... Schrittweite
![{\displaystyle R^{ST}=-(b-a)/12*h^{2}*f''(\xi )}](/index.php?title=Spezial:MathShowImage&hash=083211d2c92c22d1b72cb2dd84453853&mode=mathml)
![{\displaystyle h^{SI}=(b-a)/(2n)}](/index.php?title=Spezial:MathShowImage&hash=17828039d26d456147042271b2b66db5&mode=mathml)
![{\displaystyle R^{SI}=-(b-a)/180*h^{4}*f''''(\xi )}](/index.php?title=Spezial:MathShowImage&hash=a5f3b68af02f14acc4dc060566c84ef1&mode=mathml)
Rest: Siehe Standard Uni Formelsammlung S.26
Euler'sches Verfahren - Fehler: O(h)
; ![{\displaystyle y_{i+1}=y_{i}+hf(x_{i},y_{i})}](/index.php?title=Spezial:MathShowImage&hash=9622d4755ceefe2a05fa965c9eea955b&mode=mathml)
Verbesserter Euler - Fehler: Fehler: ![{\displaystyle O(h^{2})}](/index.php?title=Spezial:MathShowImage&hash=2a67626253be046f40c7650f783b6089&mode=mathml)
![{\displaystyle x_{i+1}=x_{i}+h}](/index.php?title=Spezial:MathShowImage&hash=b35f31e3ba4c1fe1c0d95d725ac0dee2&mode=mathml)
![{\displaystyle y_{i+1}=y_{i}+h/2*[f(x_{i},y_{i})+f(x_{i}+h,y_{i}+hf(x_{i},y_{i}))]}](/index.php?title=Spezial:MathShowImage&hash=40acdf40778daf50fc5171c3a750fce1&mode=mathml)
Klassisches Runge-Kutta - Fehler: ![{\displaystyle O(h^{4})}](/index.php?title=Spezial:MathShowImage&hash=b56a08ea423db9a934600049c0ec5345&mode=mathml)
![{\displaystyle y_{i+1}=y_{i}+h/6*(k_{1}+2(k_{2}+k_{3})+k_{4})}](/index.php?title=Spezial:MathShowImage&hash=f3512eb0e73e1c31bc96e9d84f61ca51&mode=mathml)
![{\displaystyle k_{1}=f(x_{i},y_{i})}](/index.php?title=Spezial:MathShowImage&hash=3fae1c18a96cb85c363d58889d4e2f00&mode=mathml)
![{\displaystyle k_{2}=f(x_{i}+h/2,y_{i}+h/2*k_{1})}](/index.php?title=Spezial:MathShowImage&hash=df892a04f3cfd7a87f484c445b758feb&mode=mathml)
![{\displaystyle k_{3}=f(x_{i}+h/2,y_{i}+h/2*k_{2})}](/index.php?title=Spezial:MathShowImage&hash=76f801603db40659c544e6f503a2e53e&mode=mathml)
![{\displaystyle k_{4}=f(x_{i}+h,y_{i}+h*k_{3})}](/index.php?title=Spezial:MathShowImage&hash=29e675440caaa625cb657182817a18f6&mode=mathml)
Kleinste Quadrate:
![{\displaystyle Q=\sum _{i=1}^{n}(f(x_{i},{\vec {x}})-p(x_{i}))^{2}=\sum (y_{i}-a-b*x_{i})^{2}}](/index.php?title=Spezial:MathShowImage&hash=58f4fd6d25c5d0a638574a3d2ce17837&mode=mathml)
![{\displaystyle b={\frac {\sum (x_{i}y_{i})-n*{\bar {x}}{\bar {y}}}{\sum (x_{i}^{2})-n*({\bar {x}})^{2}}}}](/index.php?title=Spezial:MathShowImage&hash=3625294daae0a2f1dcfe0bf66e428f42&mode=mathml)
Polynomfunktion:
![{\displaystyle p(x)=a_{0}+a_{1}x+a_{2}x^{2}+...}](/index.php?title=Spezial:MathShowImage&hash=6e800fe9b6c92c8dcf7a9615ec401e57&mode=mathml)
Lagrage:
![{\displaystyle \ell _{i}(x)=\prod _{j=0,j\neq i}^{n}{\frac {x-x_{j}}{x_{i}-x_{j}}}}](/index.php?title=Spezial:MathShowImage&hash=8b58ff0d029e6d4e4e01f3340ed13bf1&mode=mathml)
![{\displaystyle p(x)=y_{0}*\ell _{0}+y_{1}*\ell _{1}+...}](/index.php?title=Spezial:MathShowImage&hash=ce1713bc36a196c48c764645651cd82b&mode=mathml)
Newton:
![{\displaystyle p(x)=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})(x-x_{1})+}](/index.php?title=Spezial:MathShowImage&hash=0843ad83c90b4d4085b0cc58e411100b&mode=mathml)
![{\displaystyle ...+a_{n}(x-x_{0})*...*(x-x_{n-1})}](/index.php?title=Spezial:MathShowImage&hash=5c230a91a10f86abce5290c7e579ab13&mode=mathml)
![{\displaystyle f_{n}(t)={\frac {a_{0}}{2}}+\sum _{k=1}^{n}(a_{k}\cdot \cos(k\omega t)+b_{k}\cdot \sin(k\omega t))}](/index.php?title=Spezial:MathShowImage&hash=7943911a7f1dd24f79a9cd1751ad1c02&mode=mathml)
![{\displaystyle =\sum _{n=-\infty }^{\infty }c_{n}\mathrm {e} ^{\mathrm {i} n\omega t}}](/index.php?title=Spezial:MathShowImage&hash=40bdec867bedb31ccd62837aa7121975&mode=mathml)
![{\displaystyle b_{k}={\frac {2}{T}}\int _{c}^{c+T}f(t)\cdot \sin(k\omega t)\,\mathrm {d} t}](/index.php?title=Spezial:MathShowImage&hash=b2298e87d7648e0ace9fde492a9f4939&mode=mathml)
![{\displaystyle \omega =e^{(2\Pi *\iota )/N}}](/index.php?title=Spezial:MathShowImage&hash=77af2e8ab4e59439d909f47e55b6d736&mode=mathml)
DFT:
Heißt transformierbar wenn folgendes konvergiert
![{\displaystyle F(s)={\mathcal {L}}\left\{f(t)\right\}=\int _{0}^{\infty }e^{-st}f(t)\,dt}](/index.php?title=Spezial:MathShowImage&hash=1bd45673ddf72e3950dd441e9af6fd7a&mode=mathml)
![{\displaystyle {\mathcal {L}}\{y''\}=s^{2}Y(s)-sy(0)-f'(0)}](/index.php?title=Spezial:MathShowImage&hash=7f8389806408ecf752589aad1ec63e5d&mode=mathml)
![{\displaystyle au_{x}+bu_{y}=f(x,y)}](/index.php?title=Spezial:MathShowImage&hash=9dd0b6fd5769ee93b97632e695ea1001&mode=mathml)
![{\displaystyle \xi =bx+ay;\eta =bx-ay}](/index.php?title=Spezial:MathShowImage&hash=bdd779a4226160b848f9d31f747aa75d&mode=mathml)
![{\displaystyle u_{tt}-c^{2}u_{xx}=f(x,t)}](/index.php?title=Spezial:MathShowImage&hash=8115a06c1e74495595c973ad11df2483&mode=mathml)
![{\displaystyle \xi =x-c*t;\tau =x+c*t}](/index.php?title=Spezial:MathShowImage&hash=4ec8e04ef93691c64144494f3475daf5&mode=mathml)
![{\displaystyle u^{p}(x,t)=U(\xi ,\tau )=-1/(4c^{2})*\int \int F(\xi ,\tau )\mathrm {d} \xi \mathrm {d} \tau }](/index.php?title=Spezial:MathShowImage&hash=317369414b4e9e987d0717a1504789d2&mode=mathml)