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pytutorial/advanced_programming/slides_t2/out.md
David Rotermund 0758343605
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Signed-off-by: David Rotermund <54365609+davrot@users.noreply.github.com>
2024-02-19 17:55:29 +01:00

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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>
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