Fehlerrechnung.sagews

# Einfaches Beispiel
R1,R2=var ('R1,R2') # Variablen deklarieren


R(R1,R2)=R1*R2/(R1+R2) # Gesamtwiderstand


dR1,dR2=var ('dR1,dR2') # absolute Fehler
totdiff=diff(R,R1)*dR1+diff(R,R2)*dR2 # totales Differenzial,
show(totdiff)


# Werte einsetzen
dR1=2.5
dR2=10
# partielle Ableitungen
partR1=diff(R,R1)
partR2=diff(R,R2)
#absoluter Gesamtfehler, Werte positiv (daher Beträge weggelassen)
dR=partR1(100,400)*dR1+partR2(100,400)*dR2
show(dR) # absoluter Fehler von R

rel=dR/R(100,400)*100 #relativer Fehler in Prozent
show(rel)

# Beispiel aus dem Praktikum (Achtung, noch keine Beträge enthalten!, Betragsfunktion ist abs)
mFl,Tm,T1,K,m2,T2,cW,cFl=var ('mFl,Tm,T1,K,m2,T2,cW,cFl')

qs(mFl,Tm,T1,K,m2,T2,cW,cFl)=(mFl*cFl*(T1-Tm)+K*(T1-Tm)-m2*cW*(Tm-T2))/m2

dmFl,dTm,dT1,dK,dm2,dT2,dcW,dcFl=var ('dmFl,dTm,dT1,dK,dm2,dT2,dcW,dcFl')

# totales Differenzial
totdiff=diff(qs,mFl)*dmFl+diff(qs,Tm)*dTm+diff(qs,T1)*dT1+diff(qs,K)*dK+diff(qs,m2)*dm2+diff(qs,T2)*dT2+diff(qs,cFl)*dcFl+diff(qs,cW)*dcW
show(totdiff)
# Partielle Ableitungen
partmFl=diff(qs,mFl)
partTm=diff(qs,Tm)
partT1=diff(qs,T1)
partK=diff(qs,K)
partm2=diff(qs,m2)
partT2=diff(qs,T2)
partcW=diff(qs,cW)
partcFl=diff(qs,cFl)

# Fehler
dmFl=0.05
dTm=0.05
dT1=0.05
dK=12
dm2=0.05
dT2=0.05
dcW=0
dcFl=0

# Das sind die gemessenen Werte
mFl_mes=254.1
Tm_mes=286.25
T1_mes=304.15
K_mes=100.13
m2_mes=54.1
T2_mes=273.15
cW_mes=4.187
cFl_mes=4.187
# hier müssen als Argumente in den Funktionen jeweils alle Variablen eingetragen werden, Resultat ist der absolute Gesamtfehler:
dqs=partmFl(mFl_mes,Tm_mes,T1_mes,K_mes,m2_mes,T2_mes,cW_mes,cFl_mes)*dmFl+partTm(mFl_mes,Tm_mes,T1_mes,K_mes,m2_mes,T2_mes,cW_mes,cFl_mes)*dTm+partT1(mFl_mes,Tm_mes,T1_mes,K_mes,m2_mes,T2_mes,cW_mes,cFl_mes)*dT1+partK(mFl_mes,Tm_mes,T1_mes,K_mes,m2_mes,T2_mes,cW_mes,cFl_mes)*dK+partm2(mFl_mes,Tm_mes,T1_mes,K_mes,m2_mes,T2_mes,cW_mes,cFl_mes)*dm2+partT2(mFl_mes,Tm_mes,T1_mes,K_mes,m2_mes,T2_mes,cW_mes,cFl_mes)*dT2+partcW(mFl_mes,Tm_mes,T1_mes,K_mes,m2_mes,T2_mes,cW_mes,cFl_mes)*dcW+partcFl(mFl_mes,Tm_mes,T1_mes,K_mes,m2_mes,T2_mes,cW_mes,cFl_mes)*dcFl


show(dqs)

$\displaystyle \left( \mathit{mFl}, \mathit{Tm}, T_{1}, K, m_{2}, T_{2}, \mathit{cW}, \mathit{cFl} \right) \ {\mapsto} \ \mathit{cW} \mathit{dT}_{2} + {\left(T_{2} - \mathit{Tm}\right)} \mathit{dcW} + \mathit{dm}_{2} {\left(\frac{{\left(T_{2} - \mathit{Tm}\right)} \mathit{cW}}{m_{2}} - \frac{{\left(T_{2} - \mathit{Tm}\right)} \mathit{cW} m_{2} + {\left(T_{1} - \mathit{Tm}\right)} \mathit{cFl} \mathit{mFl} + K {\left(T_{1} - \mathit{Tm}\right)}}{m_{2}^{2}}\right)} + \frac{{\left(T_{1} - \mathit{Tm}\right)} \mathit{cFl} \mathit{dmFl}}{m_{2}} + \frac{{\left(T_{1} - \mathit{Tm}\right)} \mathit{dcFl} \mathit{mFl}}{m_{2}} + \frac{{\left(T_{1} - \mathit{Tm}\right)} \mathit{dK}}{m_{2}} + \frac{{\left(\mathit{cFl} \mathit{mFl} + K\right)} \mathit{dT}_{1}}{m_{2}} - \frac{{\left(\mathit{cW} m_{2} + \mathit{cFl} \mathit{mFl} + K\right)} \mathit{dTm}}{m_{2}}$

$\displaystyle 3.68373439000140$