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#: ../../source/behavior/functions/nonlinear.rst:4
msgid "Non-Linearities"
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:6
msgid ""
"Our next example involving functions demonstrates how to address issues that"
" come up when solving non-linear systems of equations that involve "
"functions."
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:10
msgid ""
"We'll start with a simple model that includes this simple mathematical "
"relationship:"
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:17
msgid ""
"where :math:`t` is time.  This model can be implemented in Modelica as "
"follows:"
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:23
msgid ""
"where the ``Quadratic`` function, which we will discuss shortly, evaluates a"
" quadratic polynomial."
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:26
msgid "Simulating this model gives the following solution for ``y``:"
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:31
msgid ""
"So far, all of this appears quite reasonable and, based on our previous "
"discussion on :ref:`polynomial-evaluation`, the implementation of "
"``Quadratic`` isn't likely to hold too many surprises.  However, let's make "
"things a little more complicated. Consider the following model:"
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:41
msgid "This model amounts to solving the following equation:"
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:47
msgid ""
"The important difference here is that the left hand side is known and we "
"must compute the value of :math:`y` that satisfies this equation. In other "
"words, instead of evaluating a quadratic polynomial, like we were doing in "
"the previous example, now we have to solve a quadratic equation."
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:53
msgid ""
"A model that requires solving a non-linear system of equations is not "
"remarkable by itself.  Modelica compilers are certainly more than capable of"
" recognizing and solving non-linear systems of equations (although these "
"methods usually depend on having a reasonable initial guess in order to "
"converge)."
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:59
msgid ""
"However, it turns out that in this case, **the Modelica compiler doesn't "
"actually need to solve a non-linear system**.  Although we couldn't know "
"this until we saw how the ``Quadratic`` function is implemented:"
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:69
msgid ""
"In particular, note the line specifying the ``inverse`` annotation. With "
"this function definition, we not only tell the Modelica compiler how to "
"evaluate the ``Quadratic`` function, but, through the ``inverse`` "
"annotation, we are also indicating that the ``InverseQuadratic`` function "
"should be used to compute ``x`` in terms of ``y``."
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:76
msgid "The ``InverseQuadratic`` function is defined as follows:"
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:84
msgid ""
"Note that the ``InverseQuadratic`` function computes only the positive root "
"in the quadratic equation.  This can be both a good thing and a bad thing.  "
"By computing only a single root, we avoid the issue of having multiple "
"solutions when we invert the quadratic relationship.  However, if the "
"negative root happens to be the one you want, this can be a problem."
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:91
msgid ""
"In the case of our ``ImplicitEvaluation`` model, the Modelica compiler can "
"then substitute this inverse function into the equations.  So, where we "
"initially had, ignoring the coefficient arguments for the moment, the "
"following equation to solve:"
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:100
msgid ""
"which must be solved as an implicit equation for :math:`y`, we can now solve"
" the following **explicit** equation:"
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:107
msgid "by using the ``InverseQuadratic`` function as the inverse function."
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:109
msgid ""
"Simulating the ``ImplicitEvaluation`` model we get the following solution "
"for ``y``:"
msgstr ""

#: ../../source/behavior/functions/nonlinear.rst:115
msgid ""
"Looking at this figure, we can see that, in fact, we got the correct result,"
" but, in general, without the need to solve the non-linear system that would"
" otherwise result from our ``ImplicitEvaluation`` model."
msgstr ""
