Bestimmen Sie eine Nullstelle der Funktion
![{\displaystyle F(x)=x^{2}-1}](/index.php?title=Spezial:MathShowImage&hash=c62dc053ce2d7298962714f063b21672&mode=mathml)
im Intervall
![{\displaystyle [0,3]}](/index.php?title=Spezial:MathShowImage&hash=fd78b2f70dafa49beb77b841be00382a&mode=mathml)
, indem Sie jeweils 3 Schritte der angegebenen Verfahrens durchfuhren, und vergleichen Sie die Ergebnisse.
(a) Bisektion, (b) Regula falsi, (c) Newtonsches Naeherungsverfahren (Startwert Intervallende).
Startwerte so dass
und
, also
.
![{\displaystyle y_{1}={\frac {-1+8}{2}}=3.5>0\implies a_{2}=a_{1},b_{2}=3.5}](/index.php?title=Spezial:MathShowImage&hash=abad5195a5fdb0829e4d6ff20b3a51c5&mode=mathml)
<br
![{\displaystyle y_{2}={\frac {-1+3.5}{2}}=1.25>0\implies a_{3}=a_{1},b_{3}=1.25}](/index.php?title=Spezial:MathShowImage&hash=36cce7e040827613c59dceb7316de69c&mode=mathml)
![{\displaystyle y_{3}={\frac {-1+1.25}{2}}=1.125}](/index.php?title=Spezial:MathShowImage&hash=7724a9cd8d712e788f887651ab826fdf&mode=mathml)
Nullstelle auf 1.125 approximiert.
![{\displaystyle x=a-f(a)*{\frac {b-a}{f(b)-f(a)}}}](/index.php?title=Spezial:MathShowImage&hash=5624be5cb9d7b3e0eac665261dd4ed75&mode=mathml)
![{\displaystyle f(x)<0\implies a=x,f(x)=0\implies finish,f(x)>0\implies b=x}](/index.php?title=Spezial:MathShowImage&hash=49a01a3c1bcbbfd49aadd1ffcf2fc073&mode=mathml)
1: ![{\displaystyle x=0+1*{\frac {3-0}{8+1}}=1/3,f(1/3)=-0.889<0\implies a=x}](/index.php?title=Spezial:MathShowImage&hash=50da2f38eb0bad7431b028ef9aa3ee80&mode=mathml)
2: ![{\displaystyle x=1/3+0.889*{\frac {3-1/3}{8+0.889}}=0.5998,f(0.5998)=-0.64<0\implies a=x}](/index.php?title=Spezial:MathShowImage&hash=585e2cd7d3e1c6812aa7698202d3ab1c&mode=mathml)
3: ![{\displaystyle x=0.5998+0.6403*{\frac {3-0.5998}{8+0.6403}}=0.776}](/index.php?title=Spezial:MathShowImage&hash=c49b9d38009c58dec7a49823aedb0afe&mode=mathml)
![{\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}}](/index.php?title=Spezial:MathShowImage&hash=b1c822cfef17ce6483037a03493a9d70&mode=mathml)
![{\displaystyle f'(x)=2x}](/index.php?title=Spezial:MathShowImage&hash=cf71d5c2e34a556e24d06a3d657715ea&mode=mathml)
![{\displaystyle x_{1}=3-{\frac {8}{6}}=1.66}](/index.php?title=Spezial:MathShowImage&hash=e8979b736bbd3a58d2da3e51cb3466a7&mode=mathml)
![{\displaystyle x_{2}=1.66-{\frac {1.7780}{3.3340}}=1.1264}](/index.php?title=Spezial:MathShowImage&hash=982ec6e5c9d3962815d184b8707855c5&mode=mathml)
![{\displaystyle x_{3}=1.1264-{\frac {0.2689}{2.2528}}=1.0070}](/index.php?title=Spezial:MathShowImage&hash=74c4845e9f5b835172c337847055a323&mode=mathml)