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advanced_programming/slides_t2/out.md

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+<span style="color:#FFFFCC"> __Topic \#2__ </span>  <span style="color:#FFCC00"> __Numerical__ </span>  <span style="color:#FFCC00"> __ __ </span>  <span style="color:#FFCC00"> __analysis__ </span>  <span style="color:#FFCC00"> __ __ </span>  <span style="color:#FFCC00"> __and__ </span>  <span style="color:#FFCC00"> __ __ </span>  <span style="color:#FFCC00"> __symbolic__ </span>  <span style="color:#FFCC00"> __ __ </span>  <span style="color:#FFCC00"> __computation__ </span>
+
+![](img/T2_NumAnalysis-SymComp0.png)
+
+__What__  __ __  __is__  __ __  __it__  __?__
+
+Numerical analysis
+
+Symbolic computation
+
+__Which__  __ __  __tools__  __ __  __can__  __ __  __we__  __ __  __use__  __?__
+
+ scipy
+
+ sympy
+
+![](img/T2_NumAnalysis-SymComp1.png)
+
+__Background __  __info__  __ – __  __David‘s__  __ __  __compendium__  __ __  __reloaded__  __\!__
+
+[https://davrot\.github\.io/pytutorial](https://davrot.github.io/pytutorial/)[/](https://davrot.github.io/pytutorial/)
+
+__Topics:__
+
+Sympy
+
+Numerical Integration\, Differentiation\, and Differential Equations
+
+__Which__  __ __  __mathematical__  __ __  __problems__  __ __  __are__  __ __  __we__  __ __  __interested__  __ in?__
+
+Solving equations \(only symbolic\)
+
+Integrals over functions
+
+Derivatives of functions
+
+Solving differential equations
+
+__Numerical__  __ __  __solutions__  __ will \(__  __almost__  __\) __  __always__  __ __  __be__  __ __  __approximations__  __\! __
+
+Precision is limited
+
+Range is limited
+
+Algorithm is approximating
+
+Errors can accumulate dramatically \(stability of algorithms\)
+
+__Examples__  __ __  __of__  __ __  __errors__  __:__
+
+Multiplication\, one decimal place: 2\.5 \* 2\.5 = 6\.25
+
+Addition\, 8\-bit unsigned int: 200\+200 = 400
+
+Euler integration of ODE   <span style="color:#FF0000"> __\(__ </span>  <span style="color:#FF0000"> __ Whiteboard\)__ </span>
+
+<span style="color:#0070C0"> __Integrals __ </span>  <span style="color:#0070C0"> __over__ </span>  <span style="color:#0070C0"> __ __ </span>  <span style="color:#0070C0"> __functions__ </span>  <span style="color:#0070C0"> __ \(‚__ </span>  <span style="color:#0070C0"> __quadrature__ </span>  <span style="color:#0070C0"> __‘\)__ </span>
+
+![](img/T2_NumAnalysis-SymComp2.png)
+
+__Numerical__  __ __  __methods__
+
+Integral = area under curve
+
+Approximate area by many small boxes\, e\.g\. by  _midpoint_  _ _  _rule_ :
+
+![](img/T2_NumAnalysis-SymComp3.png)
+
+![](img/T2_NumAnalysis-SymComp4.png)
+
+_Trapezoidal_  _ _  _rule_  _: _
+
+<span style="color:#FF0000"> __worse__ </span>  <span style="color:#FF0000"> __ __ </span>  <span style="color:#FF0000"> __than__ </span>  <span style="color:#FF0000"> __ __ </span>  <span style="color:#FF0000"> __midpoint__ </span>  <span style="color:#FF0000"> __\!__ </span>
+
+![](img/T2_NumAnalysis-SymComp5.png)
+
+approximate by parabolas
+
+_Simpson‘s_  _ _  _rule_  _: _
+
+__Numerical__  __ __  __methods__  __:__
+
+![](img/T2_NumAnalysis-SymComp6.png)
+
+__Symbolic__  __ __  __Methods__
+
+We will use module  <span style="color:#0070C0"> __sympy__ </span> \.
+
+For symbolic operations \(i\.e\.\, without concrete numbers\)\, we have to  __declare__  __ variables/__  __symbols__  \(and later functions…\)\.
+
+For  __mathematical__  __ __  __functions__  __ such __  __as__  __ cos\(…\)__ \, use the sympy equivalents \(not from math or numpy modules\!\)
+
+![](img/T2_NumAnalysis-SymComp7.png)
+
+For  __definite __  __integrals__ \, we can specify boundaries a and b by  __creating__  __ a __  __tuple__   <span style="color:#0070C0"> __\(x\, a\, b\)__ </span>  for the second argument\.
+
+The solution can be  __evaluated__  by using the methods  <span style="color:#0070C0"> __\.__ </span>  <span style="color:#0070C0"> __subs__ </span>  <span style="color:#0070C0"> __\(variable\, __ </span>  <span style="color:#0070C0"> __value__ </span>  <span style="color:#0070C0"> __\) __ </span> to substitute a value for a variable and  <span style="color:#0070C0"> __\.__ </span>  <span style="color:#0070C0"> __evalf__ </span>  <span style="color:#0070C0"> __\(\) __ </span> to get a numerical output\.
+
+<span style="color:#0070C0"> __„Genug für heute?“__ </span>
+
+<span style="color:#000000">[https://davrot\.github\.io/pytutorial/sympy/intro](https://davrot.github.io/pytutorial/sympy/intro/)</span>  <span style="color:#000000">[/](https://davrot.github.io/pytutorial/sympy/intro/)</span>
+
+<span style="color:#000000">[https://davrot\.github\.io/pytutorial/numpy/7](https://davrot.github.io/pytutorial/numpy/7/)</span>  <span style="color:#000000">[/](https://davrot.github.io/pytutorial/numpy/7/)</span>
+
+<span style="color:#000000">[https://davrot\.github\.io/pytutorial/numpy/8](https://davrot.github.io/pytutorial/numpy/8/)</span>  <span style="color:#000000">[/](https://davrot.github.io/pytutorial/numpy/8/)</span>
+
+<span style="color:#FF0000"> __Example__ </span>  <span style="color:#FF0000"> __ live\-__ </span>  <span style="color:#FF0000"> __coding__ </span>  <span style="color:#FF0000"> __:__ </span>  <span style="color:#FF0000">integration</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">and</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">differentiation</span>  <span style="color:#FF0000">\, </span>  <span style="color:#FF0000">stability</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">and</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">instability</span>
+
+<span style="color:#0070C0"> __Differentiation __ </span>  <span style="color:#0070C0"> __of__ </span>  <span style="color:#0070C0"> __ __ </span>  <span style="color:#0070C0"> __functions__ </span>
+
+__Numerical__  __ __  __methods__  __:__
+
+__centered__  __ __  __differentiation__
+
+__right\-sided__  __ __  __differentiation__
+
+![](img/T2_NumAnalysis-SymComp8.png)
+
+---
+
+Note: also important for integration of DEQs, since differential approximated by the same equations
+
+
+__Symbolic__  __ __  __methods__  __:__
+
+For differentiation\, the corresponding command is  <span style="color:#0070C0"> __diff__ </span> :
+
+![](img/T2_NumAnalysis-SymComp9.png)
+
+<span style="color:#0070C0"> __Integration __ </span>  <span style="color:#0070C0"> __of__ </span>  <span style="color:#0070C0"> __ differential __ </span>  <span style="color:#0070C0"> __equations__ </span>
+
+__Differential __  __quotient__  __ __  __approximated__  __ __  __by__  __ finite __  __difference__ \, like in previous example\. Solution constructed by considering the following aspects:
+
+What do we want to know\, what is known?
+
+Where do we start?   __Initial __  __value__  __ __  __problem__ …
+
+How far do we step?  Smaller than fastest timescale implies  __maximum__  __ __  __step__  __ __  __size__
+
+<span style="color:#FF0000"> __Warning__ </span>  <span style="color:#FF0000"> __:__ </span>
+
+<span style="color:#FF0000">differentiation</span>  <span style="color:#FF0000">/</span>  <span style="color:#FF0000">integration</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">of</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">functions</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">can</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">be</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">performed</span>  <span style="color:#FF0000"> in parallel\, </span>
+
+<span style="color:#FF0000">differential </span>  <span style="color:#FF0000">equations</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">require</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">an iterative </span>  <span style="color:#FF0000">solution</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">which</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">can</span>  <span style="color:#FF0000"> not </span>  <span style="color:#FF0000">be</span>  <span style="color:#FF0000"> </span>  <span style="color:#FF0000">parallelized</span>  <span style="color:#FF0000">\!</span>
+
+<span style="color:#0070C0"> __What__ </span>  <span style="color:#0070C0"> __ __ </span>  <span style="color:#0070C0"> __about__ </span>  <span style="color:#0070C0"> __ __ </span>  <span style="color:#0070C0"> __systems__ </span>  <span style="color:#0070C0"> __ __ </span>  <span style="color:#0070C0"> __of__ </span>  <span style="color:#0070C0"> __ differential __ </span>  <span style="color:#0070C0"> __equations__ </span>  <span style="color:#0070C0"> __?__ </span>
+
+…just solve them in parallel \(see previous slide\)
+
+__Higher\-order __  __methods__
+
+Idea: approximate differential quotient more precisely…
+
+__Solution \(Runge\-__  __Kutta__  __ 2nd __  __order__  __\):__
+
+Go ahead with Euler by half of the stepsize…
+
+…use slope at that position for an Euler with the full stepsize\.
+
+__Numerical__  __ __  __methods__  __:__
+
+![](img/T2_NumAnalysis-SymComp10.png)
+
+![](img/T2_NumAnalysis-SymComp11.png)
+
+__Symbolic__  __ __  __methods__  __:__
+
+In addition to declaring variables\, you need…
+
+…to  __declare__  __ __  __functions__  \(for the solution we are looking for\)
+
+…to  __define__  __ __  __the__  __ \(differential\) __  __equation__
+
+…and the  __command__  __ __  <span style="color:#0070C0"> __dsolve__ </span>  __ __ for \(trying to\) solve the DEQ:
+
+![](img/T2_NumAnalysis-SymComp12.png)
+
+__Symbolic__  __ __  __methods__  __\, __  __cont‘d__  __…__
+
+For including initial conditions\,  <span style="color:#0070C0"> __dsolve__ </span>  <span style="color:#0070C0"> __ __ </span> has the  __optional __  __argument__  __ __  <span style="color:#0070C0"> __ics__ </span> \.
+
+With <span style="color:#0070C0"> __ __ </span>  <span style="color:#0070C0"> __lambdify__ </span> \, You can  __convert__  __ __  __the__  __ RHS __  __of__  __ __  __the__  __ __  __solution__  __ __  __to__  __ a normal __  __numpy__  __ __  __function__ :
+
+Query the new function as to  __which__  __ __  __arguments__  __ __  __it__  __ __  __takes__ \, and in which order \( <span style="color:#0070C0"> __import__ </span>  <span style="color:#0070C0"> __ __ </span>  <span style="color:#0070C0"> __inspect__ </span>  <span style="color:#0070C0"> __ __ </span> for that purpose\)
+
+![](img/T2_NumAnalysis-SymComp13.png)
+
+__What__  __ __  __about__  __ partial differential __  __equations__  __?__
+
+For example\, the cable equation:
+
+![](img/T2_NumAnalysis-SymComp14.png)
+
+<span style="color:#0070C0"> __More __ </span>  <span style="color:#0070C0"> __information__ </span>  <span style="color:#0070C0"> __:__ </span>
+
+<span style="color:#000000">[https://davrot\.github\.io/pytutorial/sympy/intro](https://davrot.github.io/pytutorial/sympy/intro/)</span>  <span style="color:#000000">[/](https://davrot.github.io/pytutorial/sympy/intro/)</span>
+
+<span style="color:#000000">[https://davrot\.github\.io/pytutorial/numpy/7](https://davrot.github.io/pytutorial/numpy/7/)</span>  <span style="color:#000000">[/](https://davrot.github.io/pytutorial/numpy/7/)</span>
+
+<span style="color:#000000">[https://davrot\.github\.io/pytutorial/numpy/8](https://davrot.github.io/pytutorial/numpy/8/)</span>  <span style="color:#000000">[/](https://davrot.github.io/pytutorial/numpy/8/)</span>
+