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<h1>Produce predictions from a kinetic model using specific parameters</h1>
<small class="dont-index">Source: <a href='http://github.com/jranke/mkin/blob/master/R/mkinpredict.R'><code>R/mkinpredict.R</code></a></small>
<div class="hidden name"><code>mkinpredict.Rd</code></div>
</div>
<div class="ref-description">
<p>This function produces a time series for all the observed variables in a
kinetic model as specified by <code><a href='mkinmod.html'>mkinmod</a></code>, using a specific set of
kinetic parameters and initial values for the state variables.</p>
</div>
<pre class="usage"><span class='fu'>mkinpredict</span>(
<span class='no'>x</span>,
<span class='no'>odeparms</span>,
<span class='no'>odeini</span>,
<span class='kw'>outtimes</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/seq.html'>seq</a></span>(<span class='fl'>0</span>, <span class='fl'>120</span>, <span class='kw'>by</span> <span class='kw'>=</span> <span class='fl'>0.1</span>),
<span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"deSolve"</span>,
<span class='kw'>use_compiled</span> <span class='kw'>=</span> <span class='st'>"auto"</span>,
<span class='kw'>method.ode</span> <span class='kw'>=</span> <span class='st'>"lsoda"</span>,
<span class='kw'>atol</span> <span class='kw'>=</span> <span class='fl'>1e-08</span>,
<span class='kw'>rtol</span> <span class='kw'>=</span> <span class='fl'>1e-10</span>,
<span class='kw'>map_output</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>,
<span class='no'>...</span>
)
<span class='co'># S3 method for mkinmod</span>
<span class='fu'>mkinpredict</span>(
<span class='no'>x</span>,
<span class='kw'>odeparms</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_parent_sink</span> <span class='kw'>=</span> <span class='fl'>0.1</span>),
<span class='kw'>odeini</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>parent</span> <span class='kw'>=</span> <span class='fl'>100</span>),
<span class='kw'>outtimes</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/seq.html'>seq</a></span>(<span class='fl'>0</span>, <span class='fl'>120</span>, <span class='kw'>by</span> <span class='kw'>=</span> <span class='fl'>0.1</span>),
<span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"deSolve"</span>,
<span class='kw'>use_compiled</span> <span class='kw'>=</span> <span class='st'>"auto"</span>,
<span class='kw'>method.ode</span> <span class='kw'>=</span> <span class='st'>"lsoda"</span>,
<span class='kw'>atol</span> <span class='kw'>=</span> <span class='fl'>1e-08</span>,
<span class='kw'>rtol</span> <span class='kw'>=</span> <span class='fl'>1e-10</span>,
<span class='kw'>map_output</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>,
<span class='no'>...</span>
)
<span class='co'># S3 method for mkinfit</span>
<span class='fu'>mkinpredict</span>(
<span class='no'>x</span>,
<span class='kw'>odeparms</span> <span class='kw'>=</span> <span class='no'>x</span>$<span class='no'>bparms.ode</span>,
<span class='kw'>odeini</span> <span class='kw'>=</span> <span class='no'>x</span>$<span class='no'>bparms.state</span>,
<span class='kw'>outtimes</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/seq.html'>seq</a></span>(<span class='fl'>0</span>, <span class='fl'>120</span>, <span class='kw'>by</span> <span class='kw'>=</span> <span class='fl'>0.1</span>),
<span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"deSolve"</span>,
<span class='kw'>use_compiled</span> <span class='kw'>=</span> <span class='st'>"auto"</span>,
<span class='kw'>method.ode</span> <span class='kw'>=</span> <span class='st'>"lsoda"</span>,
<span class='kw'>atol</span> <span class='kw'>=</span> <span class='fl'>1e-08</span>,
<span class='kw'>rtol</span> <span class='kw'>=</span> <span class='fl'>1e-10</span>,
<span class='kw'>map_output</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>,
<span class='no'>...</span>
)</pre>
<h2 class="hasAnchor" id="arguments"><a class="anchor" href="#arguments"></a>Arguments</h2>
<table class="ref-arguments">
<colgroup><col class="name" /><col class="desc" /></colgroup>
<tr>
<th>x</th>
<td><p>A kinetic model as produced by <code><a href='mkinmod.html'>mkinmod</a></code>, or a kinetic
fit as fitted by <code><a href='mkinfit.html'>mkinfit</a></code>. In the latter case, the fitted
parameters are used for the prediction.</p></td>
</tr>
<tr>
<th>odeparms</th>
<td><p>A numeric vector specifying the parameters used in the
kinetic model, which is generally defined as a set of ordinary
differential equations.</p></td>
</tr>
<tr>
<th>odeini</th>
<td><p>A numeric vectory containing the initial values of the state
variables of the model. Note that the state variables can differ from the
observed variables, for example in the case of the SFORB model.</p></td>
</tr>
<tr>
<th>outtimes</th>
<td><p>A numeric vector specifying the time points for which model
predictions should be generated.</p></td>
</tr>
<tr>
<th>solution_type</th>
<td><p>The method that should be used for producing the
predictions. This should generally be "analytical" if there is only one
observed variable, and usually "deSolve" in the case of several observed
variables. The third possibility "eigen" is faster but not applicable to
some models e.g. using FOMC for the parent compound.</p></td>
</tr>
<tr>
<th>use_compiled</th>
<td><p>If set to <code>FALSE</code>, no compiled version of the
<code><a href='mkinmod.html'>mkinmod</a></code> model is used, even if is present.</p></td>
</tr>
<tr>
<th>method.ode</th>
<td><p>The solution method passed via <code>mkinpredict</code>
to <code>ode</code> in case the solution type is "deSolve". The default
"lsoda" is performant, but sometimes fails to converge.</p></td>
</tr>
<tr>
<th>atol</th>
<td><p>Absolute error tolerance, passed to <code>ode</code>. Default
is 1e-8, lower than in <code>lsoda</code>.</p></td>
</tr>
<tr>
<th>rtol</th>
<td><p>Absolute error tolerance, passed to <code>ode</code>. Default
is 1e-10, much lower than in <code>lsoda</code>.</p></td>
</tr>
<tr>
<th>map_output</th>
<td><p>Boolean to specify if the output should list values for
the observed variables (default) or for all state variables (if set to
FALSE).</p></td>
</tr>
<tr>
<th>...</th>
<td><p>Further arguments passed to the ode solver in case such a
solver is used.</p></td>
</tr>
</table>
<h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</h2>
<p>A matrix in the same format as the output of <code>ode</code>.</p>
<h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2>
<pre class="examples"><div class='input'>
<span class='no'>SFO</span> <span class='kw'><-</span> <span class='fu'><a href='mkinmod.html'>mkinmod</a></span>(<span class='kw'>degradinol</span> <span class='kw'>=</span> <span class='fu'><a href='mkinsub.html'>mkinsub</a></span>(<span class='st'>"SFO"</span>))
<span class='co'># Compare solution types</span>
<span class='fu'>mkinpredict</span>(<span class='no'>SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_degradinol_sink</span> <span class='kw'>=</span> <span class='fl'>0.3</span>), <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>degradinol</span> <span class='kw'>=</span> <span class='fl'>100</span>), <span class='fl'>0</span>:<span class='fl'>20</span>,
<span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"analytical"</span>)</div><div class='output co'>#> <span class='error'>Error in (function (t, parent_0, k) { parent = parent_0 * exp(-k * t)})(t = 0:20, parent.0 = c(degradinol = 100), k = 0.3): unbenutztes Argument (parent.0 = 100)</span></div><div class='input'><span class='fu'>mkinpredict</span>(<span class='no'>SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_degradinol_sink</span> <span class='kw'>=</span> <span class='fl'>0.3</span>), <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>degradinol</span> <span class='kw'>=</span> <span class='fl'>100</span>), <span class='fl'>0</span>:<span class='fl'>20</span>,
<span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"deSolve"</span>)</div><div class='output co'>#> time degradinol
#> 1 0 100.0000000
#> 2 1 74.0818221
#> 3 2 54.8811636
#> 4 3 40.6569660
#> 5 4 30.1194212
#> 6 5 22.3130160
#> 7 6 16.5298888
#> 8 7 12.2456428
#> 9 8 9.0717953
#> 10 9 6.7205513
#> 11 10 4.9787068
#> 12 11 3.6883167
#> 13 12 2.7323722
#> 14 13 2.0241911
#> 15 14 1.4995577
#> 16 15 1.1108996
#> 17 16 0.8229747
#> 18 17 0.6096747
#> 19 18 0.4516581
#> 20 19 0.3345965
#> 21 20 0.2478752</div><div class='input'><span class='fu'>mkinpredict</span>(<span class='no'>SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_degradinol_sink</span> <span class='kw'>=</span> <span class='fl'>0.3</span>), <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>degradinol</span> <span class='kw'>=</span> <span class='fl'>100</span>), <span class='fl'>0</span>:<span class='fl'>20</span>,
<span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"deSolve"</span>, <span class='kw'>use_compiled</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>)</div><div class='output co'>#> time degradinol
#> 1 0 100.0000000
#> 2 1 74.0818221
#> 3 2 54.8811636
#> 4 3 40.6569660
#> 5 4 30.1194212
#> 6 5 22.3130160
#> 7 6 16.5298888
#> 8 7 12.2456428
#> 9 8 9.0717953
#> 10 9 6.7205513
#> 11 10 4.9787068
#> 12 11 3.6883167
#> 13 12 2.7323722
#> 14 13 2.0241911
#> 15 14 1.4995577
#> 16 15 1.1108996
#> 17 16 0.8229747
#> 18 17 0.6096747
#> 19 18 0.4516581
#> 20 19 0.3345965
#> 21 20 0.2478752</div><div class='input'><span class='fu'>mkinpredict</span>(<span class='no'>SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_degradinol_sink</span> <span class='kw'>=</span> <span class='fl'>0.3</span>), <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>degradinol</span> <span class='kw'>=</span> <span class='fl'>100</span>), <span class='fl'>0</span>:<span class='fl'>20</span>,
<span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"eigen"</span>)</div><div class='output co'>#> time degradinol
#> 1 0 100.0000000
#> 2 1 74.0818221
#> 3 2 54.8811636
#> 4 3 40.6569660
#> 5 4 30.1194212
#> 6 5 22.3130160
#> 7 6 16.5298888
#> 8 7 12.2456428
#> 9 8 9.0717953
#> 10 9 6.7205513
#> 11 10 4.9787068
#> 12 11 3.6883167
#> 13 12 2.7323722
#> 14 13 2.0241911
#> 15 14 1.4995577
#> 16 15 1.1108997
#> 17 16 0.8229747
#> 18 17 0.6096747
#> 19 18 0.4516581
#> 20 19 0.3345965
#> 21 20 0.2478752</div><div class='input'>
<span class='co'># Compare integration methods to analytical solution</span>
<span class='fu'>mkinpredict</span>(<span class='no'>SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_degradinol_sink</span> <span class='kw'>=</span> <span class='fl'>0.3</span>), <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>degradinol</span> <span class='kw'>=</span> <span class='fl'>100</span>), <span class='fl'>0</span>:<span class='fl'>20</span>,
<span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"analytical"</span>)[<span class='fl'>21</span>,]</div><div class='output co'>#> <span class='error'>Error in (function (t, parent_0, k) { parent = parent_0 * exp(-k * t)})(t = 0:20, parent.0 = c(degradinol = 100), k = 0.3): unbenutztes Argument (parent.0 = 100)</span></div><div class='input'><span class='fu'>mkinpredict</span>(<span class='no'>SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_degradinol_sink</span> <span class='kw'>=</span> <span class='fl'>0.3</span>), <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>degradinol</span> <span class='kw'>=</span> <span class='fl'>100</span>), <span class='fl'>0</span>:<span class='fl'>20</span>,
<span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"lsoda"</span>)[<span class='fl'>21</span>,]</div><div class='output co'>#> time degradinol
#> 21 20 0.2478752</div><div class='input'><span class='fu'>mkinpredict</span>(<span class='no'>SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_degradinol_sink</span> <span class='kw'>=</span> <span class='fl'>0.3</span>), <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>degradinol</span> <span class='kw'>=</span> <span class='fl'>100</span>), <span class='fl'>0</span>:<span class='fl'>20</span>,
<span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"ode45"</span>)[<span class='fl'>21</span>,]</div><div class='output co'>#> time degradinol
#> 21 20 0.2478752</div><div class='input'><span class='fu'>mkinpredict</span>(<span class='no'>SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_degradinol_sink</span> <span class='kw'>=</span> <span class='fl'>0.3</span>), <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>degradinol</span> <span class='kw'>=</span> <span class='fl'>100</span>), <span class='fl'>0</span>:<span class='fl'>20</span>,
<span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"rk4"</span>)[<span class='fl'>21</span>,]</div><div class='output co'>#> time degradinol
#> 21 20 0.2480043</div><div class='input'><span class='co'># rk4 is not as precise here</span>
<span class='co'># The number of output times used to make a lot of difference until the</span>
<span class='co'># default for atol was adjusted</span>
<span class='fu'>mkinpredict</span>(<span class='no'>SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_degradinol_sink</span> <span class='kw'>=</span> <span class='fl'>0.3</span>), <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>degradinol</span> <span class='kw'>=</span> <span class='fl'>100</span>),
<span class='fu'><a href='https://rdrr.io/r/base/seq.html'>seq</a></span>(<span class='fl'>0</span>, <span class='fl'>20</span>, <span class='kw'>by</span> <span class='kw'>=</span> <span class='fl'>0.1</span>))[<span class='fl'>201</span>,]</div><div class='output co'>#> time degradinol
#> 201 20 0.2478752</div><div class='input'><span class='fu'>mkinpredict</span>(<span class='no'>SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_degradinol_sink</span> <span class='kw'>=</span> <span class='fl'>0.3</span>), <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>degradinol</span> <span class='kw'>=</span> <span class='fl'>100</span>),
<span class='fu'><a href='https://rdrr.io/r/base/seq.html'>seq</a></span>(<span class='fl'>0</span>, <span class='fl'>20</span>, <span class='kw'>by</span> <span class='kw'>=</span> <span class='fl'>0.01</span>))[<span class='fl'>2001</span>,]</div><div class='output co'>#> time degradinol
#> 2001 20 0.2478752</div><div class='input'>
<span class='co'># Check compiled model versions - they are faster than the eigenvalue based solutions!</span>
<span class='no'>SFO_SFO</span> <span class='kw'>=</span> <span class='fu'><a href='mkinmod.html'>mkinmod</a></span>(<span class='kw'>parent</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/list.html'>list</a></span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='st'>"m1"</span>),
<span class='kw'>m1</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/list.html'>list</a></span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</span>))</div><div class='output co'>#> <span class='message'>Successfully compiled differential equation model from auto-generated C code.</span></div><div class='input'><span class='kw'>if</span>(<span class='fu'><a href='https://rdrr.io/r/base/library.html'>require</a></span>(<span class='no'>rbenchmark</span>)) {
<span class='fu'><a href='https://rdrr.io/pkg/rbenchmark/man/benchmark.html'>benchmark</a></span>(
<span class='kw'>eigen</span> <span class='kw'>=</span> <span class='fu'>mkinpredict</span>(<span class='no'>SFO_SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_parent_m1</span> <span class='kw'>=</span> <span class='fl'>0.05</span>, <span class='kw'>k_parent_sink</span> <span class='kw'>=</span> <span class='fl'>0.1</span>, <span class='kw'>k_m1_sink</span> <span class='kw'>=</span> <span class='fl'>0.01</span>),
<span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>parent</span> <span class='kw'>=</span> <span class='fl'>100</span>, <span class='kw'>m1</span> <span class='kw'>=</span> <span class='fl'>0</span>), <span class='fu'><a href='https://rdrr.io/r/base/seq.html'>seq</a></span>(<span class='fl'>0</span>, <span class='fl'>20</span>, <span class='kw'>by</span> <span class='kw'>=</span> <span class='fl'>0.1</span>),
<span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"eigen"</span>)[<span class='fl'>201</span>,],
<span class='kw'>deSolve_compiled</span> <span class='kw'>=</span> <span class='fu'>mkinpredict</span>(<span class='no'>SFO_SFO</span>,
<span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_parent_m1</span> <span class='kw'>=</span> <span class='fl'>0.05</span>, <span class='kw'>k_parent_sink</span> <span class='kw'>=</span> <span class='fl'>0.1</span>, <span class='kw'>k_m1_sink</span> <span class='kw'>=</span> <span class='fl'>0.01</span>),
<span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>parent</span> <span class='kw'>=</span> <span class='fl'>100</span>, <span class='kw'>m1</span> <span class='kw'>=</span> <span class='fl'>0</span>), <span class='fu'><a href='https://rdrr.io/r/base/seq.html'>seq</a></span>(<span class='fl'>0</span>, <span class='fl'>20</span>, <span class='kw'>by</span> <span class='kw'>=</span> <span class='fl'>0.1</span>),
<span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"deSolve"</span>)[<span class='fl'>201</span>,],
<span class='kw'>deSolve</span> <span class='kw'>=</span> <span class='fu'>mkinpredict</span>(<span class='no'>SFO_SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>k_parent_m1</span> <span class='kw'>=</span> <span class='fl'>0.05</span>, <span class='kw'>k_parent_sink</span> <span class='kw'>=</span> <span class='fl'>0.1</span>, <span class='kw'>k_m1_sink</span> <span class='kw'>=</span> <span class='fl'>0.01</span>),
<span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='kw'>parent</span> <span class='kw'>=</span> <span class='fl'>100</span>, <span class='kw'>m1</span> <span class='kw'>=</span> <span class='fl'>0</span>), <span class='fu'><a href='https://rdrr.io/r/base/seq.html'>seq</a></span>(<span class='fl'>0</span>, <span class='fl'>20</span>, <span class='kw'>by</span> <span class='kw'>=</span> <span class='fl'>0.1</span>),
<span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"deSolve"</span>, <span class='kw'>use_compiled</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>)[<span class='fl'>201</span>,],
<span class='kw'>replications</span> <span class='kw'>=</span> <span class='fl'>10</span>)
}</div><div class='output co'>#> <span class='message'>Lade nötiges Paket: rbenchmark</span></div><div class='output co'>#> test replications elapsed relative user.self sys.self user.child
#> 3 deSolve 10 0.229 28.625 0.229 0 0
#> 2 deSolve_compiled 10 0.008 1.000 0.008 0 0
#> 1 eigen 10 0.025 3.125 0.026 0 0
#> sys.child
#> 3 0
#> 2 0
#> 1 0</div><div class='input'>
<span class='co'># Since mkin 0.9.49.11 we also have analytical solutions for some models, including SFO-SFO</span>
<span class='co'># deSolve = mkinpredict(SFO_SFO, c(k_parent_m1 = 0.05, k_parent_sink = 0.1, k_m1_sink = 0.01),</span>
<span class='co'># c(parent = 100, m1 = 0), seq(0, 20, by = 0.1),</span>
<span class='co'># solution_type = "analytical", use_compiled = FALSE)[201,],</span>
<span class='co'># \dontrun{</span>
<span class='co'># Predict from a fitted model</span>
<span class='no'>f</span> <span class='kw'><-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>SFO_SFO</span>, <span class='no'>FOCUS_2006_C</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='fu'><a href='https://rdrr.io/r/utils/head.html'>head</a></span>(<span class='fu'>mkinpredict</span>(<span class='no'>f</span>))</div><div class='output co'>#> time parent m1
#> 1 0.0 82.49216 0.000000
#> 2 0.1 80.00563 1.179963
#> 3 0.2 77.59404 2.312596
#> 4 0.3 75.25515 3.399443
#> 5 0.4 72.98675 4.442000
#> 6 0.5 70.78673 5.441717</div><div class='input'># }
</div></pre>
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