting"><span class="comment">// create initial conditions and omegas on host:</span>
|
||||||
<span class="identifier">vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">x_host</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">);</span>
|
<span class="identifier">vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">x_host</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">);</span>
|
||||||
@ -486,7 +476,6 @@
|
|||||||
<span class="special">></span></code>.
|
<span class="special">></span></code>.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">beta_host</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">beta_host</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">);</span>
|
||||||
<span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">beta_min</span> <span class="special">=</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">beta_max</span> <span class="special">=</span> <span class="number">56.0</span><span class="special">;</span>
|
<span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">beta_min</span> <span class="special">=</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">beta_max</span> <span class="special">=</span> <span class="number">56.0</span><span class="special">;</span>
|
||||||
@ -504,7 +493,6 @@
|
|||||||
calculates one particular realization of the Lorenz ensemble
|
calculates one particular realization of the Lorenz ensemble
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz_system</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz_system</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -586,7 +574,6 @@
|
|||||||
which is used within this observer is defined
|
which is used within this observer is defined
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lyap_functor</span>
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lyap_functor</span>
|
||||||
<span class="special">{</span>
|
<span class="special">{</span>
|
||||||
@ -626,7 +613,6 @@
|
|||||||
and written to <code class="computeroutput"><span class="identifier">stdout</span></code>.
|
and written to <code class="computeroutput"><span class="identifier">stdout</span></code>.
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
|
|
||||||
</p>
|
</p>
|
||||||
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">(</span> <span class="number">6</span> <span class="special">*</span> <span class="identifier">N</span> <span class="special">);</span>
|
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">(</span> <span class="number">6</span> <span class="special">*</span> <span class="identifier">N</span> <span class="special">);</span>
|
||||||
|
|
||||||
|
|||||||
@ -24,7 +24,7 @@
|
|||||||
<div><p class="copyright">Copyright © 2009-2011 Karsten Ahnert
|
<div><p class="copyright">Copyright © 2009-2011 Karsten Ahnert
|
||||||
and Mario Mulansky</p></div>
|
and Mario Mulansky</p></div>
|
||||||
<div><div class="legalnotice">
|
<div><div class="legalnotice">
|
||||||
<a name="id532853"></a><p>
|
<a name="odeint.legal"></a><p>
|
||||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||||
</p>
|
</p>
|
||||||
@ -125,7 +125,7 @@
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||||
<td align="left"><p><small>Last revised: March 29, 2012 at 12:51:03 GMT</small></p></td>
|
<td align="left"><p><small>Last revised: April 02, 2012 at 06:51:21 GMT</small></p></td>
|
||||||
<td align="right"><div class="copyright-footer"></div></td>
|
<td align="right"><div class="copyright-footer"></div></td>
|
||||||
</tr></table>
|
</tr></table>
|
||||||
<hr>
|
<hr>
|
||||||
|
|||||||
@ -221,7 +221,7 @@ We will illustrate how to write your own stepper with the example of the stochas
|
|||||||
|
|
||||||
['dx/dt = f(x) + g(x) __xi(t)]
|
['dx/dt = f(x) + g(x) __xi(t)]
|
||||||
|
|
||||||
where ['__xi] is Gaussian white noise with zero mean and a standard deviation ['__sigma]. ['f(x)] is said to be the deterministic part while [' g(x) __xi] is the noisy part. In case ['g(x)] is independent of ['x] the SDE is said to have additive noise. It is not possible to solve SDE with the classical solvers for ODEs since the noisy part of the SDE has to be scaled differently then the deterministic part with respect to the time step. BUt there exist many solvers for SDEs. A classical and easy method is the stochastic euler solver. It works by iterating
|
where ['__xi] is Gaussian white noise with zero mean and a standard deviation ['__sigma(t)]. ['f(x)] is said to be the deterministic part while [' g(x) __xi] is the noisy part. In case ['g(x)] is independent of ['x] the SDE is said to have additive noise. It is not possible to solve SDE with the classical solvers for ODEs since the noisy part of the SDE has to be scaled differently then the deterministic part with respect to the time step. BUt there exist many solvers for SDEs. A classical and easy method is the stochastic euler solver. It works by iterating
|
||||||
|
|
||||||
['x(t+__Delta t) = x(t) + __Delta t f(x(t)) + __Delta t[super 1/2] g(x) __xi(t)]
|
['x(t+__Delta t) = x(t) + __Delta t f(x(t)) + __Delta t[super 1/2] g(x) __xi(t)]
|
||||||
|
|
||||||
@ -231,17 +231,26 @@ Now we will implement this method. We will call the stepper `stochastic_euler`.
|
|||||||
|
|
||||||
[stochastic_euler_class_definition]
|
[stochastic_euler_class_definition]
|
||||||
|
|
||||||
The types are needed in order to fullfil the stepper concept. An internal state is not needed in the stochastic euler, there are no temporaries needed during each step. So, in principle `state_type` and `deriv_type` are not needed here. Nevertheless, they must be present in order to fullfil the __stepper_concept and `std::vector< double >` is a good general candidate for the state and deriv type.
|
The types are needed in order to fulfill the stepper concept. As internal state and deriv type we use simple arrays in the stochastic euler, they are needed for the temporaries. The stepper has the order one which is returned from the `order()` function.
|
||||||
|
|
||||||
|
The system functions needs to calculate the deterministic and the stochastic part of our stochastic differential equation. So it might be suitable that the system function itself is a pair of functions, one for computing the deterministic and one for the stochastic part. The first element of the pair simply computes the deterministic part while the second the stochastic one. Then, the second part also needs to calculate the random numbers in order to simulate the stochastic process. We can now implement the `do_step` method
|
||||||
|
|
||||||
|
[stochastic_euler_do_step]
|
||||||
|
|
||||||
|
This is all. It is quite simple and the stochastic euler stepper implement here is quite general. Of course it can be enhanced, for example
|
||||||
|
|
||||||
|
* use of operations and algebras as well as the resizing mechanism for maximal flexibility and portability
|
||||||
|
* use of `boost::ref` for the system functions
|
||||||
|
* use of `boost::range` for the state type in the `do_step` method
|
||||||
|
* ...
|
||||||
|
|
||||||
System function
|
Now, lets look how we use the new stepper. A nice example is the Ornstein-Uhlenbeck process. It consists of a simple Brownian motion overlapped with an relaxation process. Its SDE reads
|
||||||
|
|
||||||
steppers body
|
['dx/dt = - x + __xi]
|
||||||
|
|
||||||
do_step method
|
where __xi is Gaussian white noise with standard deviation ['__sigma].
|
||||||
|
|
||||||
|
[stochastic_euler_ornstein_uhlenbeck_det]
|
||||||
|
|
||||||
examples, with integrate function
|
examples, with integrate function
|
||||||
|
|
||||||
|
|||||||
@ -6,40 +6,59 @@
|
|||||||
#include <boost/numeric/odeint.hpp>
|
#include <boost/numeric/odeint.hpp>
|
||||||
|
|
||||||
|
|
||||||
struct sys
|
/*
|
||||||
{
|
|
||||||
template< class State >
|
|
||||||
void operator()( const State &x , State &dxdt )
|
|
||||||
{
|
|
||||||
dxdt[0] = -x[0];
|
|
||||||
}
|
|
||||||
};
|
|
||||||
|
|
||||||
//[ stochastic_euler_class_definition
|
//[ stochastic_euler_class_definition
|
||||||
class stochastic_euler
|
template< size_t N > class stochastic_euler
|
||||||
{
|
{
|
||||||
public:
|
public:
|
||||||
|
|
||||||
typedef std::vector< double > state_type;
|
typedef std::array< double , N > state_type;
|
||||||
typedef std::vector< double > deriv_type;
|
typedef std::array< double , N > deriv_type;
|
||||||
typedef double value_type;
|
typedef double value_type;
|
||||||
typedef double time_type;
|
typedef double time_type;
|
||||||
typedef unsigned short order_type;
|
typedef unsigned short order_type;
|
||||||
typedef boost::numeric::odeint::stepper_tag stepper_category;
|
typedef boost::numeric::odeint::stepper_tag stepper_category;
|
||||||
|
|
||||||
|
static order_type order( void ) { return 1; }
|
||||||
|
|
||||||
// ...
|
// ...
|
||||||
};
|
};
|
||||||
//]
|
//]
|
||||||
|
*/
|
||||||
|
|
||||||
|
|
||||||
|
/*
|
||||||
|
//[ stochastic_euler_do_step
|
||||||
|
template< size_t N > class stochastic_euler
|
||||||
|
{
|
||||||
|
public:
|
||||||
|
|
||||||
|
// ...
|
||||||
|
|
||||||
|
template< class System >
|
||||||
|
void do_step( System system , state_type &x , time_type t , time_type dt ) const
|
||||||
|
{
|
||||||
|
deriv_type det , stoch ;
|
||||||
|
system.first( x , det );
|
||||||
|
system.second( x , stoch );
|
||||||
|
for( size_t i=0 ; i<x.size() ; ++i )
|
||||||
|
x[i] += dt * det[i] + sqrt( dt ) * stoch[i];
|
||||||
|
}
|
||||||
|
};
|
||||||
|
//]
|
||||||
|
*/
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
//[ stochastic_euler_class
|
//[ stochastic_euler_class
|
||||||
|
template< size_t N >
|
||||||
class stochastic_euler
|
class stochastic_euler
|
||||||
{
|
{
|
||||||
public:
|
public:
|
||||||
|
|
||||||
typedef std::vector< double > state_type;
|
typedef std::array< double , N > state_type;
|
||||||
typedef std::vector< double > deriv_type;
|
typedef std::array< double , N > deriv_type;
|
||||||
typedef double value_type;
|
typedef double value_type;
|
||||||
typedef double time_type;
|
typedef double time_type;
|
||||||
typedef unsigned short order_type;
|
typedef unsigned short order_type;
|
||||||
@ -48,32 +67,45 @@ public:
|
|||||||
|
|
||||||
static order_type order( void ) { return 1; }
|
static order_type order( void ) { return 1; }
|
||||||
|
|
||||||
template< class System , class State , class Time >
|
template< class System >
|
||||||
void do_step( System system , State &x , Time t , Time dt ) const
|
void do_step( System system , state_type &x , time_type t , time_type dt ) const
|
||||||
{
|
{
|
||||||
State dxdt;
|
deriv_type det , stoch ;
|
||||||
system.first( x , dxdt );
|
system.first( x , det );
|
||||||
|
system.second( x , stoch );
|
||||||
for( size_t i=0 ; i<x.size() ; ++i )
|
for( size_t i=0 ; i<x.size() ; ++i )
|
||||||
x[i] += dt * dxdt[i] + sqrt( dt ) * system.second();
|
x[i] += dt * det[i] + sqrt( dt ) * stoch[i];
|
||||||
}
|
}
|
||||||
};
|
};
|
||||||
//]
|
//]
|
||||||
|
|
||||||
template< class Rng , class Dist >
|
|
||||||
struct gen
|
|
||||||
|
//[ stochastic_euler_ornstein_uhlenbeck_def
|
||||||
|
const static size_t N = 1;
|
||||||
|
typedef std::array< double , N > state_type;
|
||||||
|
|
||||||
|
struct ornstein_det
|
||||||
{
|
{
|
||||||
Rng &rng;
|
void operator()( const state_type &x , state_type &dxdt ) const
|
||||||
Dist &dist;
|
{
|
||||||
gen( Rng &rng_ , Dist &dist_ ) : rng( rng_ ) , dist( dist_ ) { }
|
dxdt[0] = -x[0];
|
||||||
double operator()( void ) { return dist( rng ); }
|
}
|
||||||
};
|
};
|
||||||
|
|
||||||
template< class Rng , class Dist >
|
struct ornstein_stoch
|
||||||
gen< Rng , Dist > make_gen( Rng &rng , Dist &dist )
|
|
||||||
{
|
{
|
||||||
return gen< Rng , Dist >( rng , dist );
|
std::mt19937 m_rng;
|
||||||
}
|
std::normal_distribution<> m_dist;
|
||||||
|
|
||||||
|
ornstein_stoch( double sigma ) : m_rng() , m_dist( 0.0 , sigma ) { }
|
||||||
|
|
||||||
|
void operator()( const state_type &x , state_type &dxdt )
|
||||||
|
{
|
||||||
|
dxdt[0] = m_dist( m_rng );
|
||||||
|
}
|
||||||
|
};
|
||||||
|
//]
|
||||||
|
|
||||||
struct streaming_observer
|
struct streaming_observer
|
||||||
{
|
{
|
||||||
@ -84,23 +116,14 @@ struct streaming_observer
|
|||||||
}
|
}
|
||||||
};
|
};
|
||||||
|
|
||||||
|
int main( int argc , char **argv )
|
||||||
|
{
|
||||||
using namespace std;
|
using namespace std;
|
||||||
using namespace boost::numeric::odeint;
|
using namespace boost::numeric::odeint;
|
||||||
|
|
||||||
typedef std::array< double , 1 > state_type;
|
|
||||||
|
|
||||||
int main( int argc , char **argv )
|
|
||||||
{
|
|
||||||
|
|
||||||
mt19937 rng;
|
|
||||||
normal_distribution<> dist( 0.0 , 1.0 );
|
|
||||||
|
|
||||||
|
|
||||||
double dt = 0.1;
|
double dt = 0.1;
|
||||||
state_type x = {{ 1.0 }};
|
state_type x = {{ 1.0 }};
|
||||||
streaming_observer obs;
|
integrate_const( stochastic_euler< N >() , make_pair( ornstein_det() , ornstein_stoch( 1.0 ) ) , x , 0.0 , 10.0 , dt , streaming_observer() );
|
||||||
cout << boost::is_same< streaming_observer , boost::numeric::odeint::null_observer >::value << endl;
|
|
||||||
integrate_const( stochastic_euler() , make_pair( sys() , make_gen( rng , dist ) ) , x , 0.0 , 10.0 , dt , obs );
|
|
||||||
|
|
||||||
return 0;
|
return 0;
|
||||||
}
|
}
|
||||||
|
|||||||
Loading…
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Reference in New Issue
Block a user