TMA4100 Matematikk 1 – høsten 2019

Oversiktsforelesning 8

Nøkkelbegreper — uke 41


  • Midtpunktsmetoden

  • Trapesmetoden

  • Simpsons metode

  • Newtons metode

Midtpunktsmetoden


\[ \int_a^b f(x)\mathop{}\! dx \approx M_n= h \left( f(x_{\frac12}) + f(x_{\frac32}) + \cdots + f(x_{n - \frac12})\right)\]

\(\quad h = \dfrac{b - a}n\)

\(\quad x_{\frac12}=a+\frac h2, \ \dots\ , \ x_{n-\frac12}=a+(n-\frac12)h\)

\[\text{FEIL}= \left| \int_a^b f(x)\mathop{}\! dx - M_n \right| \leq \frac{K(b - a)}{24} h^2 = \frac{K(b - a)^3}{24n^2} \qquad \text{når}\qquad \max_{x\in[a,b]}|f''(x)| \leq K.\]

Trapesmetoden


\[ \int_a^b f(x)\mathop{}\! dx \approx T_n= h \left( \frac12 f(x_0) + f(x_1) + \cdots + f(x_{n - 1}) + \frac12 f(x_n)\right)\]

\(\quad h = \dfrac{b - a}n\)

\(\quad x_0=a, \ \dots\ , \ x_i=a+ih, \ \dots\ ,\ x_n=b\)

\[\text{FEIL}= \left| \int_a^b f(x)\mathop{}\! dx - T_n \right| \leq \frac{K(b - a)}{12} h^2 = \frac{K(b - a)^3}{12n^2} \qquad \text{når}\qquad \max_{x\in[a,b]}|f''(x)| \leq K.\]

Simpsons metode


\[\int_a^b f(x)\mathop{}\! dx \approx S_{2n}= \frac{h}3 \bigg( f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) \cdots + 2f(x_{2n - 2}) + 4f(x_{2n - 1}) + f(x_{2n})\bigg) \]

\(\quad h = \dfrac{b - a}{2n}\)          (OBS  \(2n\) !)

\(\quad x_0=a, \ \dots\ , \ x_i=a+ih, \ \dots\ , \ x_{2n}=b\)

\[\text{FEIL} =\left| \int_a^b f(x)\mathop{}\! dx - S_{2n} \right| \leq \frac{K(b - a)}{180} h^4 = \frac{K(b - a)^5}{180 (2n)^4}\qquad \text{når}\qquad \max_{x\in[a,b]}|f^{(4)}(x)| \leq K.\]

Newtons metode



Newtons metode

\[ x_{n + 1} = x_n - \frac{f(x_n)}{f'(x_n)},\quad n=0,1,2,\dots\]